scholarly journals Positive p-summing operators, vector measures and tensor products

1988 ◽  
Vol 31 (2) ◽  
pp. 179-184 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Lattice ◽  
Tensor Products ◽  
Boundary Values ◽  
Vector Measures ◽  
Absolutely Summing Operators

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals Some stability properties of c0-saturated spaces

1995 ◽  
Vol 118 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Tensor Products ◽  
General Version ◽  
Direct Sums ◽  
Stability Properties ◽  
Infinite Dimensional ◽  
The Stability

AbstractA Banach space is c0-saturated if all of its closed infinite-dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0-saturated spaces are c0-saturated; (2) C(K, E) is c0-saturated if both C(K) and E are; (3) the tensor product is c0-saturated, where JH is the James Hagler space

scholarly journals Tensor products of l2-valued measures

1976 ◽  
Vol 21 (2) ◽  
pp. 241-246 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Tensor Products ◽  
The Other ◽  
Product Topology ◽  
Arbitrary Vector ◽  
Vector Measures ◽  
Projective Tensor

Duchon (1967, 1969) and Duchon and Kluvánek (1967) considered the problems of the existence of countably additive tensor products for vector measures. Duchon and Kluvánek (1967) showed that a countably additive product with respect to the inductive tensor topology always exists while Kluvánek (1970) presented an example which showed that this was not the case for the projective tensor topology. Kluvánek considered yet another tensor product topology which is stronger than the inductive topology but weaker than the projective tensor topology and showed that a countably additive product for two vector measures always exists for this particular tensor topology, see Kluvánek (1973). He has conjectured that this topology is the strongest tensor topology (given by a cross norm) which always admits products for any two arbitrary vector measures. In this note we use an example of Kluvánek (1974) to show that this conjecture is indeed true when one of the factors in the tensor product is l2 and the other factor is metrizable. The construction used also clarifies a conjecture made by Swartz (to appear) concerning products of Hilbert space valued measures.

scholarly journals Lipschitz (q, p, E)-summing operators on injective Lipschitz tensor products of spaces

2020 ◽  
Vol 6 (1) ◽  
pp. 127-142
Author(s):  
Abdelhamid Tallab
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Product Space ◽  
Tensor Products ◽  
Injective Tensor Product ◽  
Mixing Operators ◽  
Tensor Product Space ◽  
Special Case

AbstractIn this paper, we introduce the notion of (q, p)-mixing operators from the injective tensor product space E ̂⊗∈F into a Banach space G which we call (q, p, F)-mixing. In particular, we extend the notion of (q, p, E)-summing operators which is a special case of (q, p, F)-mixing operators to Lipschitz case by studying their properties and showing some results for this notion.

scholarly journals Properties (V) and (wV) in Projective Tensor Products

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ioana Ghenciu
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Weakly Compact ◽  
Projective Tensor Product ◽  
Projective Tensor

We give sufficient conditions for a subset of K(X,Y*)=L(X,Y*) to be relatively weakly compact. A Banach space X has property (V) (resp., (wV)) if every V-subset of X* is relatively weakly compact (resp., weakly precompact). We prove that the projective tensor product X⊗πY has property (V) (resp., (wV)), when X has property (V) (resp., (wV)), Y has property (V), and W(X,Y*)=K(X,Y*).

scholarly journals ON TENSOR PRODUCTS OF WEAK MIXING VECTOR SEQUENCES AND THEIR APPLICATIONS TO UNIQUELY E-WEAK MIXING C*-DYNAMICAL SYSTEMS

2011 ◽  
Vol 85 (1) ◽  
pp. 46-59 ◽  
Author(s):  
FARRUKH MUKHAMEDOV
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Tensor Product ◽  
Tensor Products ◽  
Weak Mixing ◽  
Vector Sequences ◽  
Bounded Sequences

AbstractWe prove that, under certain conditions, uniform weak mixing (to zero) of the bounded sequences in Banach space implies uniform weak mixing of their tensor product. Moreover, we prove that ergodicity of tensor product of the sequences in Banach space implies their weak mixing. As applications of the results obtained, we prove that the tensor product of uniquely E-weak mixing C*-dynamical systems is also uniquely E-weak mixing.

scholarly journals Convergence tensor products and a strict topology

1980 ◽  
Vol 21 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Bernd Müller
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Approximation Property ◽  
Tensor Products ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Strict Topology ◽  
Topology Of Pointwise Convergence

We are interested in the strict topology τ on , the set L(E, F) of all continuous linear mappings from E into a Banach space F endowed with the topology of pointwise convergence. The T3-completion of the convergence tensor product E ⊗cLc F is the set of all τ-continuous linear functionals on L(E, F) and τ is the topology of uniform convergence on the compact subsets of . In the case that E is a nuclear Fréchet space, a nuclear (DF)-space or a Banach space with the bounded approximation property the topology τ agrees with the topology of Lco (E, F).

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

Tauberian theorems in a Banach lattice, with applications to the LP spaces

1954 ◽  
Vol 50 (2) ◽  
pp. 242-249
Author(s):  
D. C. J. Burgess
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Laplace Transforms ◽  
Tauberian Theorems ◽  
General Argument ◽  
Arbitrary Banach Space ◽  
Lp Spaces ◽  
Special Case

In a previous paper (2) of the author, there was given a treatment of Tauberian theorems for Laplace transforms with values in an arbitrary Banach space. Now, in § 2 of the present paper, this kind of technique is applied to the more special case of Laplace transforms with values in a Banach lattice, and investigations are made on what additional results can be obtained by taking into account the existence of an ordering relation in the space. The general argument is again based on Widder (5) to which frequent references are made.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

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