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Extension of operators on spaces of vector-valued continuous functions
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Extension of operators on spaces of vector-valued continuous functions Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, University of Zielona G´ora, ul. Szafrana 4A, 65–516 Zielona G´ora, Poland Received 25 October 2013; accepted 6 July 2015 Communicated by B. de Pagter
Abstract Let X be a completely regular Hausdorff space and let Ba stand for the Baire σ -algebra of sets in X . For a Banach space E let Cr c (X, E) be the space of all continuous functions f : X → E such that f (X ) is a relatively compact set in E, and let B(Ba, E) be the space of all totally Ba-measurable functions f : X → E. For a Banach space Y we study the problem of extension of some natural classes of bounded linear operators T : Cr c (X, E) → Y to operators T : B(Ba, E) → Y . c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝
Keywords: Spaces of vector-valued continuous functions; Strict topologies; Vector measures; Integration operators
1. Introduction and terminology Throughout the paper let (E, ∥ · ∥ E ) and (Y, ∥ · ∥Y ) be real Banach spaces, and E ′ and Y ′ denote the Banach duals of E and Y respectively. Let B E , BY and BY ′ denote the closed unit balls in E, Y and Y ′ , respectively. By L(E, Y ) we denote the Banach space of all bounded linear operators U : E → Y , equipped with the uniform norm ∥·∥. Given a locally convex space (Z , ξ ) by (Z , ξ )′ we will denote its topological dual. By σ (Z , K ) we denote the weak topology on Z with respect to a dual pair ⟨Z , K ⟩. Let F be an algebra of subsets of a nonempty set X . By B(F, E) we denote the space of all totally F-measurable functions f : X → E (the uniform limits of sequences of E-valued E-mail address: [email protected]. http://dx.doi.org/10.1016/j.indag.2015.07.003 c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
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F-simple functions), provided with the uniform norm ∥ · ∥ (see [5,6]). In case E = R (the set of real numbers), we will write B(F) instead of B(F, R). Now we recall basic terminology concerning operator-valued measures (see [5,6,2,16,7]). An additive mapping m : F → L(E, Y ) is called an operator-valued measure. We define the (A) on A ∈ F by m (A) := sup ∥Σ m(Ai )(ei )∥Y , where the supremum is taken semivariation m over all finite F-partitions (Ai ) of A and ei ∈ B E for each i. For y ′ ∈ Y ′ let m y ′ : F → E ′ be a finitely additive measure defined by m y ′ (A)(e) = ′ y (m(A)(e)) for A ∈ B, e ∈ E. Then (A) = sup {|m y ′ |(A) : y ′ ∈ BY ′ }, m (X ) < ∞ if and only if where |m y ′ |(A) stands for the variation of m y ′ on A. Moreover, m |m y ′ |(X ) < ∞ for each y ′ ∈ Y ′ (see [2, Theorem 5]). (X ) < ∞, then T ( f ) = X f d m If m : F → L(E, Y ) is an operator measure with m (the immediate integral; see [5], [6, Chap. 1, Sections G–H]) defines a bounded linear operator T : B(F, E) → Y , and any bounded linear operator T : B(F, E) → Y is given this way. (X ), and for y ′ ∈ Y , Moreover, ∥T ∥ = m y ′ (T ( f )) = f d m y ′ for f ∈ B(F, E). X
Now assume that X is a completely regular Hausdorff space. Let Cr c (X, E) stand for the Banach space of all continuous functions f : X → E such that f (X ) is a relatively compact set in E, provided with the uniform norm ∥ · ∥. By Cr c (X, E)′ and Cr c (X, E)′′ we denote the Banach dual and the Banach bidual of Cr c (X, E), respectively. We will write Cb (X ) instead of Cr c (X, R) let f (x) = ∥ f (x)∥ E
for x ∈ X.
Let B (resp. Ba) be the algebra (resp. σ -algebra) of Baire sets in X , which is the algebra (resp. σ -algebra) generated by the class Z of all zero sets of functions of Cb (X ). Let M(X ) stand for the space of all Baire measures on B. Then M(X ) with the norm ∥ν∥ = |ν|(X ) (= the total variation of ν) is a Dedekind complete Banach lattice (see [20]). Due to the Alexandrov representation theorem (see [20, Theorem 5.1]) Cb (X )′ can be identified with M(X ) through the lattice isomorphism M(X ) ∋ ν → ϕν ∈ Cb (X )′ , where ϕν (u) = u d ν for u ∈ Cb (X ), X
and ∥ϕν ∥ = ∥ν∥. By M(X, E ′ ) we denote the set of all finitely additive measures µ : B → E ′ with the following properties: (i) For each e ∈ E, the function µe : B → R defined by µe (A) = µ(A)(e), belongs to M(X ), (ii) |µ|(X ) < ∞, where |µ|(A) stands for the variation of µ on A ∈ B. Due to Katsaras [10, Theorem 2.5] Cr c (X, E)′ can be identified with M(X, E ′ ) through the linear mapping M(X, E ′ ) ∋ µ → Φµ ∈ Cr c (X, E)′ , where Φµ ( f ) = f d µ for f ∈ Cr c (X, E), X
and ∥Φµ ∥ = |µ|(X ).
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Let Mσ (X ) denote the subspace of M(X ) of all σ -additive Baire measures (see [20] for more details). Let Mσ (X, E ′ ) := {µ ∈ M(X, E ′ ) : µe ∈ Mσ (X ) for each e ∈ E}. In the topological measure theory the so-called strict topologies on Cr c (X, E) are of importance (see [11–14,19] for definitions and more details). We will now consider the strict topology βσ (X, E) on Cr c (X, E). The following result will be useful (see [11, Theorem 4.7]). Theorem 1.1. For µ ∈ M(X, E ′ ) the following statements are equivalent: (i) µ ∈ Mσ (X, E ′ ). (ii) Φµ ∈ (Cr c (X, E), βσ (X, E))′ . Katsaras and Liu established a characterization of weakly compact operators from Cr c (X, E) to a locally convex space Y , using the fact that Cr c (X, E) is the space of all continuous functions ˇ f : X → E which have continuous extensions fˆ : β X → E, where β X is the Stone–Cech compactification of X (see [14, Theorem 3]). In this paper we use the device of embedding of the space B(B, E) into Cr c (X, E)′′ , and then we apply the representation theorem for bounded linear operators from B(B, E) to Y (see [3]). In Section 2 we study general properties of bounded linear operators from Cr c (X, E) to a Banach space Y . In Section 3 we derive an extension theorem for representing operator-valued measures for bounded linear operators T : Cr c (X, E) → Y . In Section 4 we study the problem of extension of some class of bounded linear operators T : Cr c (X, E) → Y to operators T : B(Ba, E) → Y . 2. Linear operators on Crc (X, E) It is known that Cr c (X, E) ⊂ B(B, E) (see [15]), and one can embed B(B, E) into Cr c (X, E)′′ by the mapping π : B(B, E) → Cr c (X, E)′′ , where for f ∈ B(B, E), f d µ for µ ∈ M(X, E ′ ). π( f )(Φµ ) = X
Let i Y : Y → Y ′′ stand for the canonical embedding, i.e., i Y (y)(y ′ ) = y ′ (y) for y ∈ Y , y ′ ∈ Y ′ . Moreover, let jY : i Y (Y ) → Y denote the left inverse of i Y , that is, jY ◦ i Y = idY . Now assume that T : Cr c (X, E) → Y is a bounded linear operator. Let T ′ : Y ′ → Cr c (X, E)′ and T ′′ : Cr c (X, E)′′ → Y ′′ stand for the conjugate and biconjugate operators of T , respectively. Let Tˆ := T ′′ ◦ π : B(B, E) → Y ′′ . Then Tˆ is a bounded operator. For A ∈ B let us put m(A)(e) ˆ := Tˆ (1 A ⊗ e) for e ∈ E. Then mˆ : B → L(E, Y ′′ ) will be called a representing measure of T . Then Tˆ ( f ) = Tmˆ ( f ) = f d mˆ for f ∈ B(B, E) X
ˆ ). Moreover, for y ′ ∈ Y ′ , and ∥Tˆ ∥ = m(X (Tˆ ( f ))(y ′ ) = f d mˆ y ′ for f ∈ B(B, E), X
where m y ′ (A)(e) = (m(A)(e))(y ′ ) for A ∈ B, e ∈ E.
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From the general properties of the operator Tˆ it follows that Tˆ (Cr c (X, E)) ⊂ i Y (Y ) and T ( f ) = jY (Tˆ ( f )) for f ∈ Cr c (X, E). For e ∈ E we will write mˆ e (A) := m(A)(e). ˆ The following Riesz representation theorem for operators T : Cr c (X, E) → Y will be of importance (see [19, Theorem 2.1]). Theorem 2.1. Let T : Cr c (X, E) → Y be a bounded linear operator, and mˆ : B → L(E, Y ′′ ) its representing measure. Then the following statements hold: (i) mˆ y ′ ∈ M(X, E ′ ) for each y ′ ∈ Y ′ . (ii) The mapping Y ′ ∋ y ′ → mˆ y ′ ∈ M(X, E ′ ) is (σ (Y ′ , Y ), σ (M(X, E ′ ), Cr c (X, E)))continuous. (iii) For each y ′ ∈ Y ′ , y ′ (T ( f )) = f d mˆ y ′ for f ∈ Cr c (X, E). X
ˆ ). (iv) ∥T ∥ = m(X Conversely, let mˆ : B → L(E, Y ′′ ) be a vector measure satisfying (i) and (ii). Then there exists a unique bounded linear operator T : Cr c (X, E) → Y such that (iii) holds and m(A)(e) ˆ = (T ′′ ◦ π )(1 A ⊗ e) for all A ∈ B, e ∈ E. In consequence, the vector measure mˆ : B → L(E, Y ′′ ) satisfying (i)–(iii) is uniquely determined by a bounded linear operator T : Cr c (X, E) → Y . Definition 2.1. A measure mˆ : B → L(E, Y ′′ ) is said to be a representing measure if it satisfies conditions (i) and (ii) of Theorem 2.1. Let T : Cr c (X, E) → Y be a bounded linear operator, and mˆ : B → L(E, Y ) its representing measure. Assume that for each e ∈ E, {mˆ e (A) : A ∈ B} ⊂ i Y (Y ). Let us put m(A)(e) := jY (m(A)(e)) ˆ
for A ∈ B, e ∈ E.
Then m = jY ◦ mˆ : B → L(E, Y ), and for each f ∈ Cr c (X, E) we have T( f ) = f d( jY ◦ m) ˆ = f d m, X
and for each
y′
∈
X
Y ′,
y ′ (T ( f )) =
we have
f d m y′ .
f d mˆ y ′ = X
X
For e ∈ E let Te (u) := T (u ⊗ e)
for u ∈ Cb (X ).
Then Te : Cb (X ) → Y is a bounded linear operator. Let χ : B(B) → Cb (X )′′ stand for the canonical embedding, i.e., for u ∈ B(B), we have χ (u)(ϕν ) = u d ν for ν ∈ M(X ). X
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Let Tˆe := (Te )′′ ◦ χ : B(B) → Y ′′ . Then Tˆe (Cb (X )) ⊂ i Y (Y ) and
Te (u) = jY (Tˆe (u))
for u ∈ Cb (X ).
The following lemma will be useful. Lemma 2.2. Let T : Cr c (X, E) → Y be a bounded linear operator. Then for e ∈ E and A ∈ B we have T ′′ (π(1 A ⊗ e)) = (Te )′′ (χ (1 A )). Proof. Let y ′ ∈ Y ′ . Then for each u ∈ Cb (X ) we have: ′ ′ ′ (y ◦ Te )(u) = y (T (u ⊗ e)) = (u ⊗ e) d mˆ y = u d mˆ e,y ′ = ϕmˆ e,y ′ (u). X
X
Hence we have (Te )′′ (χ (1 A ))(y ′ ) = χ (1 A )(Te′ (y ′ )) = χ (1 A )(y ′ ◦ Te ) = χ (1 A )(ϕmˆ e,y ′ ) 1 A d mˆ e,y ′ = mˆ e,y ′ (1 A ) = mˆ e (1 A )(y ′ ). = X
On the other hand, for each f ∈ Cr c (X, E) we have f d mˆ y ′ = Φmˆ y ′ ( f ). (y ′ ◦ T )( f ) = X
Hence T ′′ (π(1 A ⊗ e)) = π(1 A ⊗ e)(T ′ (y ′ )) = π(1 A ⊗ e)(y ′ ◦ T ) = π(1 A ⊗ e)(Φmˆ y ′ ) = Φmˆ y ′ (1 A ⊗ e) = (1 A ⊗ e) d mˆ y ′ = mˆ y ′ (A)(e) = mˆ e (1 A )(y ′ ). X
It follows that
T ′′ (π(1
A
⊗ e)) = (Te
)′′ (χ (1
A )),
as desired.
From Lemma 2.2 for A ∈ B and e ∈ E we get mˆ e (A) := Tˆ (1 A ⊗ e) = T ′′ (π(1 A ⊗ e)) = (Te )′′ (χ (1 A )), i.e., mˆ e (A) = Tˆe (1 A ), and it follows that Tˆe (u) = u d mˆ e
for u ∈ B(B).
(2.1)
X
Recall that a measure µ : B → Y is said to be strongly bounded if µ(An ) → 0 whenever (An ) is a pairwise disjoint sequence in B. Let ba(B) stand for the Banach space of all bounded scalar measures on B, provided with the norm ∥ν∥ = |ν|(X ), and let ba(B)′ denote its Banach dual. Now we are ready to prove the following Bartle–Dunford–Schwartz type theorem for operators on Cr c (X, E) (see [4, Theorem 5, p. 153–154]).
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Theorem 2.3. Let T : Cr c (X, E) → Y be a bounded linear operator and mˆ : B → L(E, Y ′′ ) its representing measure. Then for each e ∈ E the following statements are equivalent: (i) mˆ e : B → Y ′′ is strongly bounded. (ii) Te : Cb (X ) → Y is weakly compact. (iii) {mˆ e (A) : A ∈ B} ⊂ i Y (Y ) and {m e (A) : A ∈ B} is relatively weakly compact in Y . Proof. (i)=⇒(ii) In view of the Gantmacher theorem (see [1, Theorem 17.2]) it is enough to show that the conjugate operator (Te )′ : Y ′ → Cb (X )′ is weakly compact, i.e., (Te )′ (BY ′ ) is relatively σ (Cb (X )′ , Cb (X )′′ )-compact. For u ∈ Cb (X ) and y ′ ∈ Y ′ we have ′ ′ ′ ′ (u ⊗ e) d mˆ (Te ) (y )(u) = (y ◦ Te )(u) = y X u d mˆ e,y ′ . u d mˆ e = = y′ X
X
This means that we have to prove that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively weakly compact subset of the Banach space M(X ). Indeed, let (An ) be a pairwise disjoint sequence in B. Since mˆ e : B → Y ′′ is strongly bounded, ∥mˆ e (An )∥Y ′′ → 0, and it follows that sup{|mˆ e,y ′ (An )| : y ′ ∈ BY ′ } → 0, that is, {mˆ e,y ′ : y ′ ∈ BY ′ } is uniformly strongly additive. Moreover, we have sup{|mˆ e,y ′ | (X ) : y ′ ∈ BY ′ } < ∞. This means that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively σ (ba(B), ba(B)′ )compact subset of ba(B) (see [8, Theorem 2]). Since M(X ) is a closed subset of (ba(B), ∥ · ∥), we obtain that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively σ (M(X ), M(X )′ )-compact subset of M(X ) (see [9, Corollary 3.3.3]). (ii)=⇒(iii) Assume that Te : Cb (X ) → Y is weakly compact for each e ∈ E. Then by the Gantmacher theorem, (Te )′′ (Cb (X )′′ ) ⊂ i Y (Y ) and the operator (Te )′′ : Cb (X )′′ → Y ′′ is weakly compact. Then the operator Tˆe = (Te )′′ ◦ χ : B(B) → Y ′′ is weakly compact, and Tˆe (B(B)) ⊂ i Y (Y ). In view of (2.1) mˆ e (A) = Tˆe (1 A ) for A ∈ B. It follows that mˆ e (A) ∈ i Y (Y ) and mˆ e : B → Y ′′ is strongly bounded (see [4, Theorem 1, p. 148]). Hence the measure m e : B → Y is strongly bounded, and this means that {m e (A) : A ∈ B} is relatively weakly compact (see [8, Theorem 7]). (iii)=⇒(i) It follows from [8, Theorem 7]. 3. Extension of representing measures Recall that the strong operator topology (briefly, SOT) on L(E, Y ) is the topology defined by the family { pe : e ∈ E} of seminorms, where pe (U ) = ∥U (e)∥Y for U ∈ L(E, Y ). Then a (X ) < ∞ is said to be countably additive in SOT if for each measure m : Ba → L(E, Y ) with m e ∈ E, ∥m e (An )∥Y → 0 whenever An ↓ ∅, (An ) ⊂ Ba. A weak ∗ operator topology (briefly, W∗ OT) on L(E, Y ) is the topology defined by the family of seminorms { p y ′ : y ′ ∈ Y ′ }, where (X ) < ∞ is p y ′ (U ) = ∥y ′ ◦ U ∥Y ′ for U ∈ L(E, Y ). The measure m : Ba → L(E, Y ) with m said to be countably additive in W∗OT if for each y ′ ∈ Y ′ , ∥m y ′ (An )∥ E ′ → 0 whenever An ↓ ∅, (An ) ⊂ Ba. The following result due to Katsaras and Liu [14, Theorem 7] will play an important role in the proof of Theorem 3.2. (X ) < ∞, {m(A)(e) : Theorem 3.1. Assume that a measure m : B → L(E, Y ) be such that m A ∈ B} is relatively weakly compact in Y for each e ∈ E, and m y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ .
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Then there exists a unique measure m : Ba → L(E, Y ) such that the following statements hold: (a) (b) (c)
m(A) = m(A) for A ∈ B. )=m (X ). m(X For each y ′ ∈ Y ′ and e ∈ E, the measure m e,y ′ : Ba → R is countably additive and regular by zero-sets.
(X ) < ∞ is said to be Following [16] we say that a measure m : Ba → L(E, Y ) with m (An ) → 0 whenever An ↓ ∅, (An ) ⊂ Ba. variationally semi-regular if m Now we can state our main result. Theorem 3.2. Let mˆ : B → L(E, Y ′′ ) be a representing measure such that mˆ y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ and mˆ e : B → Y ′′ is strongly bounded for each e ∈ E. Then m = jY ◦ mˆ : B → )=m (X ) that is countably L(E, Y ) possesses a unique extension m : Ba → L(E, Y ) with m(X additive both in W∗ OT and in SOT. In particular, if Y contains no isomorphic copy of co , then m : Ba → L(E, Y ) is variationally semi-regular. Proof. Let mˆ : B → L(E, Y ′′ ) be a representing measure. Then by Theorem 2.3 {mˆ e (A) : A ∈ B} ⊂ i Y (Y ) and {m(A)(e) : A ∈ B} is relatively weakly compact in Y , where m = jY ◦ mˆ : B → L(E, Y ). Moreover, by Theorem 1.1, m y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ . Hence in view of Theorem 3.1, there exists a unique m : Ba → L(E, Y ) such that m(A) = m(A)
for A ∈ B
and
) = m(X ), m(X
and for each y ′ ∈ Y ′ and e ∈ E, the measure m e,y ′ : Ba → R is countably additive and regular by zero-sets. Hence for each y ′ ∈ Y ′ , |m y ′ |(X ) < ∞ and by [11, Lemma 2.1] we see that |m y ′ | ∈ ca(Ba). Since ∥m y ′ (A)∥ E ′ ≤ |m y ′ |(A) for A ∈ Ba, we obtain that m is countably additive in W∗ OT. Moreover, since m e,y ′ = y ′ ◦ m e is countably additive, in view of the Pettis–Orlicz theorem, m e : Ba → Y is countably additive, and this means that m is countably additive in SOT. Assume now that Y contains no isomorphic copy of c0 . Then by [2, Remark 7, p. 923 and Theorem 5 and 6] the set {|m y ′ | : y ′ ∈ BY ′ } is relatively weakly sequentially compact in ca(Ba), and this implies that {|m y ′ | : y ′ ∈ BY ′ } is uniformly countably additive. This means that m : Ba → L(E, Y ) is variationally semi-regular, as desired. Remark. Some related results to Theorem 3.2 concerning operators on C0 (X, E), the space of E-valued continuous functions on a locally compact space X , were established by Dobrakov (see [7, Theorem 2]). 4. Extension of bounded linear operators on Crc (X, E) The following results will be useful ([17, Proposition 3.1], [18, Proposition 2.1]). Proposition 4.1. Let L : B(Ba) → Y be a bounded linear operator and µ : Ba → Y its representing measure. Then the following statements are equivalent: (i) µ is countably additive. (ii) L(u n ) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X .
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Proposition 4.2. Let T : B(Ba, E) → Y be a bounded linear operator and m : Ba → L(E, Y ) its representing measure. Then the following statements are equivalent: (i) m is variationally semi-regular. (ii) T ( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . Now making use of Theorem 3.2 and Propositions 4.1 and 4.2 we can state the following extension theorem for operators T : Cr c (X, E) → Y . Theorem 4.3. Let T : Cr c (X, E) → Y be a bounded linear operator. Assume that y ′ ◦ T ∈ (Cr c (X, E), βσ (X, E))′ for each y ′ ∈ Y ′ and Te : Cb (X ) → Y is a weakly compact operator for each e ∈ E. Then T possesses a unique bounded linear extension T : B(Ba, E) → Y with ∥T ∥ = ∥T ∥ that satisfies the following conditions: (a) For each y ′ ∈ Y ′ , (y ′ ◦ T )( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . (b) For each e ∈ E, T (u n ⊗ e) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X . (c) If Y contains no isomorphic copy of co , then T ( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . Proof. Let mˆ : B → L(E, Y ′′ ) be the representing measure of T . Hence in view of Theorem 1.1 mˆ y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ , and by Theorem 2.3, mˆ e : B → Y ′′ is strongly bounded, and hence {mˆ e (A) : A ∈ B} ⊂ i Y (Y ), for each e ∈ E. In view of the proof of Theorem 3.2, m = jY ◦ mˆ : B → L(E, Y ) possesses a unique extension m : Ba → L(E, Y ) such that for each y ′ ∈ Y ′ , )=m (X ). |m y ′ | ∈ ca(Ba) and for each e ∈ E, m e : Ba → Y is countably additive, and m(X Define a linear operator T : B(Ba) → Y by setting T ( f ) := f d m for f ∈ B(Ba, E). X
Note that for each y ′ ∈ Y ′ and f ∈ Cr c (X, E) ⊂ B(B, E) ⊂ B(Ba, E) we have y ′ (T ( f )) = f d m y ′ , and y ′ (T ( f )) = f d mˆ y ′ = f d m y′ . X
X
X
Since m y ′ (A) = m y ′ (A) for A ∈ B, we get y ′ (T ( f )) = y ′ (T ( f )). It follows that T ( f ) = )=m (X ) = ∥T ∥. T ( f ) for f ∈ Cr c (X, E). Moreover, we have ∥T ∥ = m(X ′ ′ For each y ∈ Y , we have |y ′ (T ( f ))| = f d m y ′ ≤ f d |m y ′ | for f ∈ B(Ba, E). X
X
Assume that ( f n ) is a uniformly bounded sequence in B(Ba, E) and f n (x) → 0 for x ∈ X . Since f n d| m y ′ | → 0, |m y ′ | ∈ ca(Ba)+ , by the Lebesgue dominated convergence theorem, we have X ′ so y (T ( f n )) → 0, i.e., (a) holds. For each e ∈ E, we have for u ∈ B(Ba), (T )e (u) := T (u ⊗ e) = (u ⊗ e) d m = u d me. X
X
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Since for each e ∈ E, m e : Ba → Y is countably additive, in view of Proposition 4.1, (T )e (u n ) = T (u n ⊗ e) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X , i.e., (b) holds. Assume that a bounded linear operator S : B(Ba) → Y is another extension of T with ∥S∥ = ∥T ∥ that satisfies conditions (a) and (b). Let m o (A)(e) := S(1 A ⊗ e) for A ∈ Ba, e ∈ E. Then S( f ) = f d m o for f ∈ B(Ba, E). X
). Let An ↓ ∅, (An ) ⊂ Ba. Then 1 A (x) → 0 for x ∈ X o (X ) = ∥S∥ = ∥T ∥ = m(X Hence m n and sup ∥1 An ∥ ≤ 1. Hence for e ∈ E, (m o )e (An ) = S(1 An ⊗ e) → 0, so (m o )e : Ba → Y is countably additive. Hence for y ′ ∈ Y ′ , (m o )e,y ′ ∈ ca(Ba). We will show that m o (A) = m(A) for A ∈ B. Indeed, for y ′ ∈ Y ′ and f ∈ Cr c (X, E) we have y ′ (S( f )) = f d (m o ) y ′ and y ′ (T ( f )) = f d m y′ , X
X
where (m o ) y ′ ∈ Mσ (X, E ′ ) and m y ′ ∈ Mσ (X, E ′ ). Hence by Theorem 1.1 for each y ′ ∈ Y ′ , we have (m o ) y ′ (A) = m y ′ (A) for A ∈ B. It follows that m o (A) = m(A) for A ∈ B. Hence in view of Theorem 3.1 we get m o (A) = m(A) for A ∈ Ba, and hence S( f ) = T ( f ) for f ∈ B(Ba, E). The statement (c) follows from Theorem 3.2 and Proposition 4.2. References [1] C.D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, New York, 1985. [2] J. Batt, Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact transformations on the spaces of continuous functions, Rev. Roumaine Math. Pures Appl. 14 (1969) 907–935. [3] J.K. Brooks, P.W. Lewis, Linear operators and vector measures, Trans. Amer. Math. Soc. 192 (1972) 139–162. [4] J. Diestel, J.J. Uhl, Vector Measures, in: Math. Surveys, vol. 15, Amer. Math. Soc., Providence, RI, 1977. [5] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967. [6] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces, John Wiley and Sons Inc., 2000. [7] I. Dobrakov, On representations of linear operators on C0 (T, X ), Czechoslovak Math. J. 21 (96) (1971) 13–30. [8] J. Hoffmann-J¨org¨ensen, Vector measures, Math. Scand. 28 (1971) 5–32. [9] L.V. Kantorovitch, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford-Elmsford, New York, 1982. [10] A. Katsaras, Continuous linear functionals on spaces of vector-valued functions, Bull. Soc. Math. Gr`ece 15 (1974) 13–19. [11] A. Katsaras, Spaces of vector measures, Trans. Amer. Math. Soc. 206 (1975) 313–328. [12] A. Katsaras, Locally convex topologies on spaces of continuous functions, Math. Nachr. 71 (1976) 211–226. [13] A. Katsaras, Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1976) 367–387. [14] A. Katsaras, D.B. Liu, Integral representation of weakly compact operators, Pacific J. Math. 56 (2) (1975) 547–556. [15] S.S. Khurana, Topologies on spaces of vector-valued continuous functions, Trans. Amer. Math. Soc. 241 (1978) 195–211. [16] P.W. Lewis, Some regularity conditions on vector measures with finite semi-variation, Rev. Roumaine Math. Pures Appl. 14 (1970) 375–384. [17] M. Nowak, Vector measures and Mackey topologies, Indag. Math. 23 (2012) 113–122. [18] M. Nowak, Decomposition for weakly compact operators on the space of totally measurable functions, Indag. Math. 23 (2012) 381–378. [19] M. Nowak, Operators on spaces of vector-valued continuous functions with strict topologies, J. Korean Math. Soc. 52 (1) (2015) 177–190. [20] R. Wheeler, A servey of Baire measures and strict topologies, Expo. Math. 2 (1983) 97–190.
ScienceDirect Indagationes Mathematicae (
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Extension of operators on spaces of vector-valued continuous functions Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, University of Zielona G´ora, ul. Szafrana 4A, 65–516 Zielona G´ora, Poland Received 25 October 2013; accepted 6 July 2015 Communicated by B. de Pagter
Abstract Let X be a completely regular Hausdorff space and let Ba stand for the Baire σ -algebra of sets in X . For a Banach space E let Cr c (X, E) be the space of all continuous functions f : X → E such that f (X ) is a relatively compact set in E, and let B(Ba, E) be the space of all totally Ba-measurable functions f : X → E. For a Banach space Y we study the problem of extension of some natural classes of bounded linear operators T : Cr c (X, E) → Y to operators T : B(Ba, E) → Y . c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝
Keywords: Spaces of vector-valued continuous functions; Strict topologies; Vector measures; Integration operators
1. Introduction and terminology Throughout the paper let (E, ∥ · ∥ E ) and (Y, ∥ · ∥Y ) be real Banach spaces, and E ′ and Y ′ denote the Banach duals of E and Y respectively. Let B E , BY and BY ′ denote the closed unit balls in E, Y and Y ′ , respectively. By L(E, Y ) we denote the Banach space of all bounded linear operators U : E → Y , equipped with the uniform norm ∥·∥. Given a locally convex space (Z , ξ ) by (Z , ξ )′ we will denote its topological dual. By σ (Z , K ) we denote the weak topology on Z with respect to a dual pair ⟨Z , K ⟩. Let F be an algebra of subsets of a nonempty set X . By B(F, E) we denote the space of all totally F-measurable functions f : X → E (the uniform limits of sequences of E-valued E-mail address: [email protected]. http://dx.doi.org/10.1016/j.indag.2015.07.003 c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
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F-simple functions), provided with the uniform norm ∥ · ∥ (see [5,6]). In case E = R (the set of real numbers), we will write B(F) instead of B(F, R). Now we recall basic terminology concerning operator-valued measures (see [5,6,2,16,7]). An additive mapping m : F → L(E, Y ) is called an operator-valued measure. We define the (A) on A ∈ F by m (A) := sup ∥Σ m(Ai )(ei )∥Y , where the supremum is taken semivariation m over all finite F-partitions (Ai ) of A and ei ∈ B E for each i. For y ′ ∈ Y ′ let m y ′ : F → E ′ be a finitely additive measure defined by m y ′ (A)(e) = ′ y (m(A)(e)) for A ∈ B, e ∈ E. Then (A) = sup {|m y ′ |(A) : y ′ ∈ BY ′ }, m (X ) < ∞ if and only if where |m y ′ |(A) stands for the variation of m y ′ on A. Moreover, m |m y ′ |(X ) < ∞ for each y ′ ∈ Y ′ (see [2, Theorem 5]). (X ) < ∞, then T ( f ) = X f d m If m : F → L(E, Y ) is an operator measure with m (the immediate integral; see [5], [6, Chap. 1, Sections G–H]) defines a bounded linear operator T : B(F, E) → Y , and any bounded linear operator T : B(F, E) → Y is given this way. (X ), and for y ′ ∈ Y , Moreover, ∥T ∥ = m y ′ (T ( f )) = f d m y ′ for f ∈ B(F, E). X
Now assume that X is a completely regular Hausdorff space. Let Cr c (X, E) stand for the Banach space of all continuous functions f : X → E such that f (X ) is a relatively compact set in E, provided with the uniform norm ∥ · ∥. By Cr c (X, E)′ and Cr c (X, E)′′ we denote the Banach dual and the Banach bidual of Cr c (X, E), respectively. We will write Cb (X ) instead of Cr c (X, R) let f (x) = ∥ f (x)∥ E
for x ∈ X.
Let B (resp. Ba) be the algebra (resp. σ -algebra) of Baire sets in X , which is the algebra (resp. σ -algebra) generated by the class Z of all zero sets of functions of Cb (X ). Let M(X ) stand for the space of all Baire measures on B. Then M(X ) with the norm ∥ν∥ = |ν|(X ) (= the total variation of ν) is a Dedekind complete Banach lattice (see [20]). Due to the Alexandrov representation theorem (see [20, Theorem 5.1]) Cb (X )′ can be identified with M(X ) through the lattice isomorphism M(X ) ∋ ν → ϕν ∈ Cb (X )′ , where ϕν (u) = u d ν for u ∈ Cb (X ), X
and ∥ϕν ∥ = ∥ν∥. By M(X, E ′ ) we denote the set of all finitely additive measures µ : B → E ′ with the following properties: (i) For each e ∈ E, the function µe : B → R defined by µe (A) = µ(A)(e), belongs to M(X ), (ii) |µ|(X ) < ∞, where |µ|(A) stands for the variation of µ on A ∈ B. Due to Katsaras [10, Theorem 2.5] Cr c (X, E)′ can be identified with M(X, E ′ ) through the linear mapping M(X, E ′ ) ∋ µ → Φµ ∈ Cr c (X, E)′ , where Φµ ( f ) = f d µ for f ∈ Cr c (X, E), X
and ∥Φµ ∥ = |µ|(X ).
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Let Mσ (X ) denote the subspace of M(X ) of all σ -additive Baire measures (see [20] for more details). Let Mσ (X, E ′ ) := {µ ∈ M(X, E ′ ) : µe ∈ Mσ (X ) for each e ∈ E}. In the topological measure theory the so-called strict topologies on Cr c (X, E) are of importance (see [11–14,19] for definitions and more details). We will now consider the strict topology βσ (X, E) on Cr c (X, E). The following result will be useful (see [11, Theorem 4.7]). Theorem 1.1. For µ ∈ M(X, E ′ ) the following statements are equivalent: (i) µ ∈ Mσ (X, E ′ ). (ii) Φµ ∈ (Cr c (X, E), βσ (X, E))′ . Katsaras and Liu established a characterization of weakly compact operators from Cr c (X, E) to a locally convex space Y , using the fact that Cr c (X, E) is the space of all continuous functions ˇ f : X → E which have continuous extensions fˆ : β X → E, where β X is the Stone–Cech compactification of X (see [14, Theorem 3]). In this paper we use the device of embedding of the space B(B, E) into Cr c (X, E)′′ , and then we apply the representation theorem for bounded linear operators from B(B, E) to Y (see [3]). In Section 2 we study general properties of bounded linear operators from Cr c (X, E) to a Banach space Y . In Section 3 we derive an extension theorem for representing operator-valued measures for bounded linear operators T : Cr c (X, E) → Y . In Section 4 we study the problem of extension of some class of bounded linear operators T : Cr c (X, E) → Y to operators T : B(Ba, E) → Y . 2. Linear operators on Crc (X, E) It is known that Cr c (X, E) ⊂ B(B, E) (see [15]), and one can embed B(B, E) into Cr c (X, E)′′ by the mapping π : B(B, E) → Cr c (X, E)′′ , where for f ∈ B(B, E), f d µ for µ ∈ M(X, E ′ ). π( f )(Φµ ) = X
Let i Y : Y → Y ′′ stand for the canonical embedding, i.e., i Y (y)(y ′ ) = y ′ (y) for y ∈ Y , y ′ ∈ Y ′ . Moreover, let jY : i Y (Y ) → Y denote the left inverse of i Y , that is, jY ◦ i Y = idY . Now assume that T : Cr c (X, E) → Y is a bounded linear operator. Let T ′ : Y ′ → Cr c (X, E)′ and T ′′ : Cr c (X, E)′′ → Y ′′ stand for the conjugate and biconjugate operators of T , respectively. Let Tˆ := T ′′ ◦ π : B(B, E) → Y ′′ . Then Tˆ is a bounded operator. For A ∈ B let us put m(A)(e) ˆ := Tˆ (1 A ⊗ e) for e ∈ E. Then mˆ : B → L(E, Y ′′ ) will be called a representing measure of T . Then Tˆ ( f ) = Tmˆ ( f ) = f d mˆ for f ∈ B(B, E) X
ˆ ). Moreover, for y ′ ∈ Y ′ , and ∥Tˆ ∥ = m(X (Tˆ ( f ))(y ′ ) = f d mˆ y ′ for f ∈ B(B, E), X
where m y ′ (A)(e) = (m(A)(e))(y ′ ) for A ∈ B, e ∈ E.
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From the general properties of the operator Tˆ it follows that Tˆ (Cr c (X, E)) ⊂ i Y (Y ) and T ( f ) = jY (Tˆ ( f )) for f ∈ Cr c (X, E). For e ∈ E we will write mˆ e (A) := m(A)(e). ˆ The following Riesz representation theorem for operators T : Cr c (X, E) → Y will be of importance (see [19, Theorem 2.1]). Theorem 2.1. Let T : Cr c (X, E) → Y be a bounded linear operator, and mˆ : B → L(E, Y ′′ ) its representing measure. Then the following statements hold: (i) mˆ y ′ ∈ M(X, E ′ ) for each y ′ ∈ Y ′ . (ii) The mapping Y ′ ∋ y ′ → mˆ y ′ ∈ M(X, E ′ ) is (σ (Y ′ , Y ), σ (M(X, E ′ ), Cr c (X, E)))continuous. (iii) For each y ′ ∈ Y ′ , y ′ (T ( f )) = f d mˆ y ′ for f ∈ Cr c (X, E). X
ˆ ). (iv) ∥T ∥ = m(X Conversely, let mˆ : B → L(E, Y ′′ ) be a vector measure satisfying (i) and (ii). Then there exists a unique bounded linear operator T : Cr c (X, E) → Y such that (iii) holds and m(A)(e) ˆ = (T ′′ ◦ π )(1 A ⊗ e) for all A ∈ B, e ∈ E. In consequence, the vector measure mˆ : B → L(E, Y ′′ ) satisfying (i)–(iii) is uniquely determined by a bounded linear operator T : Cr c (X, E) → Y . Definition 2.1. A measure mˆ : B → L(E, Y ′′ ) is said to be a representing measure if it satisfies conditions (i) and (ii) of Theorem 2.1. Let T : Cr c (X, E) → Y be a bounded linear operator, and mˆ : B → L(E, Y ) its representing measure. Assume that for each e ∈ E, {mˆ e (A) : A ∈ B} ⊂ i Y (Y ). Let us put m(A)(e) := jY (m(A)(e)) ˆ
for A ∈ B, e ∈ E.
Then m = jY ◦ mˆ : B → L(E, Y ), and for each f ∈ Cr c (X, E) we have T( f ) = f d( jY ◦ m) ˆ = f d m, X
and for each
y′
∈
X
Y ′,
y ′ (T ( f )) =
we have
f d m y′ .
f d mˆ y ′ = X
X
For e ∈ E let Te (u) := T (u ⊗ e)
for u ∈ Cb (X ).
Then Te : Cb (X ) → Y is a bounded linear operator. Let χ : B(B) → Cb (X )′′ stand for the canonical embedding, i.e., for u ∈ B(B), we have χ (u)(ϕν ) = u d ν for ν ∈ M(X ). X
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Let Tˆe := (Te )′′ ◦ χ : B(B) → Y ′′ . Then Tˆe (Cb (X )) ⊂ i Y (Y ) and
Te (u) = jY (Tˆe (u))
for u ∈ Cb (X ).
The following lemma will be useful. Lemma 2.2. Let T : Cr c (X, E) → Y be a bounded linear operator. Then for e ∈ E and A ∈ B we have T ′′ (π(1 A ⊗ e)) = (Te )′′ (χ (1 A )). Proof. Let y ′ ∈ Y ′ . Then for each u ∈ Cb (X ) we have: ′ ′ ′ (y ◦ Te )(u) = y (T (u ⊗ e)) = (u ⊗ e) d mˆ y = u d mˆ e,y ′ = ϕmˆ e,y ′ (u). X
X
Hence we have (Te )′′ (χ (1 A ))(y ′ ) = χ (1 A )(Te′ (y ′ )) = χ (1 A )(y ′ ◦ Te ) = χ (1 A )(ϕmˆ e,y ′ ) 1 A d mˆ e,y ′ = mˆ e,y ′ (1 A ) = mˆ e (1 A )(y ′ ). = X
On the other hand, for each f ∈ Cr c (X, E) we have f d mˆ y ′ = Φmˆ y ′ ( f ). (y ′ ◦ T )( f ) = X
Hence T ′′ (π(1 A ⊗ e)) = π(1 A ⊗ e)(T ′ (y ′ )) = π(1 A ⊗ e)(y ′ ◦ T ) = π(1 A ⊗ e)(Φmˆ y ′ ) = Φmˆ y ′ (1 A ⊗ e) = (1 A ⊗ e) d mˆ y ′ = mˆ y ′ (A)(e) = mˆ e (1 A )(y ′ ). X
It follows that
T ′′ (π(1
A
⊗ e)) = (Te
)′′ (χ (1
A )),
as desired.
From Lemma 2.2 for A ∈ B and e ∈ E we get mˆ e (A) := Tˆ (1 A ⊗ e) = T ′′ (π(1 A ⊗ e)) = (Te )′′ (χ (1 A )), i.e., mˆ e (A) = Tˆe (1 A ), and it follows that Tˆe (u) = u d mˆ e
for u ∈ B(B).
(2.1)
X
Recall that a measure µ : B → Y is said to be strongly bounded if µ(An ) → 0 whenever (An ) is a pairwise disjoint sequence in B. Let ba(B) stand for the Banach space of all bounded scalar measures on B, provided with the norm ∥ν∥ = |ν|(X ), and let ba(B)′ denote its Banach dual. Now we are ready to prove the following Bartle–Dunford–Schwartz type theorem for operators on Cr c (X, E) (see [4, Theorem 5, p. 153–154]).
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Theorem 2.3. Let T : Cr c (X, E) → Y be a bounded linear operator and mˆ : B → L(E, Y ′′ ) its representing measure. Then for each e ∈ E the following statements are equivalent: (i) mˆ e : B → Y ′′ is strongly bounded. (ii) Te : Cb (X ) → Y is weakly compact. (iii) {mˆ e (A) : A ∈ B} ⊂ i Y (Y ) and {m e (A) : A ∈ B} is relatively weakly compact in Y . Proof. (i)=⇒(ii) In view of the Gantmacher theorem (see [1, Theorem 17.2]) it is enough to show that the conjugate operator (Te )′ : Y ′ → Cb (X )′ is weakly compact, i.e., (Te )′ (BY ′ ) is relatively σ (Cb (X )′ , Cb (X )′′ )-compact. For u ∈ Cb (X ) and y ′ ∈ Y ′ we have ′ ′ ′ ′ (u ⊗ e) d mˆ (Te ) (y )(u) = (y ◦ Te )(u) = y X u d mˆ e,y ′ . u d mˆ e = = y′ X
X
This means that we have to prove that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively weakly compact subset of the Banach space M(X ). Indeed, let (An ) be a pairwise disjoint sequence in B. Since mˆ e : B → Y ′′ is strongly bounded, ∥mˆ e (An )∥Y ′′ → 0, and it follows that sup{|mˆ e,y ′ (An )| : y ′ ∈ BY ′ } → 0, that is, {mˆ e,y ′ : y ′ ∈ BY ′ } is uniformly strongly additive. Moreover, we have sup{|mˆ e,y ′ | (X ) : y ′ ∈ BY ′ } < ∞. This means that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively σ (ba(B), ba(B)′ )compact subset of ba(B) (see [8, Theorem 2]). Since M(X ) is a closed subset of (ba(B), ∥ · ∥), we obtain that {mˆ e,y ′ : y ′ ∈ BY ′ } is a relatively σ (M(X ), M(X )′ )-compact subset of M(X ) (see [9, Corollary 3.3.3]). (ii)=⇒(iii) Assume that Te : Cb (X ) → Y is weakly compact for each e ∈ E. Then by the Gantmacher theorem, (Te )′′ (Cb (X )′′ ) ⊂ i Y (Y ) and the operator (Te )′′ : Cb (X )′′ → Y ′′ is weakly compact. Then the operator Tˆe = (Te )′′ ◦ χ : B(B) → Y ′′ is weakly compact, and Tˆe (B(B)) ⊂ i Y (Y ). In view of (2.1) mˆ e (A) = Tˆe (1 A ) for A ∈ B. It follows that mˆ e (A) ∈ i Y (Y ) and mˆ e : B → Y ′′ is strongly bounded (see [4, Theorem 1, p. 148]). Hence the measure m e : B → Y is strongly bounded, and this means that {m e (A) : A ∈ B} is relatively weakly compact (see [8, Theorem 7]). (iii)=⇒(i) It follows from [8, Theorem 7]. 3. Extension of representing measures Recall that the strong operator topology (briefly, SOT) on L(E, Y ) is the topology defined by the family { pe : e ∈ E} of seminorms, where pe (U ) = ∥U (e)∥Y for U ∈ L(E, Y ). Then a (X ) < ∞ is said to be countably additive in SOT if for each measure m : Ba → L(E, Y ) with m e ∈ E, ∥m e (An )∥Y → 0 whenever An ↓ ∅, (An ) ⊂ Ba. A weak ∗ operator topology (briefly, W∗ OT) on L(E, Y ) is the topology defined by the family of seminorms { p y ′ : y ′ ∈ Y ′ }, where (X ) < ∞ is p y ′ (U ) = ∥y ′ ◦ U ∥Y ′ for U ∈ L(E, Y ). The measure m : Ba → L(E, Y ) with m said to be countably additive in W∗OT if for each y ′ ∈ Y ′ , ∥m y ′ (An )∥ E ′ → 0 whenever An ↓ ∅, (An ) ⊂ Ba. The following result due to Katsaras and Liu [14, Theorem 7] will play an important role in the proof of Theorem 3.2. (X ) < ∞, {m(A)(e) : Theorem 3.1. Assume that a measure m : B → L(E, Y ) be such that m A ∈ B} is relatively weakly compact in Y for each e ∈ E, and m y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ .
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Then there exists a unique measure m : Ba → L(E, Y ) such that the following statements hold: (a) (b) (c)
m(A) = m(A) for A ∈ B. )=m (X ). m(X For each y ′ ∈ Y ′ and e ∈ E, the measure m e,y ′ : Ba → R is countably additive and regular by zero-sets.
(X ) < ∞ is said to be Following [16] we say that a measure m : Ba → L(E, Y ) with m (An ) → 0 whenever An ↓ ∅, (An ) ⊂ Ba. variationally semi-regular if m Now we can state our main result. Theorem 3.2. Let mˆ : B → L(E, Y ′′ ) be a representing measure such that mˆ y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ and mˆ e : B → Y ′′ is strongly bounded for each e ∈ E. Then m = jY ◦ mˆ : B → )=m (X ) that is countably L(E, Y ) possesses a unique extension m : Ba → L(E, Y ) with m(X additive both in W∗ OT and in SOT. In particular, if Y contains no isomorphic copy of co , then m : Ba → L(E, Y ) is variationally semi-regular. Proof. Let mˆ : B → L(E, Y ′′ ) be a representing measure. Then by Theorem 2.3 {mˆ e (A) : A ∈ B} ⊂ i Y (Y ) and {m(A)(e) : A ∈ B} is relatively weakly compact in Y , where m = jY ◦ mˆ : B → L(E, Y ). Moreover, by Theorem 1.1, m y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ . Hence in view of Theorem 3.1, there exists a unique m : Ba → L(E, Y ) such that m(A) = m(A)
for A ∈ B
and
) = m(X ), m(X
and for each y ′ ∈ Y ′ and e ∈ E, the measure m e,y ′ : Ba → R is countably additive and regular by zero-sets. Hence for each y ′ ∈ Y ′ , |m y ′ |(X ) < ∞ and by [11, Lemma 2.1] we see that |m y ′ | ∈ ca(Ba). Since ∥m y ′ (A)∥ E ′ ≤ |m y ′ |(A) for A ∈ Ba, we obtain that m is countably additive in W∗ OT. Moreover, since m e,y ′ = y ′ ◦ m e is countably additive, in view of the Pettis–Orlicz theorem, m e : Ba → Y is countably additive, and this means that m is countably additive in SOT. Assume now that Y contains no isomorphic copy of c0 . Then by [2, Remark 7, p. 923 and Theorem 5 and 6] the set {|m y ′ | : y ′ ∈ BY ′ } is relatively weakly sequentially compact in ca(Ba), and this implies that {|m y ′ | : y ′ ∈ BY ′ } is uniformly countably additive. This means that m : Ba → L(E, Y ) is variationally semi-regular, as desired. Remark. Some related results to Theorem 3.2 concerning operators on C0 (X, E), the space of E-valued continuous functions on a locally compact space X , were established by Dobrakov (see [7, Theorem 2]). 4. Extension of bounded linear operators on Crc (X, E) The following results will be useful ([17, Proposition 3.1], [18, Proposition 2.1]). Proposition 4.1. Let L : B(Ba) → Y be a bounded linear operator and µ : Ba → Y its representing measure. Then the following statements are equivalent: (i) µ is countably additive. (ii) L(u n ) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X .
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Proposition 4.2. Let T : B(Ba, E) → Y be a bounded linear operator and m : Ba → L(E, Y ) its representing measure. Then the following statements are equivalent: (i) m is variationally semi-regular. (ii) T ( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . Now making use of Theorem 3.2 and Propositions 4.1 and 4.2 we can state the following extension theorem for operators T : Cr c (X, E) → Y . Theorem 4.3. Let T : Cr c (X, E) → Y be a bounded linear operator. Assume that y ′ ◦ T ∈ (Cr c (X, E), βσ (X, E))′ for each y ′ ∈ Y ′ and Te : Cb (X ) → Y is a weakly compact operator for each e ∈ E. Then T possesses a unique bounded linear extension T : B(Ba, E) → Y with ∥T ∥ = ∥T ∥ that satisfies the following conditions: (a) For each y ′ ∈ Y ′ , (y ′ ◦ T )( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . (b) For each e ∈ E, T (u n ⊗ e) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X . (c) If Y contains no isomorphic copy of co , then T ( f n ) → 0 whenever ( f n ) is a uniformly bounded sequence in B(Ba, E) such that f n (x) → 0 for x ∈ X . Proof. Let mˆ : B → L(E, Y ′′ ) be the representing measure of T . Hence in view of Theorem 1.1 mˆ y ′ ∈ Mσ (X, E ′ ) for each y ′ ∈ Y ′ , and by Theorem 2.3, mˆ e : B → Y ′′ is strongly bounded, and hence {mˆ e (A) : A ∈ B} ⊂ i Y (Y ), for each e ∈ E. In view of the proof of Theorem 3.2, m = jY ◦ mˆ : B → L(E, Y ) possesses a unique extension m : Ba → L(E, Y ) such that for each y ′ ∈ Y ′ , )=m (X ). |m y ′ | ∈ ca(Ba) and for each e ∈ E, m e : Ba → Y is countably additive, and m(X Define a linear operator T : B(Ba) → Y by setting T ( f ) := f d m for f ∈ B(Ba, E). X
Note that for each y ′ ∈ Y ′ and f ∈ Cr c (X, E) ⊂ B(B, E) ⊂ B(Ba, E) we have y ′ (T ( f )) = f d m y ′ , and y ′ (T ( f )) = f d mˆ y ′ = f d m y′ . X
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Since m y ′ (A) = m y ′ (A) for A ∈ B, we get y ′ (T ( f )) = y ′ (T ( f )). It follows that T ( f ) = )=m (X ) = ∥T ∥. T ( f ) for f ∈ Cr c (X, E). Moreover, we have ∥T ∥ = m(X ′ ′ For each y ∈ Y , we have |y ′ (T ( f ))| = f d m y ′ ≤ f d |m y ′ | for f ∈ B(Ba, E). X
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Assume that ( f n ) is a uniformly bounded sequence in B(Ba, E) and f n (x) → 0 for x ∈ X . Since f n d| m y ′ | → 0, |m y ′ | ∈ ca(Ba)+ , by the Lebesgue dominated convergence theorem, we have X ′ so y (T ( f n )) → 0, i.e., (a) holds. For each e ∈ E, we have for u ∈ B(Ba), (T )e (u) := T (u ⊗ e) = (u ⊗ e) d m = u d me. X
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Since for each e ∈ E, m e : Ba → Y is countably additive, in view of Proposition 4.1, (T )e (u n ) = T (u n ⊗ e) → 0 whenever (u n ) is a uniformly bounded sequence in B(Ba) such that u n (x) → 0 for x ∈ X , i.e., (b) holds. Assume that a bounded linear operator S : B(Ba) → Y is another extension of T with ∥S∥ = ∥T ∥ that satisfies conditions (a) and (b). Let m o (A)(e) := S(1 A ⊗ e) for A ∈ Ba, e ∈ E. Then S( f ) = f d m o for f ∈ B(Ba, E). X
). Let An ↓ ∅, (An ) ⊂ Ba. Then 1 A (x) → 0 for x ∈ X o (X ) = ∥S∥ = ∥T ∥ = m(X Hence m n and sup ∥1 An ∥ ≤ 1. Hence for e ∈ E, (m o )e (An ) = S(1 An ⊗ e) → 0, so (m o )e : Ba → Y is countably additive. Hence for y ′ ∈ Y ′ , (m o )e,y ′ ∈ ca(Ba). We will show that m o (A) = m(A) for A ∈ B. Indeed, for y ′ ∈ Y ′ and f ∈ Cr c (X, E) we have y ′ (S( f )) = f d (m o ) y ′ and y ′ (T ( f )) = f d m y′ , X
X
where (m o ) y ′ ∈ Mσ (X, E ′ ) and m y ′ ∈ Mσ (X, E ′ ). Hence by Theorem 1.1 for each y ′ ∈ Y ′ , we have (m o ) y ′ (A) = m y ′ (A) for A ∈ B. It follows that m o (A) = m(A) for A ∈ B. Hence in view of Theorem 3.1 we get m o (A) = m(A) for A ∈ Ba, and hence S( f ) = T ( f ) for f ∈ B(Ba, E). The statement (c) follows from Theorem 3.2 and Proposition 4.2. References [1] C.D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, New York, 1985. [2] J. Batt, Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact transformations on the spaces of continuous functions, Rev. Roumaine Math. Pures Appl. 14 (1969) 907–935. [3] J.K. Brooks, P.W. Lewis, Linear operators and vector measures, Trans. Amer. Math. Soc. 192 (1972) 139–162. [4] J. Diestel, J.J. Uhl, Vector Measures, in: Math. Surveys, vol. 15, Amer. Math. Soc., Providence, RI, 1977. [5] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967. [6] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces, John Wiley and Sons Inc., 2000. [7] I. Dobrakov, On representations of linear operators on C0 (T, X ), Czechoslovak Math. J. 21 (96) (1971) 13–30. [8] J. Hoffmann-J¨org¨ensen, Vector measures, Math. Scand. 28 (1971) 5–32. [9] L.V. Kantorovitch, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford-Elmsford, New York, 1982. [10] A. Katsaras, Continuous linear functionals on spaces of vector-valued functions, Bull. Soc. Math. Gr`ece 15 (1974) 13–19. [11] A. Katsaras, Spaces of vector measures, Trans. Amer. Math. Soc. 206 (1975) 313–328. [12] A. Katsaras, Locally convex topologies on spaces of continuous functions, Math. Nachr. 71 (1976) 211–226. [13] A. Katsaras, Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1976) 367–387. [14] A. Katsaras, D.B. Liu, Integral representation of weakly compact operators, Pacific J. Math. 56 (2) (1975) 547–556. [15] S.S. Khurana, Topologies on spaces of vector-valued continuous functions, Trans. Amer. Math. Soc. 241 (1978) 195–211. [16] P.W. Lewis, Some regularity conditions on vector measures with finite semi-variation, Rev. Roumaine Math. Pures Appl. 14 (1970) 375–384. [17] M. Nowak, Vector measures and Mackey topologies, Indag. Math. 23 (2012) 113–122. [18] M. Nowak, Decomposition for weakly compact operators on the space of totally measurable functions, Indag. Math. 23 (2012) 381–378. [19] M. Nowak, Operators on spaces of vector-valued continuous functions with strict topologies, J. Korean Math. Soc. 52 (1) (2015) 177–190. [20] R. Wheeler, A servey of Baire measures and strict topologies, Expo. Math. 2 (1983) 97–190.