scholarly journals ON QUANTITATIVE SCHUR AND DUNFORD–PETTIS PROPERTIES

2015 ◽  
Vol 91 (3) ◽  
pp. 471-486 ◽  
Author(s):  
ONDŘEJ F. K. KALENDA ◽  
JIŘÍ SPURNÝ
Keyword(s):  
Compact Operators ◽  
Index Set ◽  
Quantitative Version ◽  
Schur Property ◽  
Space Of Compact Operators

We show that the dual to any subspace of $c_{0}({\rm\Gamma})$ (${\rm\Gamma}$ is an arbitrary index set) has the strongest possible quantitative version of the Schur property. Further, we establish a relationship between the quantitative Schur property and quantitative versions of the Dunford–Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell _{p}$ ($1<p<\infty$) with the Dunford–Pettis property automatically satisfies both its quantitative versions.

On the Non-Existence of a Projection onto the Space of Compact Operators

1982 ◽  
Vol 25 (1) ◽  
pp. 78-81 ◽  
Author(s):  
Moshe Feder
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Space Of Compact Operators

AbstractLet X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {Ti} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

Approximation from the space of compact operators and other $M$ -ideals

1975 ◽  
Vol 42 (2) ◽  
pp. 259-269 ◽  
Author(s):  
Richard Holmes ◽  
Bruce Scranton ◽  
Joseph Ward
Keyword(s):  
Compact Operators ◽  
Space Of Compact Operators

scholarly journals Chebyshev subspaces in the space of compact operators

1975 ◽  
Vol 15 (4) ◽  
pp. 326-334 ◽  
Author(s):  
David A Legg ◽  
Bruce E Scranton ◽  
Joseph D Ward
Keyword(s):  
Compact Operators ◽  
Space Of Compact Operators

scholarly journals On the spectral radius of compact operator cocycles

2020 ◽  
pp. 2150026
Author(s):  
Lucas Backes ◽  
Davor Dragičević
Keyword(s):  
Banach Spaces ◽  
Compact Operator ◽  
Spectral Radius ◽  
Compact Operators ◽  
Space Of Compact Operators

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger–Wang’s formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle.

Uniqueness of norm-preserving extensions of functionals on the space of compact operators

2019 ◽  
Vol 125 (1) ◽  
pp. 67-83
Author(s):  
Julia Martsinkevitš ◽  
Märt Põldvere
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Functional ◽  
Dual Space ◽  
Closed Subspace ◽  
Compact Operators ◽  
Bounded Linear ◽  
Nikodym Property ◽  
Space Of Compact Operators ◽  
Bounded Linear Functional

Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.

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