Continuous linear operators in Banach spaces

1987 ◽  
pp. 114-137
Author(s):  
Stefan Rolewicz
Keyword(s):  
Banach Spaces ◽  
Linear Operators ◽  
Continuous Linear ◽  
Continuous Linear Operators

scholarly journals Automatic discontinuity of intertwining operators

1982 ◽  
Vol 25 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Spaces ◽  
Linear Transformation ◽  
Linear Operators ◽  
Intertwining Operators ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Converse Problem ◽  
Simple Modification ◽  
Continuous Linear Operators

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

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.

scholarly journals Reiterative Distributional Chaos on Banach Spaces

2019 ◽  
Vol 29 (14) ◽  
pp. 1950201 ◽  
Author(s):  
Antonio Bonilla ◽  
Marko Kostić
Keyword(s):  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Nonzero Vector ◽  
Vector Subspace ◽  
Continuous Linear ◽  
Distributional Chaos ◽  
Dynamical Properties ◽  
Definition Of ◽  
Continuous Linear Operators

If we change the upper and lower densities in the definition of distributional chaos of a continuous linear operator on a Banach space [Formula: see text] by the Banach upper and Banach lower densities, respectively, we obtain Li–Yorke chaos. Motivated by this, we introduce the notions of reiterative distributional chaos of types [Formula: see text], [Formula: see text] and [Formula: see text] for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and present the conditions for the existence of a vector subspace [Formula: see text] of [Formula: see text], such that every nonzero vector in [Formula: see text] is both irregular for [Formula: see text] and distributionally near zero for [Formula: see text].

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.

Lomonosov's hyperinvariant subspace theorem for real spaces

1981 ◽  
Vol 89 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. D. Hooker
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyperinvariant Subspace ◽  
Subspace Theorem ◽  
Hyperinvariant Subspaces ◽  
A New Technique ◽  
Continuous Linear Operators

In 1973, V.I.Lomonosov introduced a new technique for finding invariant and hyperinvariant subspaces for certain classes of (continuous, linear) operators on complex Banach spaces. Recall that a closed subspace M of the Banach space X is called hyperinvariant for the operator T if S(M) ⊂ M for every operator S which commutes with T.

Copies of l∞ in an operator space

1990 ◽  
Vol 108 (3) ◽  
pp. 523-526 ◽  
Author(s):  
Lech Drewnowski
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Operator Space ◽  
Isomorphic Copy ◽  
Continuous Linear ◽  
Weakly Continuous ◽  
Continuous Linear Operators

Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

scholarly journals Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

1988 ◽  
Vol 30 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Volker Wrobel
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
Continuous Linear Operators

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

scholarly journals Pareto Optimality for Multioptimization of Continuous Linear Operators

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 661
Author(s):  
Clemente Cobos-Sánchez ◽  
José Antonio Vilchez-Membrilla ◽  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Multiobjective Optimization ◽  
Banach Spaces ◽  
Optimization Problems ◽  
Linear Operators ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Continuous Linear ◽  
Multiobjective Optimization Problems ◽  
Continuous Linear Operators

This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form max∥T(x)∥, min∥x∥ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering.

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