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Interpolative construction and factorization of operators
Accepted Manuscript Interpolative construction and factorization of operators Mieczysław Mastyło, Radosław Szwedek PII: DOI: Reference:
S0022-247X(12)00951-1 10.1016/j.jmaa.2012.11.036 YJMAA 17186
To appear in:
Journal of Mathematical Analysis and Applications
Received date: 6 September 2012 Please cite this article as: M. Mastyło, R. Szwedek, Interpolative construction and factorization of operators, J. Math. Anal. Appl. (2012), doi:10.1016/j.jmaa.2012.11.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Interpolative construction and factorization of operators Mieczyslaw Mastylo and Radoslaw Szwedek Abstract Banach operator ideals generated by interpolative construction applied to p-summing operators are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. Relationships between Banach ideals determined by Orlicz sequence spaces are shown and a variant of the Pisier factorization theorem for (p, 1)-summing operators from C(K)-spaces is proved. Applications to Schatten classes are given. It is also shown that certain known results on (q, p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.
1
Introduction
We study the Banach operator ideals generated via an interpolative construction determined by quasi-concave functions. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. In order to prepare for the discussion that will follow, we introduce some fundamental notion and definitions. The standard references for operator ideals are the monographs [3, 16]. We shall use some standard notations and notions from Banach space theory. If X is a Banach space, then BX is its closed unit ball and X ∗ is its dual space. As usual L(E, F ) denotes the Banach space of all (bounded and linear) operators T : E → F between
Banach spaces E and F endowed with the operator norm. If E ⊂ F , then E ,→ F means
that the inclusion map J : E → F is bounded. Throughout the paper, for two real valued functions f and g we use the symbol f ≺ g whenever there exists a constant c > 0 such
that f (t) ≤ cg(t) for all t in the domain of f and g, and f g whenever f ≺ g and g ≺ f .
2010 Mathematics Subject Classification: Primary: 47B10; Secondary: 47B60, 47B38, 46B70. Key words and phrases: Factorization, Banach operator ideals, (p, q)-summing operators, Interpolation spaces, Banach lattices, generalized concavity. The first named author was supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243.
1
Let Φ denote the set of all functions ϕ : [0, ∞)×[0, ∞) → [0, ∞) which are non-decreasing
in each variable and are positively homogeneous (that is, ϕ(λs, λt) = λϕ(s, t) for all λ, s, t ≥
0). If in addition ϕ is concave, then we write ϕ ∈ P. The subset of all ϕ ∈ P for which ϕ(s, 1) → 0 and ϕ(1, t) → 0 as s → 0+ and t → 0+ is denoted by P0 .
Let ϕ ∈ P and let (A, α) and (B, β) be two Banach operator ideals. In [10] an interpola-
tive construction (A, B)ϕ was studied. For the sake of completeness, we recall that if X and Y are Banach spaces, then the space (A, B)ϕ (X, Y ) consists of all operators T ∈ L(X, Y )
such that for some Banach spaces Zj and operators Sj : X → Zj (j = 0, 1) with α(S0 ) ≤ 1, β(S1 ) ≤ 1 and for some λ > 0,
kT xkY ≤ λ ϕ(kS0 xkZ0 , kS1 xkZ1 ),
x ∈ X.
We put γϕ (T ) as the infimum of the values of λ for which the inequality holds with suitable operators S0 and S1 described above. It is easy to see that an operator T : X → Y belongs to (A, B)ϕ (X, Y ) if and only if
there exist Banach spaces Z0 , Z1 and operators S0 ∈ A(X, Z0 ) and S1 ∈ B(X, Z1 ) such that kT xkY ≤ ϕ kS0 xkZ0 , kS1 xkZ1 ,
x ∈ X.
Moreover for every T ∈ (A, B)ϕ (X, Y ), we have
γϕ (T ) inf max{α(S0 ), β(S1 )}, where the infimum is taken over all operators S0 and S1 admitted the above estimate. If B = L, we put (Aϕ , αϕ ) = (A, L)ϕ . In the case ϕ(s, t) = s1−θ tθ , 0 < θ < 1, we obtain
the construction (α, β)θ studied in [12, 13].
Let A be a Banach operator ideal. We recall that an injective hull Ai of A is defined by
T ∈ Ai (X, Y ) if JY T ∈ A(X, `∞ (BY ∗ )), where JY : Y → `∞ (BY ∗ ) is the canonical isometric
embedding. An ideal A is said to be injective if A = Ai . We write A for the ideal of all
operators T : X → Y which satisfy kTn − T k → 0 as n → ∞ for some sequence (Tn ) in
A(X, Y ). If A = A, then A is said to be closed.
We notice here that for every Banach operator ideals (A, α), (B, β) and ϕ ∈ P, (A, B)ϕ
is an injective Banach operator ideal. If A is a closed and injective ideal, then Aϕ = A.
ϕ(s, 1) → 0 as s → 0 and Aϕ = A implies that A is a closed ideal (for details we refer to [10]). Here, for every ϕ ∈ Φ, we define ϕ by ϕ(αs, βt) ϕ(s, t) = sup ; α, β > 0 , ϕ(α, β) 2
s, t > 0.
We now describe the main results of the paper.
In section 2 interpolative proce-
dure is used to the Banach ideal of p-summing operators to generate the Banach ideal of (p, ϕ)-absolutely continuous operators. A description of this ideal via factorization through Lorentz space between L∞ (µ) and Lp (µ) is presented. It is also shown that under certain concavity condition (2, ϕ)-absolutely continuous operators between Hilbert spaces coincides with the Schatten class determined by an Orlicz sequence space. In Section 3 of the paper is devoted to the Banach operator ideal of (ϕ, ψ)-summing operators generated by Orlicz sequence spaces. In the case of power functions this ideal coincides with the well-known ideal of (q, p)-summing operators. A connection with (ϕ, p)-summing operators is shown. Based on the ideas from [18], we prove a variant of Pisier’s description for operators on C(K)-spaces. This result enables us to prove a variant of remarkable Pisier’s factorization theorem for (ϕ, ψ)-summing operators on C(K)-spaces. In Section 4, we introduce the notion of (ϕ, ψ)-concave operators (for normalized Orlicz functions ϕ and ψ), which generalizes the classical concept of (q, p)-concave operators (see [3]). We show relationships between (ϕ, ψ)-summing and (ϕ, ψ)-concave operators. We prove that some classical results on (q, p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.
2
(p, ϕ)-absolutely continuous operators
The following definition is a generalization of the classical concept of (q, p)-summing operators. Let E, F be two Banach sequence lattices on N and let (en ) be the standard unit vector basis. An operator T ∈ L(X, Y ) is (F, E)-summing if there exists a constant C > 0 such that for any finite set {x1 , ..., xn } of X, n
X
kT xk kY ek ≤ C
k=1
F
sup kx∗ kX ∗ ≤1
n
X
x∗ (xk )ek .
k=1
E
We denote by πF,E (T ) the least constant with this property.
It is easy to see that if there exists some non-trivial operator T : X → Y that is (F, E)-
summing, then E ,→ F . Moreover the space ΠF,E (X, Y ) of all (F, E)-summing operators is
a Banach space equipped with the norm πF,E and (ΠF,E , πF,E ) is a Banach operator ideal whenever the norm of the inclusion map E ,→ F equals 1. Notice that when F = `q and E = `p with 1 ≤ p ≤ q ≤ ∞, then we recover the ideal
(Πq,p , πq,p ) of (q, p)-summing operators (see [3]). Of particular importance is the choice of 3
p = q, which is related to the Banach ideal (Πp , πp ) of the so-called p-summing operators. We refer to [1, 2, 7] where (F, E)-summing operators are studied and applications in various parts of analysis are shown. If ϕ ∈ Φ, then operators from (Πp )ϕ := (Πp , πp )ϕ are called (p, ϕ)-absolutely continuous.
If ϕ(s, t) = s1−θ tθ (0 ≤ θ < 1), then (p, ϕ)-absolutely continuous operators are called (p, θ)absolutely continuous (see [12]).
Before the proof of the mentioned factorization theorem it is be convenient to give some preliminaries. Let ϕ ∈ Φ and X = (X0 , X1 ) be a compatible couple of Banach spaces.
Following [15], the abstract Lorentz space Λϕ (X) consists of all x ∈ X0 + X1 such that x=
X
un
(convergence in X0 + X1 ),
n∈Z
with un ∈ X0 ∩ X1 , and
P
n∈Z
ϕ(kun kX0 , kun kX1 ) < ∞. The norm on Λϕ (X) is given by
kxkΛϕ (X) = inf
nX
n∈Z
ϕ(kun kX0 , kun kX1 ); x =
where infimum is taken over all series described above.
X
n∈Z
o un ,
Let X and Y be Banach spaces. Further, we define for every operator S ∈ L(X, Y ) the
image space S(X) as the Banach space {Sx; x ∈ X} equipped with the norm kykS(X) := inf{kxkX ; Sx = y},
y ∈ S(X).
Notice that S(X) ,→ Y and so (S(X), Y ) forms an ordered Banach couple. We will need the following factorization result from [10]. Theorem 2.1. Suppose ϕ ∈ Φ is such that ϕ(1, t) → 0 as t → 0, and that operators
T ∈ L(X, Y ), S ∈ L(X, Z) satisfy the inequality kT xkY ≤ ϕ(kxkX , kSxkZ ) for all x ∈ X.
Then there exists an operator R : Λϕ (S(X), Z) → Y with kRk ≤ γϕ (T ) such that T admits a factorization:
S
J
R
T : X −→ S(X) −→ Λϕ (S(X), Z) −→ Y, where J is the inclusion map. The following theorem is a unified factorization theorem for the case of any function ϕ ∈ Φ with ϕ(1, t) → 0 as t → 0 (cf. [13] for the power functions ϕθ (s, t) = s1−θ tθ , θ ∈ (0, 1)).
4
Theorem 2.2. Let ϕ ∈ P be such that ϕ(1, t) → 0 as t → 0. An operator T ∈ L(X, Y )
is (p, ϕ)-absolutely continuous with (πp )ϕ (T ) ≤ C if and only if for every (equivalently for some) isometric embedding i : X → L∞ (µ) there exists a Borel probability measure µ on
K = (BX ∗ , σ(X ∗ , X)), a constant C > 0 and a bounded operator S : Λϕ (jp (X), Lp (µ)) → Y with kSk ≤ C such that T admits the factorization: jp
S
T : X −→ jp (X) ,→ Λϕ (jp (X), Lp (µ)) −→ Y, with jp = ji, where j : L∞ (µ) → Lp (µ) is the inclusion map. Proof. Assume that T : X → Y is (p, ϕ)-absolutely summing with (πp )ϕ (T ) ≤ C. Then there exists a Banach space Z and an operator u ∈ Πp (X, Z) with πp (u) ≤ 1 such that kT xkY ≤ C ϕ(kxkX , kuxkZ ),
x ∈ X.
The Pietsch Domination Theorem provides an isometric embedding i : X → L∞ (µ) and
a regular Borel probability measure µ on K = (BX ∗ , σ(X ∗ , X)) for which Z 1/p kuxkZ ≤ |ix(x∗ )|p dµ , x ∈ X. K
In view of above inequalities
kT xkY ≤ C ϕ(kxkX , kjp xkLp (µ ),
x∈X
with jp = ij : X → Lp (µ), where j : L∞ (µ) → Lp (µ) is the inclusion map. Now we conclude by Theorem 2.1 that T admits the required factorization with kSk ≤ C.
Conversely, assume that there exist a Borel probability measure µ on K, an isometric
embedding i : X → L∞ (µ) and a bounded operator S with kSk ≤ C such that T : X → Y admits the factorization
jp
J
S
T : X −→ jp (X) −→ Λϕ (jp (X), Lp (µ)) −→ Y, where J is the inclusion map. By the construction of the abstract Lorentz space, it follows that for any x ∈ X we have kJ(jp x)k ≤ ϕ(kjp (x)kjp (X) , kjp xkLp (µ) ),
x ∈ L∞ (µ).
Since jp = ji and the inclusion map j : L∞ (µ) → Lp (µ) is p-absolutely summing with πp (j) ≤ 1, we have πp (jp ) ≤ 1. By kSk ≤ C this implies that (πp )ϕ (T ) ≤ kik πp (jp ) kSk ≤ C, and the proof is complete. 5
We conclude the section by showing applications to Schatten classes. We show that under certain condition (2, ϕ)-absolutely continuous operators between Hilbert spaces coincides with the Schatten class determined by an Orlicz sequence space. We recall that a continuous and convex function ϕ : [0, ∞) → [0, ∞) with ϕ−1 ({0}) = {0} is called an Orlicz function. For an Orlicz function ϕ the Banach sequence lattice ∞ n X `ϕ = x = (xn )∞ ; ϕ(|xn |/λ) < ∞ n=1 n=1
o for some λ > 0
equipped with the norm
∞ n o X kxkϕ := kxk`ϕ = inf λ > 0; ϕ(|xn |/λ) ≤ 1 n=1
is called an Orlicz sequence space.
It is well known that the unit vector basis (en ) is a basis in `ϕ (or equivalently `ϕ is a separable space) if and only if and only if ϕ satisfies the δ2 -condition at 0, i.e., ϕ(2t) < ∞. 0
We note that the behaviour of ϕ outside a neighbourhood of 0 is irrelevant to the definition of ϕ, and so we may assume, without loss of generality, that ϕ satisfies the ∆2 -condition (for short ϕ ∈ (∆2 )), i.e., there exists C > 0 such that ϕ(2t) ≤ Cϕ(t) for all t > 0.
Notice that if an Orlicz space `ϕ is separable, then the dual (`ϕ )∗ is isometrically isomorP phic to the K¨ othe dual sequence space (`ϕ )0 of all sequences (xn ) such that ∞ n=1 |xn yn | < ∞ for all y = (yn ) ∈ `ϕ equipped with the norm kxk(`ϕ )0 = sup
∞ nX
n=1
o |xn yn |; k(yn )kϕ ≤ 1 .
Recall that if T : H1 → H2 is a compact operator between Hilbert spaces, then it has
the Schmidt representation, i.e., T has the form T =
∞ X
τn (·|en )fn ,
(∗)
n=1 ∞ where (en )∞ n=1 is an orthonormal sequence in H1 , (fn )n=1 is an orthonormal sequence in
H2 , and τn = τn (T ) is a null sequence which satisfy 0 ≤ τn+1 ≤ τn for all n ∈ N.
Notice that if T is a finite rank operator, (∗) is a finite sum, and in this case it will
convenient to consider finite sequences as sequences of infinite length by adding zeros. Otherwise, (∗) represents a series which converges with respect to the operator norm k · k in L(H1 , H2 ).
6
If E is a symmetric Banach sequence space on N, then the Schatten class SE (H1 , H2 )
consists of all operators T : H1 → H2 which admit the Schmidt representation with (τn ) ∈ E. It is well known that SE (H1 , H2 ) is a Banach space equipped with the norm σE (T ) = k(τn (T ))kE which does not depend on the specific representation. In particular σE (T ) = k(an (T ))kE , where for an operator u : H1 → H2 and each n ∈ N, an (u) is the n-th approximation number an (u) := inf{kv − uk; v ∈ L(H1 , H2 ),
dim v(H1 ) < n}.
If E = `ϕ is an Orlicz sequence space, we put Sϕ = SE and σϕ = σE . Of particular
importance is the choice ϕ(t) = tp for all t ≥ 0 where 1 ≤ p < ∞, which yields the well known Schatten class (Sp , σp ).
Theorem 2.3. Let ψ ∈ P0 and let ϕ(t) = ψ −1 (t, 1)2 for all t > 0. If H1 and H2 are any
Hilbert spaces, then the following statements are true:
(i) (Π2 )ψ (H1 , H2 ) ,→ Sϕ (H1 , H2 ) with σϕ (T ) ≤ (π2 )ψ (T ). √ (ii) If ϕ(1) = 1 and the function t 7→ ψ 2 ( t, 1) is concave on the interval [0, 1], then Sϕ (H1 , H2 ) ,→ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) ≤ σϕ (T ). Proof. (i): Let T ∈ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) < 1. By the Pietsch Domination Theorem
there exits a Hilbert space G and an operator S ∈ Π2 (G, H2 ) such that π2 (S) ≤ 1 and kT xkH2 ≤ ψ(kSxkG , 1),
x ∈ BH1 .
Since ψ ∈ P0 and S is compact, it follows immediately from the above inequality that T is also compact. Thus the operators S and T admit Schmidt representations: S=
∞ X
n=1
an (S) ( · |xn )yn 7
and T =
∞ X
n=1
The above inequality implies that ∞ X
n=1
an (T )2 |(x|en )|2
1/2
an (T ) ( · |en )fn .
≤ψ
∞ X
n=1
an (S)2 |(x|xn )|2
1/2 ,1
for every x ∈ BH1 , and so for each k ∈ N, ak (T ) ≤ ψ
∞ X
n=1
an (S)2 |(ek |xn )|2
1/2 ,1 .
Thus, by π2 (S) = σ2 (S) ≤ 1, we obtain ∞ X k=1
ϕ(ak (T )) ≤ =
∞ X
n=1 ∞ X
n=1
an (S)2
∞ X k=1
|(ek , xn )|2
an (S)2 ≤ 1.
In consequence T ∈ Sϕ with σϕ (T ) ≤ 1, and this completes the proof of (i).
(ii): For the converse, take any T ∈ Sϕ (H1 , H2 ) with σϕ (T ) ≤ 1. Since `ϕ ,→ c0 ,
an (T ) → 0 as n → ∞. Thus T is compact and hence it admits the Schmidt representation P∞ T = n=1 τn ( · |xn )yn , where (τn ) ∈ B`ϕ with τn ≥ 0 for all n ≥ 0 and (xn ), (yn ) are orthonormal sequences in H1 and H2 , respectively. Define the operator S : H1 → H2 by S=
∞ X
n=1
ϕ(τn )1/2 ( · |xn )yn .
Then S ∈ S2 (H1 , H2 ) = Π2 (H1 , H2 ) with π2 (S) = σ2 (S) ≤ 1. Since 0 ≤ τn ≤ 1 for all √ n ∈ N, our hypothesis on concavity of the function ρ(t) := ψ 2 ( t, 1) on the interval (0, 1] P 2 gives that for any x ∈ H1 with kxkH1 = ∞ n=1 |(x|xn )| = 1, we have kT xk2H2 =
∞ X
n=1
τn2 |(x|xn )|2 =
∞ X
n=1
ρ(ϕ(τn )) |(x|xn )|2
∞ ∞ X X 1/2 ≤ρ ϕ(τn ) |(x|xn )|2 = ψ 2 ϕ(τn ) |(x|xn )|2 ,1 . n=1
n=1
By homogeneity, it then follows that kT xkH2 ≤ ψ(kSxkH2 , kxkH1 ),
x ∈ H1 ,
and so we finally arrive at T ∈ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) ≤ 1. 8
Corollary 2.1. Let ψ ∈ P0 and let ϕ(t) := ψ −1 (t, 1)2 for t ≥ 0. If the function t 7→ √ ψ 2 ( t, 1) is equivalent to a concave function in a neighborhood of zero, then for arbitrary Hilbert spaces G and H the following formula holds: (Π2 )ψ (G, H) = Sϕ (G, H). We note that Corollary 2.1 says that the ideal (Π)ψ is an extension of the Schatten class Sϕ . Notice here that extensions of Schatten classes have been studied in different contexts in the literature (see, e.g., [17]).
3
(ϕ, ψ)-summing operators on C(K)-spaces
In this section we study the Banach operator ideal of (ϕ, ψ)-summing operators, i.e., (`ϕ , `ψ )-summing operators generated by Orlicz sequence spaces `ϕ and `ψ such that `ψ ,→ `ϕ . In the case of power functions this ideal coincides with the well-known ideal of (q, p)summing operators which found many deep applications in various aspects of functional analysis. We mention only that the Banach ideal of (q, 2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces (see [5, 17] and references therein). We also refer to [7] where Weyl numbers and eigenvalues of (E, 2)-summing operators are studied. In [18] Pisier proved a factorization theorem for (q, p)-summing operators on C(K)spaces, 1 ≤ p < q < ∞. Inspired by the ideas from [18], we prove a variant of Pisier’s
factorization theorem.
Let K be a compact Hausdorff space and let (fk )nk=1 be a finite sequence in C(K). In P n fk ek denotes the function on K given by what follows k=1 ψ n n
X
X
(t) := f (t)e f e
, k k k k k=1
ψ
k=1
ψ
t ∈ K.
Theorem 3.1. Let K be a compact Hausdorff space, and let ϕ and ψ be Orlicz functions such that ϕ ≺ ψ, ϕ satisfies the ∆2 -condition and ψ is normalized. The following are equivalent statements about an operator T from C(K) to a Banach space Y : (i) T is (ϕ, ψ)-summing.
9
(ii) There exists a Borel probability measure µ on K and a constant c > 0 such that Z ψ(|f |) dµ. ϕ(kT f kY ) ≤ c K
for every f ∈ C(K) with kf k∞ ≤ 1. Proof. (i)⇒ (ii): We may assume without loss of generality that t 7→ ϕ(t1/r ) is a concave function for some 1 ≤ r < ∞ and ψ(1) = 1.
Assume that T : C(K) → Y is (ϕ, ψ)-summing with πϕ,ψ (T ) = 1. For each 0 < ε < 1,
choose f1 , ..., fn in C(K) such that
n
X
fk ek
ψ ≤ 1 + ε ∞
k=1
(`ϕ )0
k=1
Notice that this implies that
hyk∗ , T fk i
ϕ
k=1
We choose functionals y1∗ , ..., yn∗ ∈ Y ∗ so that n
X
kyk∗ kY ∗ ek
n
X
and kT fk kY ek = 1.
=1
and
n X k=1
hyk∗ , T fk i = 1.
≥ 0 for each 1 ≤ k ≤ n.
Define a functional µε ∈ C(K) → K by µε (g) =
n X k=1
hyk∗ , T (gfk )i.
Combining the above relations with πϕ,ψ (T ) = 1 yields
X n n
X
|hµε , gi| ≤ kT (gfk )kY ek ≤ gf e k k
ϕ
k=1
k=1
ψ
∞
≤ (1 + ε)kgk∞
for all g ∈ C(K). This shows that µε is a continuous functional on C(K) with kµε k ≤ 1 + ε.
We need further properties of µε . We show that under the assumption that t 7→ ϕ(t1/r )
is a concave function, for every f ∈ C(K) with kf k∞ = 1, we have
1 r kT (f )k Y ≤1− µε (1 − ψ(|f |) . ϕ 1+ε 1+ε
To do this fix f ∈ C(K) with kf k∞ ≤ 1 and consider the continuous functions on K
defined by
gk := |fk | 1 − ψ(|f |) ,
gn+1 := |f |.
By convexity of ψ, we have ψ(cgk ) ≤ ψ(c|fk |) (1 − ψ(|f |)) for all c > 0 and 1 ≤ k ≤ n. Since
n
n
X
X
f e ≤ 1 + ε, ψ(|f |/(1 + ε))
≤ 1, k k k
k=1
ψ ∞
k=1
10
∞
and hence
Pn
k=1
ψ(|fk |/(1 + ε)) ≤ 1. Combining these inequalities, we get
n+1
X
ψ(gk /(1 + ε))
∞
k=1
≤ 1 − ψ(|f |) + ψ(|f |/(1 + ε)) ∞ ≤ 1,
Pn+1 and so k=1 gk ek ψ ∞ ≤ 1 + ε. Thus by πϕ,ψ (T ) = 1, we obtain
n+1
X
kT gk kY ek ≤ 1 + ε,
ϕ
k=1
which gives the following estimate
n+1 X k=1
P Let s > 0 be such that n
k=1
ϕ kT gk kY /(1 + ε) ≤ 1.
P kT gk kY /(1+ε) ek ϕ > s. Then nk=1 ϕ kT gk kY /(1+ε) ≥
1. The above inequality yields s ≤ 1, and so concavity of the function t 7→ ϕ(t1/r ) implies n X k=1
n X r ϕ kT gk kY /(1 + ε) ≥ kT gk kY /(1 + ε) ek ϕ . k=1
In consequence, we obtain
n n X
X r
kT gk kY /(1 + ε) ek ϕ ϕ kT gk kY /(1 + ε) ≤ 1 − ϕ kT f kY /(1 + ε) ≤ 1 − k=1
k=1
≤1−
1 (1 + ε)
n X k=1
hyk∗ , T gk i
r
=1−
r 1 ≤1− µε 1 − ψ(|f |)) . (1 + ε)
r 1 µε (1 − ψ(|f |)) (1 + ε)
Let µ be the weak∗ limit of µε as ε → 0. Clearly µ is a positive functional of norm 1, by
kµε k ≤ 1 + ε and µε (1) = 1. Thus, by the Riesz representation theorem, we may assume R that µ is a Borel probability measure on K such that µ(f ) = K f dµ for all f ∈ C(K). In consequence, it follows from the above inequality that for all f ∈ C(K) with kf k∞ ≤ 1, ϕ kT f kY ) ≤ 1 − (µ(1 − ψ(|f |))r ≤ rµ(ψ(|f |)) Z =r ψ(|f |) dµ, K
and this complete the proof of (ii). (ii) ⇒(i): Assume that we can associate with T : C(K) → Y a Borel probability measure
µ on K and a constant c such that for any f ∈ C(K) with kf k∞ ≤ 1, Z ϕ(kT f kY ) ≤ c ψ(|f |) dµ. K
11
Pn Pn Select f1 , ..., fn ∈ C(K) with k=1 fk ek ψ ∞ ≤ 1. Since k k=1 ψ(|fk |)k∞ ≤ 1, we have
kfk k∞ ≤ 1 for 1 ≤ k ≤ n. Thus using the above inequality, we get n X k=1
ϕ(kT fk kY ) ≤ c
Z X n K k=1
ψ(|fk |) dµ ≤ c.
This shows that T is (ϕ, ψ)-summing with πϕ,ψ (T ) ≤ max{1, c}. Corollary 3.1. Let K be a compact Hausdorff space and Y be a Banach space. Assume that ρ(s, t) := s (ϕ−1 ◦ ψ)(t/s) ∈ Φ, where ϕ, ψ are Orlicz functions such that ϕ ∈ (∆2 )
and ψ is sub-multiplicative in a neighbourhood of zero (i.e., there exist C > 0 such that ψ(st) ≤ Cψ(s)ψ(t) when 0 < s, t ≤ a for some a > 0). If an operator T : C(K) → Y is
(ϕ, ψ)-summing, then there exist a Borel probability measure µ on K and a constant c such that kT f kY ≤ cρ(kf kC(K) , kf kLψ (µ) ),
f ∈ C(K).
Proof. We note that the condition ρ ∈ Φ yields ϕ ≺ ψ, and whence `ψ ,→ `ϕ . By a compactness argument and normalization, we may suppose without loss of generality that ψ
is a sub-multiplicative function on [0, 1] and ψ(1) = 1. Thus, it follows by Theorem 3.1 that there exist a Borel probability measure µ on K and a constant c such that for any f ∈ C(K) with kf k∞ ≤ 1, we obtain ϕ(kT f kY ) ≤ c
Z
K
ψ(|f |) dµ.
Now let 0 6= f ∈ C(K). Combining the above estimate with inequalities kf kLψ (µ) ≤ kf k∞ , R K ψ(|f |/kf kLψ (µ) ) dµ ≤ 1 and sub-multiplicativity of ψ on [0, 1] yields Z kf kLψ (µ) kf kLψ (µ) kT f kY |f | ϕ ≤ λψ ψ dµ ≤ λ ψ , kf k∞ kf k∞ kf kLψ (µ) kf k∞ K
for some constant λ. Clearly this implies the required estimate.
Theorem 3.2. Let K be a compact Hausdorff space and Y be a Banach space. If an Orlicz function ϕ ∈ (∆2 ) and 1 ≤ p < ∞ are such that ρ(s, t) := s ϕ−1 (tp /sp ) ∈ Φ, then (Πp )ρ (C(K), Y ) = Πϕ,p (C(K), Y ).
12
Proof. Assume that T ∈ Πϕ,p (C(K), Y ). Corollary 3.1 with ψ(t) = tp gives that there exist a Borel probability measure µ on K and a positive constant c such that kT f kY ≤ cρ(kf kC(K) , kJf kLψ (µ) ),
f ∈ C(K),
where J : C(K) → Lp (µ) is the inclusion map. Since J is p-summing, T ∈ (Πp )ρ (C(K), Y ).
The proof of the converse is easy. In fact, assume u ∈ (Πp )ρ (C(K), Y ) with (πp )ρ (u) < 1.
Then there is a Banach space Z and a p-summing operator v : C(K) → Z with πp (v) ≤ 1 satisfying
kuf kY ≤ ρ(kf kC(K) , kvf kZ ), f ∈ C(K).
Pn p 1/p Select f1 , ..., fn ∈ C(K) with ≤ 1. As v is p-summing with πp (v) ≤ 1, k=1 |fk | ∞ n X k=1
ϕ(kufk kY ) ≤
n X k=1
kvfk kpZ ≤ 1.
This shows that u ∈ Πϕ,p (C(K), Y ) and πϕ,p (u) ≤ 1. Below we will use Matuszewszka-Orlicz index (cf. [6, 8]). Recall that if a function ρ : R+ → R+ is non-nondecreasing and ρ¯(t) := lim sups→0+ ρ(ts)/ρ(s) , then the upper index β0 (ρ) is given by
β0 (ρ) = lim
t→∞
log ρ¯(t) . log t
We are now ready to prove a variant of Pisier’s factorization theorem for (ϕ, p)-summing operators. We shall make use the Lorentz space. Recall that if (Ω, µ) is the measure space and ϕ : [0, µ(Ω)) → [0, ∞) is a non-decreasing concave function, with ϕ(0) = 0, the r.i.
Lorentz space Λ(ϕ) on (Ω, µ) consists of all (µ-a.e. equivalence classes) whose decreasing
rearrangement f ∗ : [0, µ(Ω)) → [0, ∞] given by f ∗ (t) = inf{s > 0; µ({|f | > s} ≤ t} for all
t > 0 satisfies
kf kψ :=
Z
0
µ(Ω)
f ∗ (t) dψ(t) < ∞.
Theorem 3.3. Let an Orlicz function ϕ ∈ (∆2 ), K be a compact Hausdorff space, and Y be a Banach space. Consider the following statements about the operator T : C(K) → Y : (i) T is (ϕ, 1)-summing. (ii) There exist a Borel probability measure µ on K and a factorization: J
S
T : C(K) −→ Λψ (µ) −→ Y, where J is the inclusion map, and ψ = ϕ−1 . 13
(iii) If `p ,→ `ϕ with 1 < p < ∞, then T is (ϕ, p)-summing. Then the statements (i) and (ii) are equivalent. If we assume additionally that p < β0 (ϕ), then all three statements are equivalent. Proof of Theorem 3.3. (i) ⇒ (ii): The preceding corollary yields that we can find a Borel probability measure µ on K and a constant c such that kT f kY ≤ cρ(kf kC(K) , kf kL1 (µ) ),
f ∈ C(K),
where ρ ∈ Φ is defined by ρ(s, t) = sϕ−1 (t/s) for s, t > 0 and ρ(0, 0) = 0.
We will show that T can be extended to a bounded linear operator S : L∞ (µ) → Y . To
see this fix f ∈ L∞ (µ). Now, applying Lusin’s Theorem, we can find a sequence (gn ) in
C(K) with kgn kC(K) ≤ kf kL∞ (µ) for all n and f = limn→∞ gn µ-a.e. Since ρ(s, t) → 0 as
t → 0, the Lebesgue dominated convergence theorem and the above inequality yield that
(T gn ) is a Cauchy sequence in Y . Thus the limit Tef := limn→∞ T gn defines the required bounded extension of T , which satisfies
kTef kY ≤ c ρ(kf kL∞ (µ) , kf kL1 (µ) ),
f ∈ L∞ (µ).
In particular for f = χA , we have
kTe(χA )kY ≤ c ϕ−1 (µ(A)) = c ψ(µ(A)).
Since C(K) is dense in Λψ (µ), it follows that Te has an unique continuous extension S from
Λψ (µ) into Y , and this completes the proof (i) ⇒(ii).
(ii) ⇒ (i): This follows by the fact that for any finite measure space (Ω, µ) the natural
inclusion map J : L∞ (µ) → Λψ (µ) is (ϕ, 1)-summing (see [2, Proposition 2.1]).
(ii) ⇒ (iii): In view of Theorem 3.1, we need only to show that for all f ∈ C(K) with
kf k∞ ≤ 1,
ϕ(kT f kY ) ≤ c
Z
K
|f |p dµ.
By our hypothesis, it follows that it is enough to prove that for each f ∈ C(K) with kf k∞ ≤ 1, we have
ϕ(kf kΛψ (µ) ) ≤ c1
where c1 is another constant.
14
Z
K
|f |p dµ,
Let f ∈ C(K) with kf k∞ ≤ 1. Then, setting u :=
R
K
|f |p dµ, we get f ∗ (t) ≤ t−1/p u1/p ≤
1 for all 0 ≤ t ≤ 1. Since p < β0 (ϕ), we have α0 (ψ) = α0 (ϕ−1 ) = 1/β0 (ϕ) < 1/p. Thus combining the remarks with the well known properties of indices (see, e.g., [6]), we obtain Z 1 Z 1 dt ∗ f ∗ (t) ψ(t) f (t) dψ(t) ≤ kf kΛψ (µ) = t 0 0 Z 1 Z u dt 1/p dt t−1/p ψ(t) u = ψ(t) + t t u 0 ≺ ψ(u) + (u−1/p ψ(u))u1/p ≺ ψ(u),
whence the required inequality follows. Since the implication (iii) ⇒ (i) is trivial, the proof is complete.
Remark. Notice here that in the case ϕ ∈ (∆2 ) the equivalence (i) ⇔ (ii) has been
established by Montgomery-Smith [14]. In the case when ϕ(t) = t1/q with 1 ≤ p < q < ∞, we obtain the factorization theorem which states that (q, p)-summing operators defined on
C(K) factorizes through the Lorentz space Lq,1 (µ) := Λψ (µ) (with ψ(t) = t1/q for t ≥ 0) proved by Pisier in his remarkable paper [18].
4
Generalized concavity of operators
In this section we continue with the study of (ϕ, ψ)-summing operators. For given normalized Orlicz functions ϕ and ψ we introduce the notion of (ϕ, ψ)-concave operators, which generalizes the classical concept of (q, p)-concave operators (see [3]). We show relationships between (ϕ, ψ)-summing and (ϕ, ψ)-concave operators. We prove that some classical results on (q, p)-concave operators from Banach lattices can be lifted to the class of generalized concave operators. Let X be a Banach lattice, and let ϕ be a normalized Orlicz function. Following the functional calculus (in the case ϕ(t) = tp for all t ≥ 0, see [3, Section 16]), for a given finite
set {x1 , ..., xn } from X, we define
n n nX o X n n )0 . x e := sup a x ; (a ) ∈ B k k k k k (` k=1 ϕ k=1
ϕ
k=1
Here `nϕ , and respectively, (`nϕ )0 is Rn equipped with the norm
n n o X k(xk )nk=1 kϕ := inf λ > 0; ϕ(|xk |/λ) ≤ 1 , k=1
15
and respectively, k(xk )nk=1 k(`nϕ )0 := max
n nX k=1
o |xk yk |; k(yk )nk=1 kϕ ≤ 1 . p 1/p k=1 |xk |
Pn
If 1 ≤ p < ∞ and ϕ(t) = tp for all t ≥ 0, then as usual we write Pn of k=1 xk ek ϕ . Thus n X k=1
|xk |p
1/p
instead
n nX o = sup ak xk ; (ak )nk=1 ∈ B`n0 , p
k=1
where 1/p + 1/p0 = 1. In this section we suppose that ϕ and ψ are normalized Orlicz functions. An operator T from a Banach lattice X to a Banach space Y is said to be (ϕ, ψ)-concave if there is a positive constant C such that, for every finite sequence (xk )nk=1 in X,
n n
X
X
kT xk kY ek ≤ C f e
k k
ϕ
k=1
k=1
. ψ X
We write Kϕ,ψ (T ) for the least constant C that works in the inequality above. If 1 ≤ p < ∞ and ψ(t) = tp (resp., ϕ(t) = tq and ψ(t) = tp for all t ≥ 0 with 1 ≤ q ≤ p),
we say that T is (ϕ, p)-concave (resp., (q, p)-concave) and write Kϕ,p (T ) (resp., Kq,p (T )). As usual, the (q, q)-concave operators are called q-concave, and Kq (T ) is used instead of Kq,q (T ). For general examples of (ϕ, ψ)-concave operators we refer to [11].
Combining our results with the techniques applied to (q, p)-concave operators (see [3, pp. 330–332]) we prove that some results on (q, p)-concave operators can be lifted to a general setting. These results show relationships between (ϕ, ψ)-concave and (ϕ, ψ)-summing operators from Banach lattices to Banach spaces. Theorem 4.1. Let X be a Banach lattice and let Y be a Banach space. Assume ψ, ϕ are functions such that ϕ ≺ ψ. An operator T : X → Y is (ϕ, ψ)-concave with Kϕ,ψ (T ) ≤ C if and only if for each compact Hausdorff space K and every positive operator V : C(K) → X, the composition T V : C(K) → Y is (ϕ, ψ)-summing with πϕ,ψ (T V ) ≤ CkV k.
Proof. Let V : C(K) → X be a positive operator. Fix f1 , ..., fn ∈ C(K). Then we have n n n nX o X X n 0 f e (V f )e = sup a f ; a ∈ B ≤ V k k k k k k (`ψ ) . k=1
ψ
k=1
k=1
16
ψ
If we assume that T : X → Y is (ϕ, ψ)-concave with Kϕ,ψ (T ) ≤ C, then
X n n
X
kT V (fk )kY ek ≤ C V f e
k k ϕ
k=1
k=1
≤ CkV k
sup
ψ
X
kx∗ kC(K)∗ ≤1
and so T V is (ϕ, ψ)-summing with πϕ,ψ (T V ) ≤ CkV k.
n
X
x∗ (fk )ek
ϕ
k=1
To prove the converse, fix x1 , ..., xn ∈ X. Without loss of generality we may assume
that kxkX = 1, where
n X . x := x e k k ψ
k=1
Let I(x) be the linear span of the order interval [−x, x] in X. Taking [−x, x] as the unit ball of I(x), I(x) is an abstract M -space, and by the well known theorem of Kakutani (see, e.g., [8, p.13]) there is a compact Hausdorff space K so that I(x) is isometrically lattice isomorphic to C(K). Let J : C(K) → X be the associated lattice isomorphism which maps
the unit ball of C(K) onto the order interval [−x, x]. Then we have xk ∈ [−x, x] for all
1 ≤ k ≤ n. Thus for each k ∈ {1, ..., n} there exists fk ∈ C(K) such that kfk kC(K) ≤ 1 and
xk = J(fk ). Since J is an isometrical lattice isomorphism, |xk | = J(|fk |) and
n
X
fk ek
ψ
C(K)
k=1
Consequently,
n
X
−1
= J xk ek ψ
C(K)
k=1
≤ kxkX = 1.
X n n n
X
X
kT xk kY ek = kT Jfk kY ek ≤ πϕ,ψ (T J) f e
k k k=1
ϕ
ϕ
k=1
≤ πϕ,ψ (T J) xkX = πϕ,ψ (T J).
k=1
ψ
C(K)
and this completes the proof.
Corollary 4.1. Let 1 ≤ p < β0 (ϕ) < ∞. An operator from a Banach space to a Banach space is (ϕ, p)-concave if and only if it is (ϕ, 1)-concave.
Proof. It follows from Theorem 3.3 that, when 1 ≤ p < ∞, an operator from C(K)-space
is (ϕ, p)-summing if and only if it is (ϕ, 1)-summing. To conclude it is enough to apply Theorem 4.1.
17
As usual we denote by(rn ) the sequence of Rademacher functions on [0, 1] (or, equivalently, any sequence of independent random variables taking the values ±1 with probabilities 1/2).
We conclude with the following corollary. Corollary 4.2. Let ϕ be an Orlicz function such that 2 < β0 (ϕ) < ∞. The following are equivalent statements about an operator T from a Banach lattice X to a Banach space Y. (i) T is (ϕ, 1)-summing. (ii) T is (ϕ, p)-summing for some (and then all ) 1 ≤ p < β0 (ϕ). (iii) There is a positive constant C such that for every finite set {x1 , ..., xn } in X n
X
Z
kT xk kY ek ≤ C
ϕ
k=1
0
n 1 X
k=1
2 1/2
rk (t)xk dt . X
Proof. Since (ϕ, 1)-summing operator T : X → Y is (ϕ, 1)-concave, (i) ⇒ (ii) follows by Corollary 4.1.
(ii) ⇒ (iii). Let {x1 , ..., xn } be a finite subset of elements from X. Then the following
inequality (see [3, Corollary 16.4])
n
X 1/2
A1 |xk |2
X
k=1
≤
Z
0
yields the required estimate
n 1 X
k=1
2 1/2
rk (t)xk dt X
n n
X
X 1/2
2 kT x k e ≤ K (T ) |x |
ϕ,2 k Y k k ϕ
k=1
X
k=1
Z ≤ A−1 K (T ) ϕ,2 1
0
The implication (iii) ⇒ (i) is obvious by Z
0
n 1 X
k=1
2 1/2
≤ rk (t)xk dt X
18
n 1 X
k=1
sup
2 1/2
. rk (t)xk dt X
n X
kx∗ kX ∗ ≤1 k=1
|x∗ (xk )| .
References [1] A. Defant, M. Mastylo and C. Michels, Summing inclusion maps between symmetric sequence spaces, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4473–4492. [2] A. Defant and M. Mastylo, Composition of (E, 2)-summing operators, Studia Math. 159 (2003), no. 1, 51–65. [3] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge, Univ. Press, 1995. [4] H. Jarchow and U. Matter, Interpolative constructions for operator ideals, Note di Mat. 8 (1988), 45–56. [5] H. K¨onig, Eigenvalue Distribution of Compact Operators, Birkh¨auser, 1986. [6] S. G. Krein, Yu. I. Petunin and E. M. Semenov. Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, 1982. [7] T. K¨ uhn and M. Mastylo, Weyl numbers and eigenvalues of abstract summing operators, J. Math. Anal. Appl. 369 (2010), no. 1, 408–422. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, SpringerVerlag, 1977. [9] G.Ya. Lozanovskii, On some Banach lattices IV , Sibirsk. Math. Z. 14 (1973), 140-155 (in Russian); English transl. in Siberian Math. J. 1 (1973), 97–108. [10] M. Mastylo, Interpolative construction and the generalized cotype of abstract Lorentz spaces, J. Math. Anal. Appl. 319 (2006), no. 2, 460–474. [11] M. Mastylo and E. A. S´ anchez P´erez, Maurey-Rosenthal factorization of operators through Orlicz spaces, Preprint. [12] U. Matter, Absolutely continuous operators and super-reflexivity, Math. Nachr. 134 (1987), 193–216. [13] U. Matter, Factoring through interpolation spaces and super-reflexive Banach spaces, Rev. Roumanie Math. Pures el Appl. 34 (1989), no. 2, 147–156. 19
[14] S. J. Montgomery-Smith, The cotype of operators from C(K), Ph.D. Dissertation, Cambridge University 1988. [15] V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep. 1 (1984), 349–516. [16] A. Pietsch, Operator Ideals, North-Holland, 1980. [17] A. Pietsch, Eigenvalues and s-numbers, Cambridge Studies in Advanced Mathematics 13, 1987. [18] G. Pisier, Factorization of operators through Lp,∞ or Lp,1 and non-commutative generalizations, Math. Ann. 276 (1986), 105–136.
Faculty of Mathematics and Computer Science Adam Mickiewicz University and Institute of Mathematics Polish Academy of Science (Pozna´ n branch) Umultowska 87 61-614 Pozna´ n E-mail: [email protected] Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna´ n E-mail: [email protected]
20
S0022-247X(12)00951-1 10.1016/j.jmaa.2012.11.036 YJMAA 17186
To appear in:
Journal of Mathematical Analysis and Applications
Received date: 6 September 2012 Please cite this article as: M. Mastyło, R. Szwedek, Interpolative construction and factorization of operators, J. Math. Anal. Appl. (2012), doi:10.1016/j.jmaa.2012.11.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Interpolative construction and factorization of operators Mieczyslaw Mastylo and Radoslaw Szwedek Abstract Banach operator ideals generated by interpolative construction applied to p-summing operators are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. Relationships between Banach ideals determined by Orlicz sequence spaces are shown and a variant of the Pisier factorization theorem for (p, 1)-summing operators from C(K)-spaces is proved. Applications to Schatten classes are given. It is also shown that certain known results on (q, p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.
1
Introduction
We study the Banach operator ideals generated via an interpolative construction determined by quasi-concave functions. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. In order to prepare for the discussion that will follow, we introduce some fundamental notion and definitions. The standard references for operator ideals are the monographs [3, 16]. We shall use some standard notations and notions from Banach space theory. If X is a Banach space, then BX is its closed unit ball and X ∗ is its dual space. As usual L(E, F ) denotes the Banach space of all (bounded and linear) operators T : E → F between
Banach spaces E and F endowed with the operator norm. If E ⊂ F , then E ,→ F means
that the inclusion map J : E → F is bounded. Throughout the paper, for two real valued functions f and g we use the symbol f ≺ g whenever there exists a constant c > 0 such
that f (t) ≤ cg(t) for all t in the domain of f and g, and f g whenever f ≺ g and g ≺ f .
2010 Mathematics Subject Classification: Primary: 47B10; Secondary: 47B60, 47B38, 46B70. Key words and phrases: Factorization, Banach operator ideals, (p, q)-summing operators, Interpolation spaces, Banach lattices, generalized concavity. The first named author was supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243.
1
Let Φ denote the set of all functions ϕ : [0, ∞)×[0, ∞) → [0, ∞) which are non-decreasing
in each variable and are positively homogeneous (that is, ϕ(λs, λt) = λϕ(s, t) for all λ, s, t ≥
0). If in addition ϕ is concave, then we write ϕ ∈ P. The subset of all ϕ ∈ P for which ϕ(s, 1) → 0 and ϕ(1, t) → 0 as s → 0+ and t → 0+ is denoted by P0 .
Let ϕ ∈ P and let (A, α) and (B, β) be two Banach operator ideals. In [10] an interpola-
tive construction (A, B)ϕ was studied. For the sake of completeness, we recall that if X and Y are Banach spaces, then the space (A, B)ϕ (X, Y ) consists of all operators T ∈ L(X, Y )
such that for some Banach spaces Zj and operators Sj : X → Zj (j = 0, 1) with α(S0 ) ≤ 1, β(S1 ) ≤ 1 and for some λ > 0,
kT xkY ≤ λ ϕ(kS0 xkZ0 , kS1 xkZ1 ),
x ∈ X.
We put γϕ (T ) as the infimum of the values of λ for which the inequality holds with suitable operators S0 and S1 described above. It is easy to see that an operator T : X → Y belongs to (A, B)ϕ (X, Y ) if and only if
there exist Banach spaces Z0 , Z1 and operators S0 ∈ A(X, Z0 ) and S1 ∈ B(X, Z1 ) such that kT xkY ≤ ϕ kS0 xkZ0 , kS1 xkZ1 ,
x ∈ X.
Moreover for every T ∈ (A, B)ϕ (X, Y ), we have
γϕ (T ) inf max{α(S0 ), β(S1 )}, where the infimum is taken over all operators S0 and S1 admitted the above estimate. If B = L, we put (Aϕ , αϕ ) = (A, L)ϕ . In the case ϕ(s, t) = s1−θ tθ , 0 < θ < 1, we obtain
the construction (α, β)θ studied in [12, 13].
Let A be a Banach operator ideal. We recall that an injective hull Ai of A is defined by
T ∈ Ai (X, Y ) if JY T ∈ A(X, `∞ (BY ∗ )), where JY : Y → `∞ (BY ∗ ) is the canonical isometric
embedding. An ideal A is said to be injective if A = Ai . We write A for the ideal of all
operators T : X → Y which satisfy kTn − T k → 0 as n → ∞ for some sequence (Tn ) in
A(X, Y ). If A = A, then A is said to be closed.
We notice here that for every Banach operator ideals (A, α), (B, β) and ϕ ∈ P, (A, B)ϕ
is an injective Banach operator ideal. If A is a closed and injective ideal, then Aϕ = A.
ϕ(s, 1) → 0 as s → 0 and Aϕ = A implies that A is a closed ideal (for details we refer to [10]). Here, for every ϕ ∈ Φ, we define ϕ by ϕ(αs, βt) ϕ(s, t) = sup ; α, β > 0 , ϕ(α, β) 2
s, t > 0.
We now describe the main results of the paper.
In section 2 interpolative proce-
dure is used to the Banach ideal of p-summing operators to generate the Banach ideal of (p, ϕ)-absolutely continuous operators. A description of this ideal via factorization through Lorentz space between L∞ (µ) and Lp (µ) is presented. It is also shown that under certain concavity condition (2, ϕ)-absolutely continuous operators between Hilbert spaces coincides with the Schatten class determined by an Orlicz sequence space. In Section 3 of the paper is devoted to the Banach operator ideal of (ϕ, ψ)-summing operators generated by Orlicz sequence spaces. In the case of power functions this ideal coincides with the well-known ideal of (q, p)-summing operators. A connection with (ϕ, p)-summing operators is shown. Based on the ideas from [18], we prove a variant of Pisier’s description for operators on C(K)-spaces. This result enables us to prove a variant of remarkable Pisier’s factorization theorem for (ϕ, ψ)-summing operators on C(K)-spaces. In Section 4, we introduce the notion of (ϕ, ψ)-concave operators (for normalized Orlicz functions ϕ and ψ), which generalizes the classical concept of (q, p)-concave operators (see [3]). We show relationships between (ϕ, ψ)-summing and (ϕ, ψ)-concave operators. We prove that some classical results on (q, p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.
2
(p, ϕ)-absolutely continuous operators
The following definition is a generalization of the classical concept of (q, p)-summing operators. Let E, F be two Banach sequence lattices on N and let (en ) be the standard unit vector basis. An operator T ∈ L(X, Y ) is (F, E)-summing if there exists a constant C > 0 such that for any finite set {x1 , ..., xn } of X, n
X
kT xk kY ek ≤ C
k=1
F
sup kx∗ kX ∗ ≤1
n
X
x∗ (xk )ek .
k=1
E
We denote by πF,E (T ) the least constant with this property.
It is easy to see that if there exists some non-trivial operator T : X → Y that is (F, E)-
summing, then E ,→ F . Moreover the space ΠF,E (X, Y ) of all (F, E)-summing operators is
a Banach space equipped with the norm πF,E and (ΠF,E , πF,E ) is a Banach operator ideal whenever the norm of the inclusion map E ,→ F equals 1. Notice that when F = `q and E = `p with 1 ≤ p ≤ q ≤ ∞, then we recover the ideal
(Πq,p , πq,p ) of (q, p)-summing operators (see [3]). Of particular importance is the choice of 3
p = q, which is related to the Banach ideal (Πp , πp ) of the so-called p-summing operators. We refer to [1, 2, 7] where (F, E)-summing operators are studied and applications in various parts of analysis are shown. If ϕ ∈ Φ, then operators from (Πp )ϕ := (Πp , πp )ϕ are called (p, ϕ)-absolutely continuous.
If ϕ(s, t) = s1−θ tθ (0 ≤ θ < 1), then (p, ϕ)-absolutely continuous operators are called (p, θ)absolutely continuous (see [12]).
Before the proof of the mentioned factorization theorem it is be convenient to give some preliminaries. Let ϕ ∈ Φ and X = (X0 , X1 ) be a compatible couple of Banach spaces.
Following [15], the abstract Lorentz space Λϕ (X) consists of all x ∈ X0 + X1 such that x=
X
un
(convergence in X0 + X1 ),
n∈Z
with un ∈ X0 ∩ X1 , and
P
n∈Z
ϕ(kun kX0 , kun kX1 ) < ∞. The norm on Λϕ (X) is given by
kxkΛϕ (X) = inf
nX
n∈Z
ϕ(kun kX0 , kun kX1 ); x =
where infimum is taken over all series described above.
X
n∈Z
o un ,
Let X and Y be Banach spaces. Further, we define for every operator S ∈ L(X, Y ) the
image space S(X) as the Banach space {Sx; x ∈ X} equipped with the norm kykS(X) := inf{kxkX ; Sx = y},
y ∈ S(X).
Notice that S(X) ,→ Y and so (S(X), Y ) forms an ordered Banach couple. We will need the following factorization result from [10]. Theorem 2.1. Suppose ϕ ∈ Φ is such that ϕ(1, t) → 0 as t → 0, and that operators
T ∈ L(X, Y ), S ∈ L(X, Z) satisfy the inequality kT xkY ≤ ϕ(kxkX , kSxkZ ) for all x ∈ X.
Then there exists an operator R : Λϕ (S(X), Z) → Y with kRk ≤ γϕ (T ) such that T admits a factorization:
S
J
R
T : X −→ S(X) −→ Λϕ (S(X), Z) −→ Y, where J is the inclusion map. The following theorem is a unified factorization theorem for the case of any function ϕ ∈ Φ with ϕ(1, t) → 0 as t → 0 (cf. [13] for the power functions ϕθ (s, t) = s1−θ tθ , θ ∈ (0, 1)).
4
Theorem 2.2. Let ϕ ∈ P be such that ϕ(1, t) → 0 as t → 0. An operator T ∈ L(X, Y )
is (p, ϕ)-absolutely continuous with (πp )ϕ (T ) ≤ C if and only if for every (equivalently for some) isometric embedding i : X → L∞ (µ) there exists a Borel probability measure µ on
K = (BX ∗ , σ(X ∗ , X)), a constant C > 0 and a bounded operator S : Λϕ (jp (X), Lp (µ)) → Y with kSk ≤ C such that T admits the factorization: jp
S
T : X −→ jp (X) ,→ Λϕ (jp (X), Lp (µ)) −→ Y, with jp = ji, where j : L∞ (µ) → Lp (µ) is the inclusion map. Proof. Assume that T : X → Y is (p, ϕ)-absolutely summing with (πp )ϕ (T ) ≤ C. Then there exists a Banach space Z and an operator u ∈ Πp (X, Z) with πp (u) ≤ 1 such that kT xkY ≤ C ϕ(kxkX , kuxkZ ),
x ∈ X.
The Pietsch Domination Theorem provides an isometric embedding i : X → L∞ (µ) and
a regular Borel probability measure µ on K = (BX ∗ , σ(X ∗ , X)) for which Z 1/p kuxkZ ≤ |ix(x∗ )|p dµ , x ∈ X. K
In view of above inequalities
kT xkY ≤ C ϕ(kxkX , kjp xkLp (µ ),
x∈X
with jp = ij : X → Lp (µ), where j : L∞ (µ) → Lp (µ) is the inclusion map. Now we conclude by Theorem 2.1 that T admits the required factorization with kSk ≤ C.
Conversely, assume that there exist a Borel probability measure µ on K, an isometric
embedding i : X → L∞ (µ) and a bounded operator S with kSk ≤ C such that T : X → Y admits the factorization
jp
J
S
T : X −→ jp (X) −→ Λϕ (jp (X), Lp (µ)) −→ Y, where J is the inclusion map. By the construction of the abstract Lorentz space, it follows that for any x ∈ X we have kJ(jp x)k ≤ ϕ(kjp (x)kjp (X) , kjp xkLp (µ) ),
x ∈ L∞ (µ).
Since jp = ji and the inclusion map j : L∞ (µ) → Lp (µ) is p-absolutely summing with πp (j) ≤ 1, we have πp (jp ) ≤ 1. By kSk ≤ C this implies that (πp )ϕ (T ) ≤ kik πp (jp ) kSk ≤ C, and the proof is complete. 5
We conclude the section by showing applications to Schatten classes. We show that under certain condition (2, ϕ)-absolutely continuous operators between Hilbert spaces coincides with the Schatten class determined by an Orlicz sequence space. We recall that a continuous and convex function ϕ : [0, ∞) → [0, ∞) with ϕ−1 ({0}) = {0} is called an Orlicz function. For an Orlicz function ϕ the Banach sequence lattice ∞ n X `ϕ = x = (xn )∞ ; ϕ(|xn |/λ) < ∞ n=1 n=1
o for some λ > 0
equipped with the norm
∞ n o X kxkϕ := kxk`ϕ = inf λ > 0; ϕ(|xn |/λ) ≤ 1 n=1
is called an Orlicz sequence space.
It is well known that the unit vector basis (en ) is a basis in `ϕ (or equivalently `ϕ is a separable space) if and only if and only if ϕ satisfies the δ2 -condition at 0, i.e., ϕ(2t) < ∞. 0
We note that the behaviour of ϕ outside a neighbourhood of 0 is irrelevant to the definition of ϕ, and so we may assume, without loss of generality, that ϕ satisfies the ∆2 -condition (for short ϕ ∈ (∆2 )), i.e., there exists C > 0 such that ϕ(2t) ≤ Cϕ(t) for all t > 0.
Notice that if an Orlicz space `ϕ is separable, then the dual (`ϕ )∗ is isometrically isomorP phic to the K¨ othe dual sequence space (`ϕ )0 of all sequences (xn ) such that ∞ n=1 |xn yn | < ∞ for all y = (yn ) ∈ `ϕ equipped with the norm kxk(`ϕ )0 = sup
∞ nX
n=1
o |xn yn |; k(yn )kϕ ≤ 1 .
Recall that if T : H1 → H2 is a compact operator between Hilbert spaces, then it has
the Schmidt representation, i.e., T has the form T =
∞ X
τn (·|en )fn ,
(∗)
n=1 ∞ where (en )∞ n=1 is an orthonormal sequence in H1 , (fn )n=1 is an orthonormal sequence in
H2 , and τn = τn (T ) is a null sequence which satisfy 0 ≤ τn+1 ≤ τn for all n ∈ N.
Notice that if T is a finite rank operator, (∗) is a finite sum, and in this case it will
convenient to consider finite sequences as sequences of infinite length by adding zeros. Otherwise, (∗) represents a series which converges with respect to the operator norm k · k in L(H1 , H2 ).
6
If E is a symmetric Banach sequence space on N, then the Schatten class SE (H1 , H2 )
consists of all operators T : H1 → H2 which admit the Schmidt representation with (τn ) ∈ E. It is well known that SE (H1 , H2 ) is a Banach space equipped with the norm σE (T ) = k(τn (T ))kE which does not depend on the specific representation. In particular σE (T ) = k(an (T ))kE , where for an operator u : H1 → H2 and each n ∈ N, an (u) is the n-th approximation number an (u) := inf{kv − uk; v ∈ L(H1 , H2 ),
dim v(H1 ) < n}.
If E = `ϕ is an Orlicz sequence space, we put Sϕ = SE and σϕ = σE . Of particular
importance is the choice ϕ(t) = tp for all t ≥ 0 where 1 ≤ p < ∞, which yields the well known Schatten class (Sp , σp ).
Theorem 2.3. Let ψ ∈ P0 and let ϕ(t) = ψ −1 (t, 1)2 for all t > 0. If H1 and H2 are any
Hilbert spaces, then the following statements are true:
(i) (Π2 )ψ (H1 , H2 ) ,→ Sϕ (H1 , H2 ) with σϕ (T ) ≤ (π2 )ψ (T ). √ (ii) If ϕ(1) = 1 and the function t 7→ ψ 2 ( t, 1) is concave on the interval [0, 1], then Sϕ (H1 , H2 ) ,→ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) ≤ σϕ (T ). Proof. (i): Let T ∈ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) < 1. By the Pietsch Domination Theorem
there exits a Hilbert space G and an operator S ∈ Π2 (G, H2 ) such that π2 (S) ≤ 1 and kT xkH2 ≤ ψ(kSxkG , 1),
x ∈ BH1 .
Since ψ ∈ P0 and S is compact, it follows immediately from the above inequality that T is also compact. Thus the operators S and T admit Schmidt representations: S=
∞ X
n=1
an (S) ( · |xn )yn 7
and T =
∞ X
n=1
The above inequality implies that ∞ X
n=1
an (T )2 |(x|en )|2
1/2
an (T ) ( · |en )fn .
≤ψ
∞ X
n=1
an (S)2 |(x|xn )|2
1/2 ,1
for every x ∈ BH1 , and so for each k ∈ N, ak (T ) ≤ ψ
∞ X
n=1
an (S)2 |(ek |xn )|2
1/2 ,1 .
Thus, by π2 (S) = σ2 (S) ≤ 1, we obtain ∞ X k=1
ϕ(ak (T )) ≤ =
∞ X
n=1 ∞ X
n=1
an (S)2
∞ X k=1
|(ek , xn )|2
an (S)2 ≤ 1.
In consequence T ∈ Sϕ with σϕ (T ) ≤ 1, and this completes the proof of (i).
(ii): For the converse, take any T ∈ Sϕ (H1 , H2 ) with σϕ (T ) ≤ 1. Since `ϕ ,→ c0 ,
an (T ) → 0 as n → ∞. Thus T is compact and hence it admits the Schmidt representation P∞ T = n=1 τn ( · |xn )yn , where (τn ) ∈ B`ϕ with τn ≥ 0 for all n ≥ 0 and (xn ), (yn ) are orthonormal sequences in H1 and H2 , respectively. Define the operator S : H1 → H2 by S=
∞ X
n=1
ϕ(τn )1/2 ( · |xn )yn .
Then S ∈ S2 (H1 , H2 ) = Π2 (H1 , H2 ) with π2 (S) = σ2 (S) ≤ 1. Since 0 ≤ τn ≤ 1 for all √ n ∈ N, our hypothesis on concavity of the function ρ(t) := ψ 2 ( t, 1) on the interval (0, 1] P 2 gives that for any x ∈ H1 with kxkH1 = ∞ n=1 |(x|xn )| = 1, we have kT xk2H2 =
∞ X
n=1
τn2 |(x|xn )|2 =
∞ X
n=1
ρ(ϕ(τn )) |(x|xn )|2
∞ ∞ X X 1/2 ≤ρ ϕ(τn ) |(x|xn )|2 = ψ 2 ϕ(τn ) |(x|xn )|2 ,1 . n=1
n=1
By homogeneity, it then follows that kT xkH2 ≤ ψ(kSxkH2 , kxkH1 ),
x ∈ H1 ,
and so we finally arrive at T ∈ (Π2 )ψ (H1 , H2 ) with (π2 )ψ (T ) ≤ 1. 8
Corollary 2.1. Let ψ ∈ P0 and let ϕ(t) := ψ −1 (t, 1)2 for t ≥ 0. If the function t 7→ √ ψ 2 ( t, 1) is equivalent to a concave function in a neighborhood of zero, then for arbitrary Hilbert spaces G and H the following formula holds: (Π2 )ψ (G, H) = Sϕ (G, H). We note that Corollary 2.1 says that the ideal (Π)ψ is an extension of the Schatten class Sϕ . Notice here that extensions of Schatten classes have been studied in different contexts in the literature (see, e.g., [17]).
3
(ϕ, ψ)-summing operators on C(K)-spaces
In this section we study the Banach operator ideal of (ϕ, ψ)-summing operators, i.e., (`ϕ , `ψ )-summing operators generated by Orlicz sequence spaces `ϕ and `ψ such that `ψ ,→ `ϕ . In the case of power functions this ideal coincides with the well-known ideal of (q, p)summing operators which found many deep applications in various aspects of functional analysis. We mention only that the Banach ideal of (q, 2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces (see [5, 17] and references therein). We also refer to [7] where Weyl numbers and eigenvalues of (E, 2)-summing operators are studied. In [18] Pisier proved a factorization theorem for (q, p)-summing operators on C(K)spaces, 1 ≤ p < q < ∞. Inspired by the ideas from [18], we prove a variant of Pisier’s
factorization theorem.
Let K be a compact Hausdorff space and let (fk )nk=1 be a finite sequence in C(K). In P n fk ek denotes the function on K given by what follows k=1 ψ n n
X
X
(t) := f (t)e f e
, k k k k k=1
ψ
k=1
ψ
t ∈ K.
Theorem 3.1. Let K be a compact Hausdorff space, and let ϕ and ψ be Orlicz functions such that ϕ ≺ ψ, ϕ satisfies the ∆2 -condition and ψ is normalized. The following are equivalent statements about an operator T from C(K) to a Banach space Y : (i) T is (ϕ, ψ)-summing.
9
(ii) There exists a Borel probability measure µ on K and a constant c > 0 such that Z ψ(|f |) dµ. ϕ(kT f kY ) ≤ c K
for every f ∈ C(K) with kf k∞ ≤ 1. Proof. (i)⇒ (ii): We may assume without loss of generality that t 7→ ϕ(t1/r ) is a concave function for some 1 ≤ r < ∞ and ψ(1) = 1.
Assume that T : C(K) → Y is (ϕ, ψ)-summing with πϕ,ψ (T ) = 1. For each 0 < ε < 1,
choose f1 , ..., fn in C(K) such that
n
X
fk ek
ψ ≤ 1 + ε ∞
k=1
(`ϕ )0
k=1
Notice that this implies that
hyk∗ , T fk i
ϕ
k=1
We choose functionals y1∗ , ..., yn∗ ∈ Y ∗ so that n
X
kyk∗ kY ∗ ek
n
X
and kT fk kY ek = 1.
=1
and
n X k=1
hyk∗ , T fk i = 1.
≥ 0 for each 1 ≤ k ≤ n.
Define a functional µε ∈ C(K) → K by µε (g) =
n X k=1
hyk∗ , T (gfk )i.
Combining the above relations with πϕ,ψ (T ) = 1 yields
X n n
X
|hµε , gi| ≤ kT (gfk )kY ek ≤ gf e k k
ϕ
k=1
k=1
ψ
∞
≤ (1 + ε)kgk∞
for all g ∈ C(K). This shows that µε is a continuous functional on C(K) with kµε k ≤ 1 + ε.
We need further properties of µε . We show that under the assumption that t 7→ ϕ(t1/r )
is a concave function, for every f ∈ C(K) with kf k∞ = 1, we have
1 r kT (f )k Y ≤1− µε (1 − ψ(|f |) . ϕ 1+ε 1+ε
To do this fix f ∈ C(K) with kf k∞ ≤ 1 and consider the continuous functions on K
defined by
gk := |fk | 1 − ψ(|f |) ,
gn+1 := |f |.
By convexity of ψ, we have ψ(cgk ) ≤ ψ(c|fk |) (1 − ψ(|f |)) for all c > 0 and 1 ≤ k ≤ n. Since
n
n
X
X
f e ≤ 1 + ε, ψ(|f |/(1 + ε))
≤ 1, k k k
k=1
ψ ∞
k=1
10
∞
and hence
Pn
k=1
ψ(|fk |/(1 + ε)) ≤ 1. Combining these inequalities, we get
n+1
X
ψ(gk /(1 + ε))
∞
k=1
≤ 1 − ψ(|f |) + ψ(|f |/(1 + ε)) ∞ ≤ 1,
Pn+1 and so k=1 gk ek ψ ∞ ≤ 1 + ε. Thus by πϕ,ψ (T ) = 1, we obtain
n+1
X
kT gk kY ek ≤ 1 + ε,
ϕ
k=1
which gives the following estimate
n+1 X k=1
P Let s > 0 be such that n
k=1
ϕ kT gk kY /(1 + ε) ≤ 1.
P kT gk kY /(1+ε) ek ϕ > s. Then nk=1 ϕ kT gk kY /(1+ε) ≥
1. The above inequality yields s ≤ 1, and so concavity of the function t 7→ ϕ(t1/r ) implies n X k=1
n X r ϕ kT gk kY /(1 + ε) ≥ kT gk kY /(1 + ε) ek ϕ . k=1
In consequence, we obtain
n n X
X r
kT gk kY /(1 + ε) ek ϕ ϕ kT gk kY /(1 + ε) ≤ 1 − ϕ kT f kY /(1 + ε) ≤ 1 − k=1
k=1
≤1−
1 (1 + ε)
n X k=1
hyk∗ , T gk i
r
=1−
r 1 ≤1− µε 1 − ψ(|f |)) . (1 + ε)
r 1 µε (1 − ψ(|f |)) (1 + ε)
Let µ be the weak∗ limit of µε as ε → 0. Clearly µ is a positive functional of norm 1, by
kµε k ≤ 1 + ε and µε (1) = 1. Thus, by the Riesz representation theorem, we may assume R that µ is a Borel probability measure on K such that µ(f ) = K f dµ for all f ∈ C(K). In consequence, it follows from the above inequality that for all f ∈ C(K) with kf k∞ ≤ 1, ϕ kT f kY ) ≤ 1 − (µ(1 − ψ(|f |))r ≤ rµ(ψ(|f |)) Z =r ψ(|f |) dµ, K
and this complete the proof of (ii). (ii) ⇒(i): Assume that we can associate with T : C(K) → Y a Borel probability measure
µ on K and a constant c such that for any f ∈ C(K) with kf k∞ ≤ 1, Z ϕ(kT f kY ) ≤ c ψ(|f |) dµ. K
11
Pn Pn Select f1 , ..., fn ∈ C(K) with k=1 fk ek ψ ∞ ≤ 1. Since k k=1 ψ(|fk |)k∞ ≤ 1, we have
kfk k∞ ≤ 1 for 1 ≤ k ≤ n. Thus using the above inequality, we get n X k=1
ϕ(kT fk kY ) ≤ c
Z X n K k=1
ψ(|fk |) dµ ≤ c.
This shows that T is (ϕ, ψ)-summing with πϕ,ψ (T ) ≤ max{1, c}. Corollary 3.1. Let K be a compact Hausdorff space and Y be a Banach space. Assume that ρ(s, t) := s (ϕ−1 ◦ ψ)(t/s) ∈ Φ, where ϕ, ψ are Orlicz functions such that ϕ ∈ (∆2 )
and ψ is sub-multiplicative in a neighbourhood of zero (i.e., there exist C > 0 such that ψ(st) ≤ Cψ(s)ψ(t) when 0 < s, t ≤ a for some a > 0). If an operator T : C(K) → Y is
(ϕ, ψ)-summing, then there exist a Borel probability measure µ on K and a constant c such that kT f kY ≤ cρ(kf kC(K) , kf kLψ (µ) ),
f ∈ C(K).
Proof. We note that the condition ρ ∈ Φ yields ϕ ≺ ψ, and whence `ψ ,→ `ϕ . By a compactness argument and normalization, we may suppose without loss of generality that ψ
is a sub-multiplicative function on [0, 1] and ψ(1) = 1. Thus, it follows by Theorem 3.1 that there exist a Borel probability measure µ on K and a constant c such that for any f ∈ C(K) with kf k∞ ≤ 1, we obtain ϕ(kT f kY ) ≤ c
Z
K
ψ(|f |) dµ.
Now let 0 6= f ∈ C(K). Combining the above estimate with inequalities kf kLψ (µ) ≤ kf k∞ , R K ψ(|f |/kf kLψ (µ) ) dµ ≤ 1 and sub-multiplicativity of ψ on [0, 1] yields Z kf kLψ (µ) kf kLψ (µ) kT f kY |f | ϕ ≤ λψ ψ dµ ≤ λ ψ , kf k∞ kf k∞ kf kLψ (µ) kf k∞ K
for some constant λ. Clearly this implies the required estimate.
Theorem 3.2. Let K be a compact Hausdorff space and Y be a Banach space. If an Orlicz function ϕ ∈ (∆2 ) and 1 ≤ p < ∞ are such that ρ(s, t) := s ϕ−1 (tp /sp ) ∈ Φ, then (Πp )ρ (C(K), Y ) = Πϕ,p (C(K), Y ).
12
Proof. Assume that T ∈ Πϕ,p (C(K), Y ). Corollary 3.1 with ψ(t) = tp gives that there exist a Borel probability measure µ on K and a positive constant c such that kT f kY ≤ cρ(kf kC(K) , kJf kLψ (µ) ),
f ∈ C(K),
where J : C(K) → Lp (µ) is the inclusion map. Since J is p-summing, T ∈ (Πp )ρ (C(K), Y ).
The proof of the converse is easy. In fact, assume u ∈ (Πp )ρ (C(K), Y ) with (πp )ρ (u) < 1.
Then there is a Banach space Z and a p-summing operator v : C(K) → Z with πp (v) ≤ 1 satisfying
kuf kY ≤ ρ(kf kC(K) , kvf kZ ), f ∈ C(K).
Pn p 1/p Select f1 , ..., fn ∈ C(K) with ≤ 1. As v is p-summing with πp (v) ≤ 1, k=1 |fk | ∞ n X k=1
ϕ(kufk kY ) ≤
n X k=1
kvfk kpZ ≤ 1.
This shows that u ∈ Πϕ,p (C(K), Y ) and πϕ,p (u) ≤ 1. Below we will use Matuszewszka-Orlicz index (cf. [6, 8]). Recall that if a function ρ : R+ → R+ is non-nondecreasing and ρ¯(t) := lim sups→0+ ρ(ts)/ρ(s) , then the upper index β0 (ρ) is given by
β0 (ρ) = lim
t→∞
log ρ¯(t) . log t
We are now ready to prove a variant of Pisier’s factorization theorem for (ϕ, p)-summing operators. We shall make use the Lorentz space. Recall that if (Ω, µ) is the measure space and ϕ : [0, µ(Ω)) → [0, ∞) is a non-decreasing concave function, with ϕ(0) = 0, the r.i.
Lorentz space Λ(ϕ) on (Ω, µ) consists of all (µ-a.e. equivalence classes) whose decreasing
rearrangement f ∗ : [0, µ(Ω)) → [0, ∞] given by f ∗ (t) = inf{s > 0; µ({|f | > s} ≤ t} for all
t > 0 satisfies
kf kψ :=
Z
0
µ(Ω)
f ∗ (t) dψ(t) < ∞.
Theorem 3.3. Let an Orlicz function ϕ ∈ (∆2 ), K be a compact Hausdorff space, and Y be a Banach space. Consider the following statements about the operator T : C(K) → Y : (i) T is (ϕ, 1)-summing. (ii) There exist a Borel probability measure µ on K and a factorization: J
S
T : C(K) −→ Λψ (µ) −→ Y, where J is the inclusion map, and ψ = ϕ−1 . 13
(iii) If `p ,→ `ϕ with 1 < p < ∞, then T is (ϕ, p)-summing. Then the statements (i) and (ii) are equivalent. If we assume additionally that p < β0 (ϕ), then all three statements are equivalent. Proof of Theorem 3.3. (i) ⇒ (ii): The preceding corollary yields that we can find a Borel probability measure µ on K and a constant c such that kT f kY ≤ cρ(kf kC(K) , kf kL1 (µ) ),
f ∈ C(K),
where ρ ∈ Φ is defined by ρ(s, t) = sϕ−1 (t/s) for s, t > 0 and ρ(0, 0) = 0.
We will show that T can be extended to a bounded linear operator S : L∞ (µ) → Y . To
see this fix f ∈ L∞ (µ). Now, applying Lusin’s Theorem, we can find a sequence (gn ) in
C(K) with kgn kC(K) ≤ kf kL∞ (µ) for all n and f = limn→∞ gn µ-a.e. Since ρ(s, t) → 0 as
t → 0, the Lebesgue dominated convergence theorem and the above inequality yield that
(T gn ) is a Cauchy sequence in Y . Thus the limit Tef := limn→∞ T gn defines the required bounded extension of T , which satisfies
kTef kY ≤ c ρ(kf kL∞ (µ) , kf kL1 (µ) ),
f ∈ L∞ (µ).
In particular for f = χA , we have
kTe(χA )kY ≤ c ϕ−1 (µ(A)) = c ψ(µ(A)).
Since C(K) is dense in Λψ (µ), it follows that Te has an unique continuous extension S from
Λψ (µ) into Y , and this completes the proof (i) ⇒(ii).
(ii) ⇒ (i): This follows by the fact that for any finite measure space (Ω, µ) the natural
inclusion map J : L∞ (µ) → Λψ (µ) is (ϕ, 1)-summing (see [2, Proposition 2.1]).
(ii) ⇒ (iii): In view of Theorem 3.1, we need only to show that for all f ∈ C(K) with
kf k∞ ≤ 1,
ϕ(kT f kY ) ≤ c
Z
K
|f |p dµ.
By our hypothesis, it follows that it is enough to prove that for each f ∈ C(K) with kf k∞ ≤ 1, we have
ϕ(kf kΛψ (µ) ) ≤ c1
where c1 is another constant.
14
Z
K
|f |p dµ,
Let f ∈ C(K) with kf k∞ ≤ 1. Then, setting u :=
R
K
|f |p dµ, we get f ∗ (t) ≤ t−1/p u1/p ≤
1 for all 0 ≤ t ≤ 1. Since p < β0 (ϕ), we have α0 (ψ) = α0 (ϕ−1 ) = 1/β0 (ϕ) < 1/p. Thus combining the remarks with the well known properties of indices (see, e.g., [6]), we obtain Z 1 Z 1 dt ∗ f ∗ (t) ψ(t) f (t) dψ(t) ≤ kf kΛψ (µ) = t 0 0 Z 1 Z u dt 1/p dt t−1/p ψ(t) u = ψ(t) + t t u 0 ≺ ψ(u) + (u−1/p ψ(u))u1/p ≺ ψ(u),
whence the required inequality follows. Since the implication (iii) ⇒ (i) is trivial, the proof is complete.
Remark. Notice here that in the case ϕ ∈ (∆2 ) the equivalence (i) ⇔ (ii) has been
established by Montgomery-Smith [14]. In the case when ϕ(t) = t1/q with 1 ≤ p < q < ∞, we obtain the factorization theorem which states that (q, p)-summing operators defined on
C(K) factorizes through the Lorentz space Lq,1 (µ) := Λψ (µ) (with ψ(t) = t1/q for t ≥ 0) proved by Pisier in his remarkable paper [18].
4
Generalized concavity of operators
In this section we continue with the study of (ϕ, ψ)-summing operators. For given normalized Orlicz functions ϕ and ψ we introduce the notion of (ϕ, ψ)-concave operators, which generalizes the classical concept of (q, p)-concave operators (see [3]). We show relationships between (ϕ, ψ)-summing and (ϕ, ψ)-concave operators. We prove that some classical results on (q, p)-concave operators from Banach lattices can be lifted to the class of generalized concave operators. Let X be a Banach lattice, and let ϕ be a normalized Orlicz function. Following the functional calculus (in the case ϕ(t) = tp for all t ≥ 0, see [3, Section 16]), for a given finite
set {x1 , ..., xn } from X, we define
n n nX o X n n )0 . x e := sup a x ; (a ) ∈ B k k k k k (` k=1 ϕ k=1
ϕ
k=1
Here `nϕ , and respectively, (`nϕ )0 is Rn equipped with the norm
n n o X k(xk )nk=1 kϕ := inf λ > 0; ϕ(|xk |/λ) ≤ 1 , k=1
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and respectively, k(xk )nk=1 k(`nϕ )0 := max
n nX k=1
o |xk yk |; k(yk )nk=1 kϕ ≤ 1 . p 1/p k=1 |xk |
Pn
If 1 ≤ p < ∞ and ϕ(t) = tp for all t ≥ 0, then as usual we write Pn of k=1 xk ek ϕ . Thus n X k=1
|xk |p
1/p
instead
n nX o = sup ak xk ; (ak )nk=1 ∈ B`n0 , p
k=1
where 1/p + 1/p0 = 1. In this section we suppose that ϕ and ψ are normalized Orlicz functions. An operator T from a Banach lattice X to a Banach space Y is said to be (ϕ, ψ)-concave if there is a positive constant C such that, for every finite sequence (xk )nk=1 in X,
n n
X
X
kT xk kY ek ≤ C f e
k k
ϕ
k=1
k=1
. ψ X
We write Kϕ,ψ (T ) for the least constant C that works in the inequality above. If 1 ≤ p < ∞ and ψ(t) = tp (resp., ϕ(t) = tq and ψ(t) = tp for all t ≥ 0 with 1 ≤ q ≤ p),
we say that T is (ϕ, p)-concave (resp., (q, p)-concave) and write Kϕ,p (T ) (resp., Kq,p (T )). As usual, the (q, q)-concave operators are called q-concave, and Kq (T ) is used instead of Kq,q (T ). For general examples of (ϕ, ψ)-concave operators we refer to [11].
Combining our results with the techniques applied to (q, p)-concave operators (see [3, pp. 330–332]) we prove that some results on (q, p)-concave operators can be lifted to a general setting. These results show relationships between (ϕ, ψ)-concave and (ϕ, ψ)-summing operators from Banach lattices to Banach spaces. Theorem 4.1. Let X be a Banach lattice and let Y be a Banach space. Assume ψ, ϕ are functions such that ϕ ≺ ψ. An operator T : X → Y is (ϕ, ψ)-concave with Kϕ,ψ (T ) ≤ C if and only if for each compact Hausdorff space K and every positive operator V : C(K) → X, the composition T V : C(K) → Y is (ϕ, ψ)-summing with πϕ,ψ (T V ) ≤ CkV k.
Proof. Let V : C(K) → X be a positive operator. Fix f1 , ..., fn ∈ C(K). Then we have n n n nX o X X n 0 f e (V f )e = sup a f ; a ∈ B ≤ V k k k k k k (`ψ ) . k=1
ψ
k=1
k=1
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ψ
If we assume that T : X → Y is (ϕ, ψ)-concave with Kϕ,ψ (T ) ≤ C, then
X n n
X
kT V (fk )kY ek ≤ C V f e
k k ϕ
k=1
k=1
≤ CkV k
sup
ψ
X
kx∗ kC(K)∗ ≤1
and so T V is (ϕ, ψ)-summing with πϕ,ψ (T V ) ≤ CkV k.
n
X
x∗ (fk )ek
ϕ
k=1
To prove the converse, fix x1 , ..., xn ∈ X. Without loss of generality we may assume
that kxkX = 1, where
n X . x := x e k k ψ
k=1
Let I(x) be the linear span of the order interval [−x, x] in X. Taking [−x, x] as the unit ball of I(x), I(x) is an abstract M -space, and by the well known theorem of Kakutani (see, e.g., [8, p.13]) there is a compact Hausdorff space K so that I(x) is isometrically lattice isomorphic to C(K). Let J : C(K) → X be the associated lattice isomorphism which maps
the unit ball of C(K) onto the order interval [−x, x]. Then we have xk ∈ [−x, x] for all
1 ≤ k ≤ n. Thus for each k ∈ {1, ..., n} there exists fk ∈ C(K) such that kfk kC(K) ≤ 1 and
xk = J(fk ). Since J is an isometrical lattice isomorphism, |xk | = J(|fk |) and
n
X
fk ek
ψ
C(K)
k=1
Consequently,
n
X
−1
= J xk ek ψ
C(K)
k=1
≤ kxkX = 1.
X n n n
X
X
kT xk kY ek = kT Jfk kY ek ≤ πϕ,ψ (T J) f e
k k k=1
ϕ
ϕ
k=1
≤ πϕ,ψ (T J) xkX = πϕ,ψ (T J).
k=1
ψ
C(K)
and this completes the proof.
Corollary 4.1. Let 1 ≤ p < β0 (ϕ) < ∞. An operator from a Banach space to a Banach space is (ϕ, p)-concave if and only if it is (ϕ, 1)-concave.
Proof. It follows from Theorem 3.3 that, when 1 ≤ p < ∞, an operator from C(K)-space
is (ϕ, p)-summing if and only if it is (ϕ, 1)-summing. To conclude it is enough to apply Theorem 4.1.
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As usual we denote by(rn ) the sequence of Rademacher functions on [0, 1] (or, equivalently, any sequence of independent random variables taking the values ±1 with probabilities 1/2).
We conclude with the following corollary. Corollary 4.2. Let ϕ be an Orlicz function such that 2 < β0 (ϕ) < ∞. The following are equivalent statements about an operator T from a Banach lattice X to a Banach space Y. (i) T is (ϕ, 1)-summing. (ii) T is (ϕ, p)-summing for some (and then all ) 1 ≤ p < β0 (ϕ). (iii) There is a positive constant C such that for every finite set {x1 , ..., xn } in X n
X
Z
kT xk kY ek ≤ C
ϕ
k=1
0
n 1 X
k=1
2 1/2
rk (t)xk dt . X
Proof. Since (ϕ, 1)-summing operator T : X → Y is (ϕ, 1)-concave, (i) ⇒ (ii) follows by Corollary 4.1.
(ii) ⇒ (iii). Let {x1 , ..., xn } be a finite subset of elements from X. Then the following
inequality (see [3, Corollary 16.4])
n
X 1/2
A1 |xk |2
X
k=1
≤
Z
0
yields the required estimate
n 1 X
k=1
2 1/2
rk (t)xk dt X
n n
X
X 1/2
2 kT x k e ≤ K (T ) |x |
ϕ,2 k Y k k ϕ
k=1
X
k=1
Z ≤ A−1 K (T ) ϕ,2 1
0
The implication (iii) ⇒ (i) is obvious by Z
0
n 1 X
k=1
2 1/2
≤ rk (t)xk dt X
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n 1 X
k=1
sup
2 1/2
. rk (t)xk dt X
n X
kx∗ kX ∗ ≤1 k=1
|x∗ (xk )| .
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[14] S. J. Montgomery-Smith, The cotype of operators from C(K), Ph.D. Dissertation, Cambridge University 1988. [15] V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep. 1 (1984), 349–516. [16] A. Pietsch, Operator Ideals, North-Holland, 1980. [17] A. Pietsch, Eigenvalues and s-numbers, Cambridge Studies in Advanced Mathematics 13, 1987. [18] G. Pisier, Factorization of operators through Lp,∞ or Lp,1 and non-commutative generalizations, Math. Ann. 276 (1986), 105–136.
Faculty of Mathematics and Computer Science Adam Mickiewicz University and Institute of Mathematics Polish Academy of Science (Pozna´ n branch) Umultowska 87 61-614 Pozna´ n E-mail: [email protected] Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna´ n E-mail: [email protected]
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