Spaces Of Holomorphic Functions Research Articles

Preduals of spaces of holomorphic functions and the approximation property

Let U be a balanced open subset of a Hausdorff complex locally convex space E. In this paper we represent the predual G(U) of the space of holomorphic functions as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of U. As an application we prove that if E is a Banach space, then it has the approximation property if and only if G(U) has the approximation property. We also give a new proof of a result due to Aron and Schottenloher stating that if E is a complex Banach space with the approximation property then the space of holomorphic functions H(U), with the compact-open topology, has the approximation property.

On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions

We consider Bergman spaces and variations of them on domains in one or several complex variables. For certain domains , we show that the generic function in these spaces is totally unbounded in and hence non-extendable. We also show that generically these functions do not belong – not even locally – to Bergman spaces of higher order. Finally, in certain domains , we give examples of bounded non-extendable holomorphic functions – a generic result in the spaces of holomorphic functions in whose derivatives of order extend continuously to .

Reducing subspaces of non-analytic Toeplitz operators on weighted Hardy and Dirichlet spaces of the bidisk

We characterize reducing subspaces of a class of non-analytic Toeplitz operators on Hilbert spaces of holomorphic functions on the bidisk including the Hardy space and the Dirichlet space of the bidisk. These results compliment a recent characterization of the reducing subspaces for this class of non-analytic Toeplitz operators on the Bergman space of the bidisk.

Spectra of weighted composition operators on abstract Hardy spaces

In the paper, the spectra of weighted composition operators on Hardy-type spaces of holomorphic functions on the unit disc of the complex plane are studied. The spectra of invertible operators induced by elliptic and parabolic automorphisms are described, for weighted composition operators acting on abstract Hardy spaces generated by Banach lattices. We also study spectra of weighted composition operators (not necessarily invertible) on Hardy–Lorentz spaces.

On the isomorphism classification of spaces of 2π-periodic holomorphic functions on the upper half-plane

In this paper we obtain isomorphism classes of spaces of 2π-periodic holomorphic functions on the upper half-plane endowed with Lp norm with respect to a bounded positive non-atomic measure.

Invariant Subspaces for Classical Operators on Weighted Spaces of Holomorphic Functions

We develop an approach to describe invariant subspaces of the integration operator on various scales of weighted spaces of holomorphic functions on the unit disk and the complex plane. It allows us to solve the problem for wide classes of Bergman, Bloch, Dirichlet, and Fock spaces, while all previous known results concern spaces defined by some weights of a special form. In addition, we also show that an analogous method works as well for the differentiation operator on weighted spaces of entire functions.

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.

On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderon–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

Decomposition of L p (∂D a ) space and boundary value of holomorphic functions

This paper deals with two topics mentioned in the title. First, it is proved that function f in L p (∂D a ) can be decomposed into a sum g + h, where D a is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H p (D a ) and $${H^p}\left( {\overline D _a^c} \right)$$ in the sense of L p (D a ), respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained.

English

Let G be a locally compact group and let 1 ≤ p < 1. Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space L p(G) in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given.

INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS AS GENERIC LOG-CONVEX WEIGHTS

Let ${\mathcal{H}}ol(B_{d})$ denote the space of holomorphic functions on the unit ball $B_{d}$ of $\mathbb{C}^{d}$, $d\geq 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in {\mathcal{H}}ol(B_{d})$ such that the standard integral means $M_{p}(f,r)$ and $w(r)$ are equivalent for any $p$ with $0&lt;p\leq \infty$. We also obtain similar results related to volume integral means.

Density of states for random contractions

We define a linear functional, the DOS functional, on spaces of holomorphic functions on the unit disk which is associated with random ergodic contraction operators on a Hilbert space, in analogy with the density of states functional for random self-adjoint operators. The DOS functional is shown to enjoy natural integral representations on the unit circle and on the unit disk. For random contractions with suitable finite volume approximations, the DOS functional is proven to be the almost sure infinite volume limit of the trace per unit volume of functions of the finite volume restrictions. Finally, in case the normalised counting measure of the spectrum of the finite volume restrictions converges in the infinite volume limit, the DOS functional is shown to admit an integral representation on the disk in terms of the limiting measure, despite the discrepancy between the spectra of non normal operators and their finite volume restrictions. Moreover, the integral representation of the DOS functional on the unit circle is related to the Borel transform of the limiting measure.

Hadamard Multipliers on Spaces of Holomorphic Functions

Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions. The co-induced topology via the representation theorems is discussed. As an application of the representation theorems the density of non-invertible multipliers is proved. Moreover, Euler differential operators are distinguished among all multipliers.

Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain [Formula: see text] and any connected complex manifold [Formula: see text], the space [Formula: see text] contains a dense holomorphic disc. Our second result states that [Formula: see text] is an Oka manifold if and only if for any Stein space [Formula: see text] there exists a dense entire curve in every path component of [Formula: see text]. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain [Formula: see text], any fixed-point-free automorphism of [Formula: see text] and any connected complex manifold [Formula: see text], there exists a universal map [Formula: see text]. We also characterize Oka manifolds by the existence of universal maps.

ON MULTIPLIERS BETWEEN SOME SPACES OF HOLOMORPHIC FUNCTIONS

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence$\ell ^{p}$spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on$\ell ^{p}$’,Sibirsk. Mat. Zh.7(1966), 146–158].

Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

We determine the boundedness and compactness of a large class of operators, mapping from general Banach spaces of holomorphic functions into a particular type of spaces of functions determined by the growth of the functions, or the growth of the functions derivatives. The results show that the boundedness and compactness of such intrinsic operators depends only on the behaviour on the point evaluation functionals. They also generalize previous similar results about several specific classes of operators, such as the multiplication, composition and integral operators.

Functions of Exponential Growth in a Half-Plane, Sets of Uniqueness, and the Müntz–Szász Problem for the Bergman Space

We introduce and study some new spaces of holomorphic functions on the right half-plane \(\mathcal {R}.\) In a previous work, S. Krantz, C. Stoppato, and the first named author formulated the Muntz–Szasz problem for the Bergman space, that is, the problem to characterize the sets of complex powers \(\{\zeta ^{\lambda _j-1}\}\) with \({\text {Re}}\lambda _j>0\) that form a complete set in the Bergman space \(A^2(\varDelta ),\) where \(\varDelta =\{\zeta {\text {:}}\, |\zeta -1|<1\}.\) In this paper, we construct a space of holomorphic functions on the right half-plane, that we denote by \(\mathcal {M}^2_\omega (\mathcal {R}),\) whose sets of uniqueness \(\{\lambda _j\}\) correspond exactly to the sets of powers \(\{\zeta ^{\lambda _j-1}\}\) that are a complete set in \(A^2(\varDelta ).\) We show that \(\mathcal {M}^2_\omega (\mathcal {R})\) is a reproducing kernel Hilbert space, and we prove a Paley–Wiener-type theorem and several other structural properties. We determine both a necessary and a sufficient condition on a set \(\{\lambda _j\}\) to be a set of uniqueness for \(\mathcal {M}^2_\omega (\mathcal {R}),\) thus providing a condition for the solution of the Muntz–Szasz problem for the Bergman space. Finally, we prove that the orthogonal projection is unbounded on \(L^p(\mathcal {R},\,\mathrm{d}\omega )\) for all \(p\ne 2.\)

Module tensor product of subnormal modules need not be subnormal

Let κ:D×D→C be a diagonal positive definite kernel and let Hκ denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc D. Assume that zf∈H whenever f∈H. Then H is a Hilbert module over the polynomial ring C[z] with module action p⋅f↦pf. We say that Hκ is a subnormal Hilbert module if the operator Mz of multiplication by the coordinate function z on Hκ is subnormal. N. Salinas (1988) [11] asked whether the module tensor product Hκ1⊗C[z]Hκ2 of subnormal Hilbert modules Hκ1 and Hκ2 is again subnormal. In this regard, we describe all subnormal module tensor products La2(D,ws1)⊗C[z]La2(D,ws2), where La2(D,ws) denotes the weighted Bergman Hilbert module with radial weightws(z)=1sπ|z|2(1−s)s (z∈D, s>0). In particular, the module tensor product La2(D,ws)⊗C[z]La2(D,ws) is never subnormal for any s≥6. Thus the answer to this question is no.

Duality for large Bergman-Orlicz spaces and Hankel operators between Bergman-Orlicz spaces on the unit ball

For , the unit ball of , we consider Bergman–Orlicz spaces of holomorphic functions in , which are generalizations of classical Bergman spaces. We characterize the dual space of large Bergman–Orlicz space, and bounded Hankel operators between some Bergman–Orlicz spaces and , where and are either convex or concave growth functions.

Determinantal Point Processes Associated with Hilbert Spaces of Holomorphic Functions

We study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singular. In the second case, we show that all reduced Palm measures, $\textit{of all orders}$, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.