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.

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).

Weak Sequential Completeness of 𝑲(X,Y)

2013 ◽  
Vol 56 (3) ◽  
pp. 503-509 ◽  
Author(s):  
Qingying Bu
Keyword(s):  
Banach Spaces ◽  
Compact Operator ◽  
Approximation Property ◽  
Compact Operators ◽  
Weakly Compact ◽  
Sequential Completeness ◽  
Weakly Compact Operator ◽  
Compact Approximation

AbstractFor Banach spaces X and Y, we show that if X* and Y are weakly sequentially complete and every weakly compact operator from X to Y is compact, then the space of all compact operators from X to Y is weakly sequentially complete. The converse is also true if, in addition, either X* or Y has the bounded compact approximation property.

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.

Non-complemented Spaces of Operators, Vector Measures, and co

2012 ◽  
Vol 55 (3) ◽  
pp. 548-554 ◽  
Author(s):  
Paul Lewis ◽  
Polly Schulle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Operators ◽  
Vector Measures ◽  
Spaces Of Operators ◽  
Space Of Compact Operators

AbstractThe Banach spaces L(X, Y), K(X,Y), Lw* (X*, Y), and Kw* (X*, Y) are studied to determine when they contain the classical Banach spaces co or ℓ∞. The complementation of the Banach space K(X, Y) in L(X, Y) is discussed as well as what impact this complementation has on the embedding of co or ℓ∞ in K(X, Y) or L(X, Y). Results of Kalton, Feder, and Emmanuele concerning the complementation of K(X, Y) in L(X, Y) are generalized. Results concerning the complementation of the Banach space Kw* (X*, Y) in Lw* (X*, Y) are also explored as well as how that complementation affects the embedding of co or ℓ∞ in Kw* (X*, Y) or Lw* (X*, Y). The ℓp spaces for 1 = p < ∞ are studied to determine when the space of compact operators from one ℓp space to another contains co. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

scholarly journals DIFFERENCES OF COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS

2008 ◽  
Vol 84 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JOSÉ BONET ◽  
MIKAEL LINDSTRÖM ◽  
ELKE WOLF
Keyword(s):  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Holomorphic Functions ◽  
Compact Operators ◽  
Spaces Of Holomorphic Functions ◽  
Weighted Banach Spaces ◽  
Space Of Compact Operators

AbstractWe consider differences of composition operators between given weighted Banach spaces $H_v^{\infty }$ or Hv0 of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results.

Spaces of operators between Fréchet spaces

1994 ◽  
Vol 115 (1) ◽  
pp. 133-144 ◽  
Author(s):  
José Bonet ◽  
Mikael Lindström
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Linear Maps ◽  
Spaces Of Operators ◽  
Space Of Compact Operators ◽  
Compact Subsets

AbstractMotivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces (E, F) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F. As a consequence of our results we characterize pairs of Köthe echelon spaces (E, F) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F.

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.

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