scholarly journals ON WEYL AND BROWDER SPECTRA OF TENSOR PRODUCTS

2008 ◽  
Vol 50 (2) ◽  
pp. 289-302 ◽  
Author(s):  
C. S. KUBRUSLY ◽  
B. P. DUGGAL

AbstractLet A and B be Hilbert space operators. In this paper we explore the structure of parts of the spectrum of the tensor product A ⊗ B, and conclude some properties that follow from such a structure. We give conditions on A and B ensuring that σw(A ⊗ B) =σw(A)ċσ(B) ∪ σ(A)ċσw(B), where σ(ċ) and σw(ċ) stand for the spectrum and Weyl spectrum, respectively. We also investigate the problem of transferring Weyl and Browder's theorems from A and B to their tensor product A⊗B.

1990 ◽  
Vol 108 (2) ◽  
pp. 395-403 ◽  
Author(s):  
David P. Blecher

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).


1990 ◽  
Vol 13 (4) ◽  
pp. 807-810 ◽  
Author(s):  
Ram Verma ◽  
Lokenath Debnath

We prove a result on the joint numerical status of the bounded Hilbert space operators on the tensor products. The result seems to have nice applications in the multiparameter spectral theory.


1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.


2014 ◽  
Vol 25 (02) ◽  
pp. 1450019 ◽  
Author(s):  
RALF MEYER ◽  
SUTANU ROY ◽  
STANISŁAW LECH WORONOWICZ

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.


Author(s):  
Mazen Ali ◽  
Anthony Nouy

We are interested in applying SVD to more general spaces, the motivating example being the Sobolev space $H^1(\Omega)$ of weakly differentiable functions over a domain $\Omega\subset\R^d$. Controlling the truncation error in the energy norm is particularly interesting for PDE applications. To this end, one can apply SVD to tensor products in $H^1(\Omega_1)\otimes H^1(\Omega_2)$ with the induced tensor scalar product. However, the resulting space is not $H^1(\Omega_1\times\Omega_2)$ but is instead the space $H^1_{\text{mix}}(\Omega_1\times\Omega_2)$ of functions with mixed regularity. For large $d>2$ this poses a restrictive regularity requirement on $u\in H^1(\Omega)$. On the other hand, the space $H^1(\Omega)$ is not a tensor product Hilbert space, in particular $\|\cdot\|_{H^1}$ is not a reasonable crossnorm. Thus, we can not identify $H^1(\Omega)$ with the space of Hilbert Schmidt operators and apply SVD. However, it is known that $H^1(\Omega)$ is isomorph (here written for $d=2$) to the Banach intersection space $$H^1(\Omega_1\times\Omega_2)=H^1(\Omega_1)\otimes L_2(\Omega_2)\cap L_2(\Omega_1)\otimes H^1(\Omega_2)$$ with equivalent norms. Each of the spaces in the intersection is a tensor product Hilbert space where SVD applies. We investigate several approaches to construct low-rank approximations for a function $u\in H^1(\Omega_1\times\Omega_2)$.


1976 ◽  
Vol 21 (2) ◽  
pp. 241-246 ◽  
Author(s):  
Charles Swartz

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.


2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming

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 *.


2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY

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.


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