Vector-valued Holomorphic Functions Research Articles

Vector-valued holomorphic functions and abstract Fubini-type theorems

Vector-valued holomorphic functions and abstract Fubini-type theorems

A simplified approach to the holomorphic discrete series

Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.

Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball BE of a Banach space E lack weak⁎-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk [56]. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric π-property. Besides, we obtain several equivalent statements for the Banach space of vector-valued homogeneous polynomials to be reflexive, which improves the result of J. Mujica [47], J. A. Jaramillo and L. A. Moraes [39]. As a byproduct, we generalize some results on polynomial reflexivity due to J. Farmer [35].

Stević-Sharma Operator on Spaces of Vector-Valued Holomorphic Functions

Stević-Sharma Operator on Spaces of Vector-Valued Holomorphic Functions

Difference of composition operators on spaces of vector-valued holomorphic functions

In this paper, we completely characterize the boundedness of difference of weighted composition operators between weak and strong vector-valued Bergman spaces in three terms: one is a function theoretic characterization of Julia-Carathéodory type, the second is a power type characterization and the other is a measure theoretic characterization of Carleson type. Furthermore, the bounded difference of composition operators is investigated for corresponding vector-valued Fock space case, which is in sharp contrast with some phenomenon on the setting of vector-valued Bergman space.

Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C \subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h \in \End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) \subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{\pi i h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens' principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

Given a directed Cartesian product T of locally finite, leafless, rooted directed trees T1,…,Td of finite joint branching index, one may associate with T the Drury-Arveson-type C[z1,…,zd]-Hilbert module Hca(T) of vector-valued holomorphic functions on the unit ball Bd in Cd, where a>0. The main result of this paper classifies all directed Cartesian products T for which the Hilbert modules Hca(T) are isomorphic in case a is an integer. Indeed, a careful analysis of these Hilbert modules allows us to prove that the cardinality of generations of T1,…,Td are complete invariants for Hca(⋅) if ad≠1.

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.

Dirichlet Spaces Associated with Locally Finite Rooted Directed Trees

Let $$\mathscr {T}=(V, \mathcal E)$$ be a leafless, locally finite rooted directed tree. We associate with $$\mathscr {T}$$ a one parameter family of Dirichlet spaces $$\mathscr {H}_q~(q \geqslant 1)$$ , which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $$\mathbb D$$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel $$\begin{aligned} \kappa _{\mathscr {H}_q}(z, w)= & {} \sum _{n=0}^{\infty }\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf {root}}\rangle } \\&+\,\sum _{v \in V_{\prec }} \sum _{n=0}^{\infty } \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{aligned}$$ where $$V_{\prec }$$ denotes the set of branching vertices of $$\mathscr {T}$$ , $$n_v$$ denotes the depth of $$v \in V$$ in $$\mathscr {T},$$ and $$P_{\langle e_{\mathsf {root}}\rangle }$$ , $$~P_{v}~(v \in V_{\prec })$$ are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators $$\mathscr {M}^{(1)}_z$$ and $$\mathscr {M}^{(2)}_z$$ of multiplication by z on Dirichlet spaces $$\mathscr {H}_q$$ associated with directed trees $$\mathscr {T}_1$$ and $$\mathscr {T}_2$$ respectively.

A local theory for operator tuples in the Cowen–Douglas class

We present a local theory for a commuting m-tuple S=(S1,S2,⋯,Sm) of Hilbert space operators lying in the Cowen–Douglas class. By representing S on a Hilbert module M consisting of vector-valued holomorphic functions over Cm, we identify and study the localization of S on an analytic hyper-surface in Cm. We completely determine unitary equivalence of the localization and relate it to geometric invariants of the Hermitian holomorphic vector bundle associated to S. It turns out that the localization coincides with an important class of quotient Hilbert modules, and our result concludes its classification problem in full generality.

Vector-valued holomorphic and harmonic functions

Abstract Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

Coherent states and Berezin quantization for non-scalar holomorphic representations

Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).

Estimates for vector-valued holomorphic functions and Littlewood–Paley–Stein theory

In this paper we consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form \[L^p(X)\subseteq \gamma(X) \subseteq L^q(X),\] in terms of the type $p$ and cotype $q$ for the Banach space $X$. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein $g$-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

Weighted Vector-Valued Holomorphic Functions on Banach Spaces

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a functionfdefined in a subsetAof an open and connected subsetUof a Banach spaceX, with values in another Banach spaceE, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Abstract Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let UX and UY be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on UX and UY, respectively. Let HV(UX, E) (or HV0(UX, E)) and HW(UY, F) (or HW0(UY, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ϕ : UY → UX and ψ : UY → B(E, F) which generate weighted composition operators between these weighted spaces.

Extension of vector-valued holomorphic and harmonic functions

We present a unified approach to the study of extensions of vector-valued holomorphic or harmonic functions based on the existence of weak or weak$^*$-holomorphic or harmonic extensions. Several recent results due to Arendt, Nikolski, Bierstedt, Holtmanns

Extension of vector-valued holomorphic and meromorphic functions

We present a unified approach to the study of extensions of vector-valued holomorphic or harmonic functions based on the existence of weak or weak ∗ -holomorphic or harmonic extensions. Several recent results due to Arendt, Nikolski, Bierstedt, Holt- manns and Grosse-Erdmann are extended. An open problem by Grosse-Erdmann is solved in the negative. Using the extension results we prove existence of Wolff type representa- tions for the duals of certain function spaces.

Preduals of Spaces of Vector-Valued Holomorphic Functions

For U a balanced open subset of a Frechet space E and F a dual-Banach space we introduce the topology τγ on the space \(H\left( {U,F} \right)\) of holomorphic functions from U into F. This topology allows us to construct a predual for \(\left( {H\left( {U,F} \right),\tau \delta } \right)\) which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions.

Meromorphic Functions on a Riemann Surface to a Locally Semi-convex Space

In this article, vector-valued holomorphic and meromorphic functions on a Riemann surface to a complete Hausdorff locally semi-convex space are discussed. By introducing the concepts of vector-valued holomorphic and meromorphic differential forms, Cauchy's theorem and the Residue theorem of a vector-valued differential form on a Riemann surface are proved. Using the theory on the operator and the theory of a cohomology of a sheaf, we give a proof of the Mittag-Leffler theorem for vector-valued meromorphic functions on a non-compact Riemann surface to a complete Hausdorff locally semi-convex space.