Continuous Linear Operators Research Articles

On regular operators on Banach lattices

Let $E$ and $F$ be Banach lattices and $X$ and $Y$ be Banach spaces. A linear operator $T: E \rightarrow F$ is called regular if it is the difference of two positive operators. $L_{r}(E,F)$ denotes the vector space of all regular operators from $E$ into $F$. A continuous linear operator $T: E \rightarrow X$ is called $M$-weakly compact operator if for every disjoint bounded sequence $(x_{n})$ in $E$, we have $lim_{n \rightarrow\infty} \| Tx_{n} \| =0$. $W^{r}_{M}(E,F)$ denotes the regular $M$-weakly compact operators from $E$ into $F$. This paper is devoted to the study of regular operators and $M$-weakly compact operators on Banach lattices. We show that $F$ has a b-property if and only if $L_{r}(E,F)$ has b-property. Also, $W^{r}_{M}(E,F)$ is a $KB$-space if and only if $F$ is a $KB$-space.

On the closedness of the sum of subspaces of the space $B(H,Y)$ consisting of operators whose kernels contain given subspaces of $H$

Let $H$ be a Hilbert space and $Y$ be a Banach space. Denote by $B(H,Y)$ the linear space of all continuous linear operators $A:H\to Y$ endowed with the standard operator norm. For a closed subspace $H_0$ of $H$ denote by $Z(H_0;H,Y)$ the set of all o

FUZZY CONTINUOUS AND FUZZY BOUNDED LINEAR OPERATORS OVER ANTI-FUZZY FIELDS

In this paper, we have studied the concept of anti-norm and anti-inner product function on anti-fuzzy linear space over anti-fuzzy field, we have also given fuzzy continuous linear operator from an anti-normed anti-fuzzy linear space to another anti-normed anti-fuzzy linear space and also introduced three types (strong, weak and sequential) of fuzzy bounded linear operators.

Codiskcyclic sets of operators on complex topological vector spaces

Let X be a complex topological vector space and L(X) the set of all continuous linear operators on X. In this paper, we extend the notion of the codiskcyclicity of a single operator T ∈ L(X) to a set of operators Γ ⊂ L(X). We prove some results for codiskcyclic sets of operators and we establish a codiskcyclicity criterion. As an application, we study the codiskcyclicity of C0-semigroups of operators.

Group Invariant Operators and Some Applications to Norm-Attaining Theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties \(\alpha \) of Schachermayer and \(\beta \) of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, which says that if X has the Radon–Nikodým property, then X has the G-Bishop–Phelps property for G-invariant operators whenever \(G \subseteq \mathcal {L}(X)\) is a compact group of isometries on X.

Generalized weighted number operators on functionals of discrete-time normal martingales

Let M be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of M, one can construct generalized functionals of M. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of M, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of M). We also examine commutation relations between a GWN operator and a generalized annihilation (or creation) operator, and obtain several formulas expressing such commutation relations.

Backward Stochastic Differential Equations (BSDEs) Using Infinite-Dimensional Martingales with Subdifferential Operator

In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with subdifferential operators that are driven by infinite-dimensional martingales. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence and uniqueness of the solution are established using Yosida approximations. Furthermore, as an application of the main result, we shall show that the backward stochastic partial differential equation driven by infinite-dimensional martingales with a continuous linear operator has a unique solution under the special condition that the Ft-progressively measurable generator F of the model we proposed in this paper equals zero.

A NOTE ON FREQUENT HYPERCYCLICITY OF OPERATORS THAT lambda-COMMUTE WITH THE DIFFERENTIATION OPERATOR

A continuous linear operator on a Fréchet space mathcal {X} is frequently hypercyclic if there exists a vector x such that for any nonempty open subset Usubset mathcal {X} the set of nin mathbb {N}cup {0} for which T^nxin U has a positive lower density. Here we determine when an operator that commutes up to a factor with the differentiation operator D, defined on the space of entire functions, is frequently hypercyclic.

Strong and total duality for constrained composed optimization via a coupling conjugation scheme

ABSTRACT Based on a coupling conjugation scheme and the perturbational approach, we build Fenchel–Lagrange dual problem of a composed optimization model with infinite constraints in separated locally convex spaces. This paper has mainly two targets. One is to establish strong duality under a new regularity condition ( RC A ) and an extension closed-type condition ( ECRC A ). The e-convex counterpart of Fenchel–Moreau theorem plays a key role in analysing the relation between them. The other aim is to achieve the sufficient and necessary characterizations for total duality in terms of c-subdifferentials. For this purpose, a formula for ε-c-subdifferentials of a proper function composed with a linear continuous operator is proved and applied.

On Special Properties for Continuous Convex Operators and Related Linear Operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

The inner kernel theorem for a certain Segal algebra

The Segal algebra {mathbf{S}}_{0}(G) is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups G. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from {mathbf{S}}_{0}(G_1) to {mathbf{S}}'_{0}(G_2) by generalized functions in {mathbf{S}}'_{0}(G_1 times G_2) the “outer kernel theorem”. The main subject of this paper is to formulate what we call the “inner kernel theorem”. This is the characterization of those linear operators that have kernels in {mathbf{S}}_{0}(G_1 times G_2). Such operators are regularizing—in the sense that they map {mathbf{S}}'_{0}(G_1) into {mathbf{S}}_{0}(G_2) in a w^{*} to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies “integrate” to a Dirac delta distribution in a mathematically justifiable way.

Banach lattices of orthogonally additive operators

Using new notions of consistent sets and levels in Riesz spaces, we study the relationship between linear and orthogonally additive operators on Banach lattices. We define a norm on the Riesz space U(E,F) of order bounded orthogonally additive operators acting between Banach lattices E,F and prove that if F is Dedekind complete then the set UB(E,F) of all bounded with respect to this norm elements of U(E,F) is a Dedekind complete Banach lattice which is a sublattice of U(E,F). One of the results asserts that if E is an AL-space, F a Banach lattice with a Fatou-Levi norm (in particular, a KB-space) then the Banach lattice L(E,F)=Lr(E,F) of all continuous linear (regular) operators from E to F is a 1-complemented subspace of UB(E,F). If, moreover, both E and F are separable then for every S∈L(E,F)+ there exists T∈UB(E,F) such that for every S1∈L(E,F) with |S1|≤S there is a contractive projection Φ of UB(E,F) onto L(E,F) with Φ(T)=S1. Note that, being a subspace of UB(E,F), the Banach lattice Lr(E,F) is not a sublattice of UB(E,F), because the order on Lr(E,F) differs from the order on UB(E,F).

Analytical Solutions to Minimum-Norm Problems

For G∈Rm×n and g∈Rm, the minimization min∥Gψ−g∥2, with ψ∈Rn, is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min∥T(h)−k∥, where T:H→K is a continuous linear operator between Hilbert spaces H,K and h∈H,k∈K. In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min∥T(h)−k∥andmin∥h∥. We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min∥T(h)−k∥with∥h∥=α. The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils.

Integration with respect to Baire vector measures with applications to the spectral theory

Let X be a completely regular Hausdorff space and C_b(X) the space of all bounded continuous scalar functions on X , equipped with the strict topology \beta_\sigma . We develop a general integral representation theory of (\beta_\sigma,\xi) -continuous linear operators from C_b(X) to a locally convex Hausdorff space (E,\xi) . We present equivalent conditions for a (\beta_\sigma,\xi) -continuous operator T\colon C_b(X) \to E to be weakly compact. If (\mathcal{A},\xi) is a sequentially complete unital locally convex algebra, we establish an integral representation of a (\beta_\sigma,\xi) -continuous unital algebra homomorphism T\colon C_b(X) \to \mathcal{A} . As an application, we develop spectral theory for operators T\colon C_b(X) \to \mathcal{L}_s(\mathcal{Y}) , where \mathcal{L}_s(\mathcal{Y}) denotes the algebra of bounded linear operators of a Banach space \mathcal{Y} into itself, equipped with the topology \tau_s of simple convergence.

Cosine families in GDP Quojection-Fréchet spaces

We prove that if the Quojection-Fréchet space $X$ is a Grothendieck space with the Dunford-Pettis property, then every $C_ {0}$-cosine family is necessarily uniformly continuous and therefore its infinitesimal generator is a continuous linear operator.

The norm of continuous linear operator between two fuzzy quasi-normed spaces

<abstract> <p>In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous <italic>t</italic>-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.</p> </abstract>

Ultradifferentiable extension theorems: A survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney’s classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun–Meise–Taylor classes based on their representation as intersections and unions of Denjoy–Carleman classes.

Investigating Distributional Chaos for Operators on Fréchet Spaces

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

Extension of vector-valued functions and weak\u2013strong principles for differentiable functions of finite order

In this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field mathbb {K}, which has weak extensions in a weighted Banach space {mathcal {F}}nu (Omega ,mathbb {K}) of scalar-valued functions on a set Omega, to functions in a vector-valued counterpart mathcal {F}nu (Omega ,E) of {mathcal {F}}nu (Omega ,mathbb {K}). Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.

Multiobjective Convex Optimization in Real Banach Space

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.