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Some mapping properties of p-summing adjoint operators
J. Math. Anal. Appl. 303 (2005) 585–590 www.elsevier.com/locate/jmaa
Some mapping properties of p-summing adjoint operators Qingying Bu ∗ , Przemo T. Kranz 1 Department of Mathematics, University of Mississippi, University, MS 38677, USA Received 2 October 2003 Available online 19 November 2004 Submitted by J. Diestel
Abstract Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . We prove that if u∗ , the adjoint operator of u, is p-summing for some p 1, then for any q 2, u takes (almost) ˆ Y , the projective tensor product of unconditionally summable sequences in X into members of q ⊗ q and Y . 2004 Elsevier Inc. All rights reserved.
1. Introduction J.S. Cohen [4] and H. Apiola [1] have investigated the p-summing operators and psumming adjoint operators between Banach spaces. For a Banach space X and 1 < q < ∞, ˆ X, the projective tensor product of q and X, can be expressed as a Banach sequence q ⊗ space q X (see Q. Bu and J. Diestel [3]). By using the q X space together with Khinchin’s inequality, Kahane’s inequality, and Pietsch’s domination theorem, Q. Bu [2] deduced a mapping property of p-summing operators that have a Hilbert space domain. In this pa-
* Corresponding author.
E-mail addresses: [email protected] (Q. Bu), [email protected] (P.T. Kranz). 1 Current address: Department of Mathematics, Naresuan University, Phitsanuloke 65000, Thailand.
0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.08.054
586
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
per, we will apply the q X space in an attempt to deduce some mapping properties of those operators whose adjoints are p-summing for p 1.
2. Preliminary For a Banach space X and 1 p < ∞, let ∞ strong N p p (X) = x¯ = (xn )n ∈ X : xn < ∞ n=1
with the norm
x ¯ strong (X) =
∞
p
1/p xn
p
,
n=1
and let
weak (X) = p
∞ ∗ x (xn )p < ∞, ∀x ∗ ∈ X ∗ x¯ = (xn )n ∈ X :
N
n=1
with the norm
x ¯ weak (X) = sup p
∞ ∗ x (xn )p
1/p
∗
: x ∈ BX∗ .
n=1 strong
(X) are Banach spaces (see [6, pp. 32–36]). For a Banach space Then p (X) and weak p X and 1 < p, p < ∞ such that 1/p + 1/p = 1, let ∞ ∗ ∗ N weak ∗ x (xn ) < ∞, ∀ x ∈ (X ) p X = x¯ = (xn )n ∈ X : n
n n
p
n=1
with the norm
∞ ∗ ∗ x (xn ): x weak ∗ 1 . x ¯ p X = sup n n n (X ) n=1
p
Then p X is a Banach space (see [1,4]). From the definitions we have for 1 < p < ∞, strong
p X ⊆ p
(X) ⊆ weak (X) p
(1)
and · weak (X) · strong (X) · p X . p p
(2)
Moreover, in case dim X = ∞, all the containments in (1) are proper. Theorem (Bu and Diestel [3]). For a Banach space X and 1 < p < ∞, p X is isometˆ X, the projective tensor product of p and X. rically isomorphic to p ⊗
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
587
Let rn (t) denote the Rademacher functions. Recall that
a sequence (xn )n in a Banach space X is called unconditionally summable if the series n xn converges in X uncondi
tionally, or equivalently, the Rademacher series n rn (t)xn converges for every t ∈ [0, 1]; while a sequence (xn )n in X is called almost unconditionally summable if n rn (t)xn converges for (Lebesgue) almost all t ∈ [0, 1] (see [6, p. 230]). Theorem (Hoffmann-Jorgensen [7] or
[6, p. 232]). Let (xn )n be a sequence in a Banach space X. Then the Rademacher series n rn (t)xn converges in X for almost all t ∈ [0, 1]
if and only if the series n rn (·)xn converges in Lp ([0, 1], X) for all 0 < p < ∞. We should mention here Khinchin’s inequality (see [6, p. 10]) and Kahane’s inequality (see [6, p. 211]) as follows. Khinchin’s inequality. For any 0 < p < ∞, there are positive constants Ap , Bp such that for any scalars a1 , . . . , an , we have Ap ·
n
1/2 |ak |
2
k=1
p 1/p 1 n n 1/2 2 ak rk (t) dt Bp · |ak | . 0
k=1
k=1
Kahane’s inequality. If 0 < p, q < ∞, then there is a constant Kp,q > 0 for which q 1/q p 1/p 1 n 1 n rk (t)xk dt Kp,q · rk (t)xk dt 0
k=1
0
k=1
regardless of the choice of a Banach space X and of finitely many vectors x1 , . . . , xn from X. Let Rad(X) denote the space of all almost unconditionally summable sequences in a Banach space X. By Hoffmann-Jorgensen’s theorem and Kahane’s inequality, we can write 2 1 ∞ rn (t)xn dt < ∞ , Rad(X) = (xn )n ∈ X N :
0
n=1
and for each (xn )n ∈ Rad(X), define 2 1/2 1 ∞ = r (t)x . n n dt Rad(X)
(xn )n
0
n=1
Then with this norm Rad(X) is a Banach space (see [6, p. 233]).
588
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
3. Main results Theorem 1. Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . If u∗ , the adjoint operator of u, is p-summing for some p 1, then for any q 2, ˆ Y. u takes almost unconditionally summable sequences in X into members of q ⊗ Proof. By Bu and Diestel’s theorem and Hoffmann-Jorgensen’s theorem, to prove the theorem, we need only to prove that u takes members of Rad(X) into members of q Y . Since u∗ is p-summing, by Pietsch’s domination theorem (see [8] or [6, p. 44]) there is a regular probability measure µ on BY ∗∗ such that for any y ∗ ∈ Y ∗ , 1/p ∗ ∗∗ p ∗ ∗ ∗ ∗∗ y , y dµ(y ) u y πp (u ) · , BY ∗∗
(u∗ )
is the p-summing norm of u∗ . where πp Now for any x1 , . . . , xn ∈ X and any y1∗ , . . . , yn∗ ∈ Y ∗ , by Khinchin’s inequality and Kahane’s inequality, n n
∗ uxk , y ∗ = uxk , θk yk θk = sign uxk , yk∗ k k=1
k=1
1 n n ∗ ∗ = rk (t)xk , rk (t)u θk yk dt k=1
0
k=1
p 1/p 1 n 1 n p 1/p ∗ ∗ rk (t)xk dt · rk (t)θk yk dt u k=1
0
0
k=1
2 1/2 1 n K2,p · rk (t)xk dt k=1
0
1 n p 1/p ∗ ∗ · rk (t)θk yk dt u k=1 0 K2,p · (xk )n1 Rad(X) · πp (u∗ ) 1/p 1 n p ∗ ∗∗ ∗∗ rk (t)θk yk , y dµ(y ) dt · 0
BY ∗∗
k=1
= K2,p · πp (u∗ ) · (xk )n1 Rad(X) p 1 n 1/p ∗ ∗∗ ∗∗ · rk (t) θk yk , y dt dµ(y ) BY ∗∗
0
k=1
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
589
K2,p · πp (u∗ ) · (xk )n1 Rad(X) · Bp n 1/p p/2
∗ ∗∗ 2 ∗∗ θ k yk , y · dµ(y ) BY ∗∗
k=1
n Bp · K2,p · πp (u∗ ) · (xk )n1 Rad(X) · θk yk∗ 1 weak (Y ∗ ) 2 ∗ n ∗ n Bp · K2,p · πp (u ) · (xk )1 Rad(X) · yk 1 weak (Y ∗ ) . 2
This shows that for each (xn )n ∈ Rad(X) and each (yn∗ )n ∈ weak (Y ∗ ), 2 ∞
uxn , y ∗ < ∞. n
n=1 weak (Y ∗ ) for q 2. So we have for each (x ) ∈ Rad(X) and each ∗ Note that weak n n q (Y ) ⊆ 2 ∗ (yn∗ )n ∈ weak q (Y ),
∞
uxn , y ∗ < ∞. n
n=1
That is, for each (xn )n ∈ Rad(X), (uxn )n ∈ q Y .
2
Corollary 2. Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . If u∗ is p-summing for some p 1, then for any q 2, u takes unconditionally ˆ Y. summable sequences in X into members of q ⊗
Recall that (see [5]) a series n xn converges unconditionally in a Banach space X if ˇ X, the injective tensor product of 1 and only if the series n en ⊗ xn converges in 1 ⊗ and X. This yields the following Corollary 3. Let X and Y be Banach spaces and u be a continuous linear operator from ˇX X to Y . If u∗ is p-summing for some p 1, then for any q 2, u takes members of 1 ⊗ ˆ into members of q ⊗ Y . Now let us consider the inverse of Theorem 1. If u takes almost unconditionally sumˆ Y for any q 2, is u∗ p-summing for some mable sequences in X into members of q ⊗ p 1? The answer is no in general. For example, it is known from [5] that if X = L1 [0, 1] ˆ X. Thus the identity operator on L1 [0, 1] takes almost unconditionally then Rad(X) = 2 ⊗ ˆ L1 [0, 1]. But the identity operator summable sequences in L1 [0, 1] into members of 2 ⊗ on L∞ [0, 1] is not p-summing for any p 1. However, the answer is yes in case the domain space has type 2. That is the following Theorem 4. Let X and Y be Banach spaces such that X has type 2, and let u be a continuous linear operator from X to Y . Then u takes almost unconditionally summable sequences ˆ Y if and only if u∗ is 2-summing. in X into members of 2 ⊗
590
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
Proof. By Theorem 1 we need only to show that if u takes almost unconditionally sumˆ Y then u∗ is 2-summing. It is well known that mable sequences in X into members of 2 ⊗ an operator from a Banach space E to a Banach space F is p-summing (1 p < ∞) if and strong (E) into members of p (F ). So to prove the theorem, only if it takes members of weak p we need only to show that if u takes members of Rad(X) into members of 2 Y , then u∗ strong takes members of weak (Y ∗ ) into members of 2 (X ∗ ). 2 strong weak ∗ ∗ Let (xn )n ∈ 2 (X) and (yn )n ∈ 2 (Y ). Since X has type 2, it follows from [6, Proposition 12.4, p. 233] that (xn )n ∈ Rad(X) and so (uxn )n ∈ 2 Y . Thus ∞
uxn , y ∗ < ∞. n
n=1
That is, ∞
xn , u∗ y ∗ < ∞. n
n=1 strong
Since (xn )n is arbitrary in 2
(X), (u∗ yn∗ )n ∈ 2
strong
(X ∗ ).
2
References [1] H. Apiola, Duality between spaces of p-summable sequences, (p, q)-summing operators and characterization of nuclearity, Math. Ann. 219 (1976) 53–64. [2] Q. Bu, Some mapping properties of p-summing operators with Hilbertian domain, Contemp. Math. 328 (2003) 145–149. ˆ X, 1 < p < ∞, [3] Q. Bu, J. Diestel, Observations about the projective tensor product of Banach spaces, I—p ⊗ Quaestiones Math. 24 (2001) 519–533. [4] J.S. Cohen, Absolutely p-summing, p-nuclear operators, and their conjugates, Math. Ann. 201 (1973) 177– 200. [5] J. Diestel, J. Fourie, J. Swart, A theorem of Littlewood, Orlicz, and Grothendieck about sums in L1 (0, 1), J. Math. Anal. Appl. 251 (2000) 376–394. [6] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. [7] J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974) 159–186. [8] A. Pietsch, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1967) 333–353.
Some mapping properties of p-summing adjoint operators Qingying Bu ∗ , Przemo T. Kranz 1 Department of Mathematics, University of Mississippi, University, MS 38677, USA Received 2 October 2003 Available online 19 November 2004 Submitted by J. Diestel
Abstract Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . We prove that if u∗ , the adjoint operator of u, is p-summing for some p 1, then for any q 2, u takes (almost) ˆ Y , the projective tensor product of unconditionally summable sequences in X into members of q ⊗ q and Y . 2004 Elsevier Inc. All rights reserved.
1. Introduction J.S. Cohen [4] and H. Apiola [1] have investigated the p-summing operators and psumming adjoint operators between Banach spaces. For a Banach space X and 1 < q < ∞, ˆ X, the projective tensor product of q and X, can be expressed as a Banach sequence q ⊗ space q X (see Q. Bu and J. Diestel [3]). By using the q X space together with Khinchin’s inequality, Kahane’s inequality, and Pietsch’s domination theorem, Q. Bu [2] deduced a mapping property of p-summing operators that have a Hilbert space domain. In this pa-
* Corresponding author.
E-mail addresses: [email protected] (Q. Bu), [email protected] (P.T. Kranz). 1 Current address: Department of Mathematics, Naresuan University, Phitsanuloke 65000, Thailand.
0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.08.054
586
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
per, we will apply the q X space in an attempt to deduce some mapping properties of those operators whose adjoints are p-summing for p 1.
2. Preliminary For a Banach space X and 1 p < ∞, let ∞ strong N p p (X) = x¯ = (xn )n ∈ X : xn < ∞ n=1
with the norm
x ¯ strong (X) =
∞
p
1/p xn
p
,
n=1
and let
weak (X) = p
∞ ∗ x (xn )p < ∞, ∀x ∗ ∈ X ∗ x¯ = (xn )n ∈ X :
N
n=1
with the norm
x ¯ weak (X) = sup p
∞ ∗ x (xn )p
1/p
∗
: x ∈ BX∗ .
n=1 strong
(X) are Banach spaces (see [6, pp. 32–36]). For a Banach space Then p (X) and weak p X and 1 < p, p < ∞ such that 1/p + 1/p = 1, let ∞ ∗ ∗ N weak ∗ x (xn ) < ∞, ∀ x ∈ (X ) p X = x¯ = (xn )n ∈ X : n
n n
p
n=1
with the norm
∞ ∗ ∗ x (xn ): x weak ∗ 1 . x ¯ p X = sup n n n (X ) n=1
p
Then p X is a Banach space (see [1,4]). From the definitions we have for 1 < p < ∞, strong
p X ⊆ p
(X) ⊆ weak (X) p
(1)
and · weak (X) · strong (X) · p X . p p
(2)
Moreover, in case dim X = ∞, all the containments in (1) are proper. Theorem (Bu and Diestel [3]). For a Banach space X and 1 < p < ∞, p X is isometˆ X, the projective tensor product of p and X. rically isomorphic to p ⊗
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
587
Let rn (t) denote the Rademacher functions. Recall that
a sequence (xn )n in a Banach space X is called unconditionally summable if the series n xn converges in X uncondi
tionally, or equivalently, the Rademacher series n rn (t)xn converges for every t ∈ [0, 1]; while a sequence (xn )n in X is called almost unconditionally summable if n rn (t)xn converges for (Lebesgue) almost all t ∈ [0, 1] (see [6, p. 230]). Theorem (Hoffmann-Jorgensen [7] or
[6, p. 232]). Let (xn )n be a sequence in a Banach space X. Then the Rademacher series n rn (t)xn converges in X for almost all t ∈ [0, 1]
if and only if the series n rn (·)xn converges in Lp ([0, 1], X) for all 0 < p < ∞. We should mention here Khinchin’s inequality (see [6, p. 10]) and Kahane’s inequality (see [6, p. 211]) as follows. Khinchin’s inequality. For any 0 < p < ∞, there are positive constants Ap , Bp such that for any scalars a1 , . . . , an , we have Ap ·
n
1/2 |ak |
2
k=1
p 1/p 1 n n 1/2 2 ak rk (t) dt Bp · |ak | . 0
k=1
k=1
Kahane’s inequality. If 0 < p, q < ∞, then there is a constant Kp,q > 0 for which q 1/q p 1/p 1 n 1 n rk (t)xk dt Kp,q · rk (t)xk dt 0
k=1
0
k=1
regardless of the choice of a Banach space X and of finitely many vectors x1 , . . . , xn from X. Let Rad(X) denote the space of all almost unconditionally summable sequences in a Banach space X. By Hoffmann-Jorgensen’s theorem and Kahane’s inequality, we can write 2 1 ∞ rn (t)xn dt < ∞ , Rad(X) = (xn )n ∈ X N :
0
n=1
and for each (xn )n ∈ Rad(X), define 2 1/2 1 ∞ = r (t)x . n n dt Rad(X)
(xn )n
0
n=1
Then with this norm Rad(X) is a Banach space (see [6, p. 233]).
588
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
3. Main results Theorem 1. Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . If u∗ , the adjoint operator of u, is p-summing for some p 1, then for any q 2, ˆ Y. u takes almost unconditionally summable sequences in X into members of q ⊗ Proof. By Bu and Diestel’s theorem and Hoffmann-Jorgensen’s theorem, to prove the theorem, we need only to prove that u takes members of Rad(X) into members of q Y . Since u∗ is p-summing, by Pietsch’s domination theorem (see [8] or [6, p. 44]) there is a regular probability measure µ on BY ∗∗ such that for any y ∗ ∈ Y ∗ , 1/p ∗ ∗∗ p ∗ ∗ ∗ ∗∗ y , y dµ(y ) u y πp (u ) · , BY ∗∗
(u∗ )
is the p-summing norm of u∗ . where πp Now for any x1 , . . . , xn ∈ X and any y1∗ , . . . , yn∗ ∈ Y ∗ , by Khinchin’s inequality and Kahane’s inequality, n n
∗ uxk , y ∗ = uxk , θk yk θk = sign uxk , yk∗ k k=1
k=1
1 n n ∗ ∗ = rk (t)xk , rk (t)u θk yk dt k=1
0
k=1
p 1/p 1 n 1 n p 1/p ∗ ∗ rk (t)xk dt · rk (t)θk yk dt u k=1
0
0
k=1
2 1/2 1 n K2,p · rk (t)xk dt k=1
0
1 n p 1/p ∗ ∗ · rk (t)θk yk dt u k=1 0 K2,p · (xk )n1 Rad(X) · πp (u∗ ) 1/p 1 n p ∗ ∗∗ ∗∗ rk (t)θk yk , y dµ(y ) dt · 0
BY ∗∗
k=1
= K2,p · πp (u∗ ) · (xk )n1 Rad(X) p 1 n 1/p ∗ ∗∗ ∗∗ · rk (t) θk yk , y dt dµ(y ) BY ∗∗
0
k=1
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
589
K2,p · πp (u∗ ) · (xk )n1 Rad(X) · Bp n 1/p p/2
∗ ∗∗ 2 ∗∗ θ k yk , y · dµ(y ) BY ∗∗
k=1
n Bp · K2,p · πp (u∗ ) · (xk )n1 Rad(X) · θk yk∗ 1 weak (Y ∗ ) 2 ∗ n ∗ n Bp · K2,p · πp (u ) · (xk )1 Rad(X) · yk 1 weak (Y ∗ ) . 2
This shows that for each (xn )n ∈ Rad(X) and each (yn∗ )n ∈ weak (Y ∗ ), 2 ∞
uxn , y ∗ < ∞. n
n=1 weak (Y ∗ ) for q 2. So we have for each (x ) ∈ Rad(X) and each ∗ Note that weak n n q (Y ) ⊆ 2 ∗ (yn∗ )n ∈ weak q (Y ),
∞
uxn , y ∗ < ∞. n
n=1
That is, for each (xn )n ∈ Rad(X), (uxn )n ∈ q Y .
2
Corollary 2. Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . If u∗ is p-summing for some p 1, then for any q 2, u takes unconditionally ˆ Y. summable sequences in X into members of q ⊗
Recall that (see [5]) a series n xn converges unconditionally in a Banach space X if ˇ X, the injective tensor product of 1 and only if the series n en ⊗ xn converges in 1 ⊗ and X. This yields the following Corollary 3. Let X and Y be Banach spaces and u be a continuous linear operator from ˇX X to Y . If u∗ is p-summing for some p 1, then for any q 2, u takes members of 1 ⊗ ˆ into members of q ⊗ Y . Now let us consider the inverse of Theorem 1. If u takes almost unconditionally sumˆ Y for any q 2, is u∗ p-summing for some mable sequences in X into members of q ⊗ p 1? The answer is no in general. For example, it is known from [5] that if X = L1 [0, 1] ˆ X. Thus the identity operator on L1 [0, 1] takes almost unconditionally then Rad(X) = 2 ⊗ ˆ L1 [0, 1]. But the identity operator summable sequences in L1 [0, 1] into members of 2 ⊗ on L∞ [0, 1] is not p-summing for any p 1. However, the answer is yes in case the domain space has type 2. That is the following Theorem 4. Let X and Y be Banach spaces such that X has type 2, and let u be a continuous linear operator from X to Y . Then u takes almost unconditionally summable sequences ˆ Y if and only if u∗ is 2-summing. in X into members of 2 ⊗
590
Q. Bu, P.T. Kranz / J. Math. Anal. Appl. 303 (2005) 585–590
Proof. By Theorem 1 we need only to show that if u takes almost unconditionally sumˆ Y then u∗ is 2-summing. It is well known that mable sequences in X into members of 2 ⊗ an operator from a Banach space E to a Banach space F is p-summing (1 p < ∞) if and strong (E) into members of p (F ). So to prove the theorem, only if it takes members of weak p we need only to show that if u takes members of Rad(X) into members of 2 Y , then u∗ strong takes members of weak (Y ∗ ) into members of 2 (X ∗ ). 2 strong weak ∗ ∗ Let (xn )n ∈ 2 (X) and (yn )n ∈ 2 (Y ). Since X has type 2, it follows from [6, Proposition 12.4, p. 233] that (xn )n ∈ Rad(X) and so (uxn )n ∈ 2 Y . Thus ∞
uxn , y ∗ < ∞. n
n=1
That is, ∞
xn , u∗ y ∗ < ∞. n
n=1 strong
Since (xn )n is arbitrary in 2
(X), (u∗ yn∗ )n ∈ 2
strong
(X ∗ ).
2
References [1] H. Apiola, Duality between spaces of p-summable sequences, (p, q)-summing operators and characterization of nuclearity, Math. Ann. 219 (1976) 53–64. [2] Q. Bu, Some mapping properties of p-summing operators with Hilbertian domain, Contemp. Math. 328 (2003) 145–149. ˆ X, 1 < p < ∞, [3] Q. Bu, J. Diestel, Observations about the projective tensor product of Banach spaces, I—p ⊗ Quaestiones Math. 24 (2001) 519–533. [4] J.S. Cohen, Absolutely p-summing, p-nuclear operators, and their conjugates, Math. Ann. 201 (1973) 177– 200. [5] J. Diestel, J. Fourie, J. Swart, A theorem of Littlewood, Orlicz, and Grothendieck about sums in L1 (0, 1), J. Math. Anal. Appl. 251 (2000) 376–394. [6] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. [7] J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974) 159–186. [8] A. Pietsch, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1967) 333–353.