Continuous Linear Operators Research Articles

The Boundedness of Cauchy Integral Operator on a Domain Having Closed Analytic Boundary

In this paper, we prove that the Cauchy integral operators (or Cauchy transforms) define continuous linear operators on the Smirnov classes for some certain domain with closed analytic boundary.

Universality of Composition Operators and Applications to Holomorphic Dynamics

By investigating which level of universality composition operators Cf can have, where the symbol f is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of f, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.

On linear dynamics of sets of operators

Let X role=presentation> X X X be a complex topological vector space with d i m ( X ) > 1 role=presentation> d i m ( X ) > 1 d i m ( X ) > 1 dim(X)>1 and B ( X ) role=presentation>  ( X ) B ( X ) \mathcal{B}(X) the set of all continuous linear operators on X role=presentation> X X X . The concept of hypercyclicity for a subset of B ( X ) role=presentation>  ( X ) B ( X ) \mathcal{B}(X) was introduced in [1]. In this work, we introduce the notion of hypercyclic criterion for a subset of B ( X ) role=presentation>  ( X ) B ( X ) \mathcal{B}(X) . We extend some results known for a single operator and C 0 role=presentation> C 0 C 0 C_0 -semigroup to a subset of B ( X ) role=presentation>  ( X ) B ( X ) \mathcal{B}(X) and we give applications for C role=presentation> C C C -regularized groups of operators.

Teoría de semigrupos fuertemente continuos de operadores lineales continuos aplicada a una ecuación de transporte-Difusión

En este trabajo aplicamos la teoría de semigrupos de operadores lineales continuos para probar la existencia y unicidad de la solución para cierta ecuación de transporte-difusión, en concreto aquellos que son fuertemente continuos, ya que estos son generados por un operados lineal usualmente denotado por A y llamado comúnmente el generador infinitesimal del semigrupo. Este operador es de gran importancia puesto que suele ser el operador relacionado a una ecuación o sistema de ecuaciones diferenciales, es decir, relacionamos la ecuación de transporte-difusión a un operador diferencial lineal, el cual genera a un semigrupo fuertemente continuo, y así aplicamos ciertos resultados conocidos en el área de los semigrupos y lo que nos permitirá probar la existencia y unicidad de la solución para nuestra ecuación, en contraste con los métodos clásicos conocidos.

Convexity and continuous linear operators with respect to the f-norm

In this paper, the analogues of various results are obtained in terms of a generalization of the norm called the f-norm. Convex sets are defined with respect to the f-norm and properties obtained. In addition, conditions are presented for the continuity of linear operators and also for the existence of a continuous inverse of a linear operator.

Modulus hyperinvariant ideals for a finitely quasinilpotent operators

Let X be a Banach space with an unconditional basis, and let C ? {0} be a collection of continuous linear operators with modulus on X that is finitely modulus-quasinilpotent at a non-zero positive vector. Then C and its right modulus sub-commutant C'm have a common non-trivial invariant closed ideal.

ON TOPOLOGICAL ALGEBRAS OF ANALYTIC FUNCTIONALS WITH A MULTIPLICATION DEFINED BY TRANSLATIONS

We define a multiplication convolution in the dual of a countable inductive limit E of weighted Frechet spaces of entire functions of several variables. This algebra is isomorphic to the commutant of the system of partial derivatives in the algebra of all continuous linear operators in E. In the constructed algebra of analytic functionals in two pure cases a topology is defined. With this topology the mentioned algebra is topological and it is now topologically isomorphic to the considered commutant with its natural operator topology. It is proved that in this pure situations the present algebra has no zero divisors provided that polynomials are dense in E. We show that this condition is essential for the validity of the last statement.

On a Class of Operator Equations in Locally Convex Spaces

We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.

An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, $\{e^{mΔ}:\; m∈ \mathbb{N}\}$, where $Δ$ is the Laplace operator on the unbounded domain $\mathbb{R}^{n}\setminus \{0\}$.Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

On cyclic sets of operators

Let X be a complex topological vector space with $$dim(X)>1$$ and $$\mathcal {B}(X)$$ the space of all linear continuous operators on X. In this paper, we extend the concept of cyclicity for a single operator and $$C_0$$ -semigroups to a subset of $$\mathcal {B}(X).$$ We establish some results for cyclic set of operators and give some applications for $$C_0$$ -semigroups.

Layer Potentials and Poisson Problems for the Nonsmooth Coefficient Brinkman System in Sobolev and Besov Spaces

In this paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$ -based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$ . For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$ -based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$ , and the property that this operator is an isomorphism for $$p=2$$ . Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$ to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$ -based Sobolev and Besov spaces. For $$p=2$$ , such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$ -Sobolev spaces for any $$q\in (1,\infty )$$ , which is also invertible on the solution space corresponding to $$q=2$$ . Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$ to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$ -based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$ -based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$ -based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$ , by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$ -based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$ -based Sobolev and Besov spaces.

Efficient primal–dual fixed point algorithms with dynamic stepsize for composite convex optimization problems

In this paper, we propose a new primal–dual fixed point algorithm with dynamic stepsize ( $$\hbox {PDFP}^{2}O_{DS_{n}}$$ ) for solving convex minimization problems involving the sum of a smooth function with Lipschitzian gradient and the composition of a nonsmooth convex function with a continuous linear operator. Based on modified Mann iteration and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithm. Moreover, we give the connection of the proposed algorithm with other existing $$\hbox {PDFP}^{2}O$$ (Chen et al. in Inverse Probl 29:025011–025033, 2013). Finally, we illustrate the efficiency of $$\hbox {PDFP}^{2}O_{DS_{n}}$$ through some numerical examples on the CT image reconstruction problem. Numerical results show that our iterative algorithm ( $$\hbox {PDFP}^{2}O_{DS}$$ ) performs better than the original one ( $$\hbox {PDFP}^{2}O$$ ).

Fixed points of polarity type operators

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in Rn, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm must be Euclidean. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possible independent interest, among them results related to coercive bilinear forms (essentially a quantitative convex analytic converse to the celebrated Lax–Milgram theorem from partial differential equations) and a characterization of real Hilbertian spaces.

N-SOT Hypercyclic Linear Maps on Banach Algebra of Operators

Let B(X) be the algebra of bounded linear operators on a Banach space X. A subset E of B(X) is said to be n-SOT dense in B(X) if for every continuous linear operator \(\Lambda \) from B(X) onto \(X^{(n)}\), the direct sum of n copies of X, \(\Lambda (E)\) is dense in \(X^{(n)}\). We consider the n-SOT hypercyclic continuous linear maps on B(X), namely, those that have orbits that are n-SOT dense in B(X). Some nontrivial examples of such operators are provided and many of their basic properties are investigated. In particular, we show that the left multiplication operator \(L_T\) is 1-SOT hypercyclic if and only if T is hypercyclic on X.

Some properties of shift operators on algebras generated by $*$-polynomials

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces.
 We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

ADMM for monotone operators: convergence analysis and rates

We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper, convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on suitable choice of dynamical step sizes. The numerical performances, which can be obtained for different dynamical step size strategies, are compared in the context of solving an image denoising problem.

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.

Extension of Pettis integration: Pettis operators and their integrals

In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak $$^*$$ –weakly continuous linear operator F from a dual Banach space to an $$L_1$$ -space, and of its Pettis integral, understood simply as the dual operator $$F^*$$ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.

Korovkin-type approximation in spaces of vector-valued and set-valued functions

We consider some Korovkin-type approximation results for sequences of linear continuous operators in spaces of vector-valued and set-valued continuous functions without assuming the existence of the limit operator. Even in spaces of real continuous functions, where similar results have already been established, we replace the positivity assumption with a weaker condition. We also give some quantitative estimate of the convergence and some applications where previous results cannot be applied.

On the dense unbounded divergence of interpolatory product integration on Jacobi nodes

This paper deals with interpolatory product integration rules based on Jacobi nodes, associated with the Banach space of all s-times continuously differentiable functions, and with a Banach space of absolutely integrable functions, on the interval $$[-1,1]$$ of the real axis. In order to highlight the topological structure of the set of unbounded divergence for the corresponding product quadrature formulas, a family of continuous linear operators associated with these product integration procedures is pointed out, and the unboundedness of the set of their norms is established, by means of some properties involving the theory of Jacobi polynomials. The main result of the paper is based on some principles of Functional Analysis, and emphasizes the phenomenon of double condensation of singularities with respect to the considered interpolatory product quadrature formulas, by pointing out large subsets (in topological meaning) of the considered Banach spaces, on which the quadrature procedures are unboundedly divergent.