Spaces Of Vector-valued Analytic Functions Research Articles

Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions

For a fixed nonnegative integer $m$, an analytic map $\varphi$ and an analytic function $\psi$, the generalized integration operator $I^{(m)}_{\varphi,\psi}$ is defined by \[ I^{(m)}_{\varphi,\psi} f(z) = \int_0^z f^{(m)}(\varphi(\zeta)) \psi(\zeta) \, d\zeta \] for $X$-valued analytic function $f$, where $X$ is a Banach space. Some estimates for the norm of the operator $I^{(m)}_{\varphi,\psi} \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ are obtained. In particular, it is shown that the Volterra operator $J_b \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ is bounded if and only if $J_b \colon A^2_{\alpha} \to A^2_{\alpha}$ is in the Schatten class $S_p(A^2_{\alpha})$ for $2 \leq p \lt \infty$ and $\alpha \gt -1$. Some corresponding results are established for $X$-valued Hardy spaces and $X$-valued Fock spaces.

Weighted composition operators between weak spaces of vector-valued analytic functions

Abstract We consider weighted composition operators W ψ , φ : f ↦ ψ ⁢ ( f ∘ φ ) ${W_{\psi,\varphi}\colon f\mapsto\psi(f\circ\varphi)}$ on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of W ψ , φ ${W_{\psi,\varphi}}$ on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of W ψ , φ ${W_{\psi,\varphi}}$ between two different vector-valued Bloch-type spaces. This characterization appears to be new also in the scalar-valued case.

Composition operators on vector-valued analytic function spaces: a survey

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

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

The research of the first three authors was partially supported by MEC and FEDER Project MTM2010-15200, by GVA, Projects GV/2010/040 and Prometeo/2008/101, and Universidad Politecnica de Valencia, Project Ref. 2773.

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C ∞ -boundary) to the case of C 2 -boundary.

On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions

Let H be a Hilbert space of analytic functions on the unit disc D with ‖ M z ‖ ⩽ 1 , where M z denotes the operator of multiplication by the identity function on D . Under certain conditions on H it has been shown by Aleman, Richter and Sundberg that all invariant subspaces have index 1 if and only if lim k → ∞ ‖ M z k f ‖ ≠ 0 for all f ∈ H , f ≢ 0 [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (7) (2007) 3369–3407]. We show that the natural counterpart to this statement in Hilbert spaces of C n -valued analytic functions is false and prove a correct generalization of the theorem. In doing so we obtain new information on the boundary behavior of functions in such spaces, thereby improving the main result of [M. Carlsson, Boundary behavior in Hilbert spaces of vector-valued analytic functions, J. Funct. Anal. 247 (1) (2007)].

WEIGHTED COMPOSITION OPERATORS ON WEIGHTED SPACES OF VECTOR-VALUED ANALYTIC FUNCTIONS

Let V be an arbitrary system of weights on an open connected subset G of and let B (E) be the Banach algebra of all bounded linear operators on a Banach space E. Let (G, E) and (G, E) be the weighted locally convex spaces of vector-valued analytic functions. In this paper, we characterize self-analytic mappings and operator-valued analytic mappings which generate weighted composition operators and invertible weighted composition operators on the spaces (G, E) and (G, E) for different systems of weights V on G. Also, we obtained compact weighted composition operators on these spaces for some nice classes of weights

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

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.

Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D . More specifically, we give operator-theoretic conditions on M z , where M z denotes the operator of multiplication by the identity function on D , that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.

Composition Operators and Multiplication Operators on Weighted Spaces of 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).

Convolution of three functions by means of bilinear maps and applications

When dealing with spaces of vector-valued analytic functions there is a natural way to understand multipliers between them. If X and Y are Banach spaces and L(X,Y ) stands for the space of linear and continuous operators we may consider the convolution of L(X,Y )-valued analytic functions, say F (z) = ∑ n=0∞ Tnz , and X-valued polynomials, say f(z) = ∑m n=0 xnz , to get the Y -valued function F ∗ f(z) = ∑ Tn(xn)z. The second author considered such a definition and studied multipliers between H(X) and BMOA(Y ) in [5]. When the functions take values in a Banach algebra A then the natural extension of multiplier is simply that if f(z) = ∑ anz n and g(z) = ∑ bnz , then f ∗ g(z) = ∑ an.bnz n where a.b stands for the product in the algebra A. Of course, similarly one can consider an ∈ L(R), bn ∈ L(R) and the convolution an ∗ bn ∈ L(R) (where p, q, r verifies the condition on Young’s theorem). The reader is referred to [3] for results along these lines. In this paper we shall consider a much more general notion of convolutions coming from general bilinear maps and that will extend the previous examples. Assume X,Y, Z are Banach spaces and let u : X × Y → Z be a bounded bilinear map. Given a X-valued polynomial f(z) = ∑m n=0 xnz n and given a Y -valued polynomial g(z) = ∑k n=0 ynz n we define the u-convolution of f an g as the polynomial given by

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.

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

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