Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

We ask when convolution operators with scalar- or operator-valued kernel functions map between weighted L2 spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a)−m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for Lp(w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input–output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general.

A characterization of Hilbert spaces in terms of multipliers between spaces of vector-valued analytic functions.

Hadamard type operators on spaces of real analytic functions in several variables

We discuss new results concerning unbounded Toeplitz operators defined in Segal-Bargmann spaces of (vector-valued) functions, i.e. the space of all entire functions which are square summable with respect to the Gaussian measure in $\mathrm{C}^n$. The problem of finding adjoints of analytic Toeplitz operators is solved in some cases. Closedness of the range of analytic Toeplitz operators is studied. We indicate an example of an entire function inducing a Toeplitz operator, for which the space of polynomials is not a core though it is contained in its domain.

Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent.

We analyse the strong connections between spaces of vector-valued Lipschitz functions and spaces of linear continuous operators. We apply these links to study duality, Schur properties and norm attainment in the former class of spaces as well as in their canonical preduals.

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL α 1 (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range.

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions.

We correct a mistake found in our the article “Cesàro-like operators acting on spaces of analytic functions”, published in Anal. Math. Phys. 12 (2022), no. 2, Paper No. 51, 29 pp.

In an earlier paper [2]; the author used equivalence classes of analytic functions to construct an imbedding space for Schwartz Distributions. The mechanism for constructing the mapping was the bilateral Laplace Transform, in this way the traditional approach to operational calculus was preserved. In that paper a topology was imposed on the imbedding space from the space of analytic functions. We now obtain some additional results about the possible topologies defined on the space of analytic functions.

In this paper we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on a bounded domain of the complex plane.

On Newton's interpolation polynomials