Approximation numbers of nuclear operators and geometry of Banach spaces

1991 ◽  
Vol 57 (2) ◽  
pp. 155-168 ◽  
Author(s):  
Albrecht Pietsch
Keyword(s):  
Banach Spaces ◽  
Approximation Numbers ◽  
Nuclear Operators

scholarly journals Operators constructed by means of basic sequences and nuclear matrices

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ahmed Morsy ◽  
Nashat Faried ◽  
Samy A. Harisa ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Schauder Basis ◽  
Countable Number ◽  
Nuclear Operator ◽  
Approximation Numbers ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Nuclear Operators

AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.

scholarly journals Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

scholarly journals On tensor products of nuclear operators in Banach spaces

2021 ◽  
Vol 14 (3) ◽  
pp. 187-205
Author(s):  
Oleg Reinov
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Fourier Coefficients ◽  
Schatten Classes ◽  
Nuclear Operators ◽  
Vector Valued

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

Nuclear Operators in Banach Spaces

2000 ◽  
pp. 91-110
Author(s):  
Israel Gohberg ◽  
Seymour Goldberg ◽  
Nahum Krupnik
Keyword(s):  
Banach Spaces ◽  
Nuclear Operators

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

1997 ◽  
Vol 39 (2) ◽  
pp. 227-230 ◽  
Author(s):  
Dumitru Popa
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Integral Operators ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Nuclear Operators ◽  
Injective Tensor Product ◽  
Image Position

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.

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

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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