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 A generalization of Lagrange multipliers: Corrigendum

1978 ◽  
Vol 18 (1) ◽  
pp. 159-160
Author(s):  
B.D. Craven
Keyword(s):  
Banach Spaces ◽  
Lagrange Multipliers ◽  
Continuous Linear ◽  
Linear Map ◽  
Linear Maps ◽  
Valid Proof

There is a lacuna in the proof of Lemma 1 of [1]; the projector q is assumed without proof. An alternative, valid proof is as follows.LEMMA 1. Let S, U0, V0 be real Banach spaces; let A : S → U0 and B : S → V0 be continuous linear maps, whose null spaces are N(A) respectively N(B); let N(A) ⊂ N(B) : let A map S onto U0. Then there exists a continuous linear map C : U0 → V0 such that B = C ° A.

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 On the surjectivity of linear maps on locally convex spaces

1982 ◽  
Vol 32 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Basic Notion ◽  
Linear Map ◽  
Linear Maps ◽  
Locally Convex ◽  
Do So ◽  
Convex Spaces

AbstractThe aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

Automatic Continuity for Linear Functions Intertwining Continuous Linear Operators on Frechet Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 518-530 ◽  
Author(s):  
Marc P. Thomas
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Linear Operators ◽  
Linear Functions ◽  
General Situation ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Continuity Theory ◽  
Continuous Linear Operators

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

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).

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

1989 ◽  
Vol 32 (4) ◽  
pp. 450-458
Author(s):  
Takemitsu Kiyosawa
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergent Sequence ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Infinite Dimensional ◽  
Valued Field ◽  
Infinite Dimensional Banach Space ◽  
Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear 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.

scholarly journals Closed linear maps from a barrelled normed space into itself need not be continuous

1998 ◽  
Vol 57 (2) ◽  
pp. 177-179 ◽  
Author(s):  
José Bonet
Keyword(s):  
Normed Space ◽  
Continuous Linear ◽  
Closed Graph ◽  
Linear Map ◽  
Linear Maps ◽  
Barrelled Spaces

Examples of normed barrelled spacesEor quasicomplete barrelled spacesEare given such that there is a non-continuous linear map from the spaceEinto itself with closed graph.

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.

Topological decompositions of the duals of locally convex operator spaces

1983 ◽  
Vol 93 (2) ◽  
pp. 307-314 ◽  
Author(s):  
D. J. Fleming ◽  
D. M. Giarrusso
Keyword(s):  
Uniform Convergence ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Natural Topology ◽  
Usual Norm ◽  
Linear Maps ◽  
Locally Convex

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

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