Pietsch integral operators defined on injective tensor products of spaces and applications

AbstractFor X and Y Banach spaces, let X⊗εY, be the injective tensor product. If Z is also a Banach space and U ∊ L(X⊗εY,Z) we consider the operatorWe prove that if U ∊ PI(X⊗εY, Z), then U# ∊ I(X, PI(Y,Z)). This result is then applied in the case of operators defined on the space of all X-valued continuous functions on the compact Hausdorff space T. We obtain also an affirmative answer to a problem of J. Diestel and J. J. Uhl about the RNP property for the space of all nuclear operators; namely if X* and Y have the RNP and Y can be complemented in its bidual, then N(X, Y) has the RNP.

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On Additive Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-050-7 ◽

1971 ◽

Vol 23 (3) ◽

pp. 468-480 ◽

Cited By ~ 8

Author(s):

N. A. Friedman ◽

A. E. Tong

Keyword(s):

Banach Space ◽

Compact Hausdorff Space ◽

Weak Topology ◽

Hausdorff Space ◽

Continuous Functions ◽

Additive Functional ◽

Representation Theorems ◽

Image Position ◽

Vector Valued ◽

Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

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Characterization of some classes of operators on spaces of vector-valued continuous functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062678 ◽

1985 ◽

Vol 97 (1) ◽

pp. 137-146 ◽

Cited By ~ 23

Author(s):

Fernando Bombal ◽

Pilar Cembranos

Keyword(s):

Banach Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Supremum Norm ◽

Representing Measure ◽

Bounded Linear ◽

Second Dual ◽

Image Position ◽

Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

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Factoring absolutely summing operators through Hilbert-Schmidt operators

Glasgow Mathematical Journal ◽

10.1017/s0017089500007643 ◽

1989 ◽

Vol 31 (2) ◽

pp. 131-135 ◽

Cited By ~ 2

Author(s):

Hans Jarchow

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Hausdorff Space ◽

Bounded Operator ◽

Continuous Functions ◽

Space Of Continuous Functions ◽

Reflexive Banach Spaces ◽

Summing Operator ◽

Schmidt Operator ◽

Second Dual

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

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Averaging distances in certain Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700030616 ◽

1997 ◽

Vol 55 (1) ◽

pp. 147-160 ◽

Cited By ~ 5

Author(s):

Reinhard Wolf

Keyword(s):

Banach Space ◽

Hausdorff Space ◽

Finite Dimension ◽

Closed Interval ◽

Continuous Functions ◽

Real Numbers ◽

Empty Interval ◽

Positive Real ◽

Complete Discussion ◽

Image Position

Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such thatIt follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

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The centraliser of the injective tensor product

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700029841 ◽

1991 ◽

Vol 44 (3) ◽

pp. 357-365 ◽

Cited By ~ 1

Author(s):

Wend Werner

Keyword(s):

Tensor Product ◽

Banach Spaces ◽

Product Formula ◽

Injective Tensor Product ◽

Image Position

The aim of this note is to obtain an intrinsic product formula for the centraliser of the injective tensor product of a couple of Banach spaces, Z(). More precisely, we are going to prove thatHere, the spaces and depend only on X and Y, respectively, and Xk denotes the topological k-product.A Counterexaple used to demonstrate that the k-product cannot beavoided serves as an answer to a question posed by W. Rueß and D. Werner concerning the behaviour of M-ideals on Y.

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On l1 subspaces of Orlicz vector-valued function spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066445 ◽

1987 ◽

Vol 101 (1) ◽

pp. 107-112 ◽

Cited By ~ 7

Author(s):

Fernando Bombal

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Hausdorff Space ◽

Continuous Functions ◽

Function Spaces ◽

Complete Characterization ◽

Complemented Subspace ◽

Vector Valued Function ◽

Vector Valued ◽

General Banach Space

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

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Characterizations of continuous operators on $$C_b(X)$$ with the strict topology

Annals of Functional Analysis ◽

10.1007/s43034-021-00112-1 ◽

2021 ◽

Vol 12 (2) ◽

Author(s):

Marian Nowak ◽

Juliusz Stochmal

Keyword(s):

Banach Space ◽

Hausdorff Space ◽

Compact Operators ◽

Continuous Functions ◽

Linear Operators ◽

Strict Topology ◽

Completely Regular ◽

Absolutely Summing Operators ◽

Nuclear Operators ◽

Nikodym Property

AbstractLet X be a completely regular Hausdorff space and $$C_b(X)$$ C b ( X ) be the space of all bounded continuous functions on X, equipped with the strict topology $$\beta $$ β . We study some important classes of $$(\beta ,\Vert \cdot \Vert _E)$$ ( β , ‖ · ‖ E ) -continuous linear operators from $$C_b(X)$$ C b ( X ) to a Banach space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) : $$\beta $$ β -absolutely summing operators, compact operators and $$\beta $$ β -nuclear operators. We characterize compact operators and $$\beta $$ β -nuclear operators in terms of their representing measures. It is shown that dominated operators and $$\beta $$ β -absolutely summing operators $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E coincide and if, in particular, E has the Radon–Nikodym property, then $$\beta $$ β -absolutely summing operators and $$\beta $$ β -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.

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Nuclear operators on Cb(X, E) and the strict topology

Mathematica Slovaca ◽

10.1515/ms-2017-0086 ◽

2018 ◽

Vol 68 (1) ◽

pp. 135-146

Author(s):

Marian Nowak ◽

Juliusz Stochmal

Keyword(s):

Banach Spaces ◽

Hausdorff Space ◽

Continuous Functions ◽

Strict Topology ◽

Completely Regular ◽

Borel Measures ◽

Nuclear Operators ◽

Bounded Continuous Functions

AbstractLetXbe a completely regular Hausdorff space,EandFbe Banach spaces. LetCb(X,E) be the space of allE-valued bounded, continuous functions onX, equipped with the natural strict topologyβ. We study nuclear operatorsT:Cb(X,E) →Fin terms of their representing operator-valued Borel measures.

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Integral operators on the section space of a Banach bundle

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171293000560 ◽

1993 ◽

Vol 16 (3) ◽

pp. 449-458

Author(s):

J. W. kitchen ◽

D. A. Robbins

Keyword(s):

Banach Space ◽

On Banach spaces of vector valued continuous functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020852 ◽

1983 ◽

Vol 28 (2) ◽

pp. 175-186 ◽

Cited By ~ 7

Author(s):

Pilar Cembranos

Keyword(s):

Banach Space ◽

Large Class ◽

Banach Spaces ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Supremum Norm ◽

Vector Valued

Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm.Recently, Talagrand [Israel J. Math.44 (1983), 317–321] constructed a Banach space E having the Dunford-Pettis property such that C([0, 1], E) fails to have the Dunford-Pettis property. So he answered negatively a question which was posed some years ago.We prove in this paper that for a large class of compacts K (the scattered compacts), C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonné property, or property V if and only if E has the same property.Also some properties of the operators defined on C(K, E) are studied.

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