Function Theoretic Properties Research Articles

Extended Cesàro companion operators on generalized Fock spaces

In this paper we give some characterizations of the bounded and compact extended Cesàro companion operators acting between the generalized Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain integral transforms, which are related to some function-theoretic properties of the symbols that induce the operators. As an application, we prove that there exist no nontrivial compact extended Cesàro companion integral and multiplication operators on generalized Fock spaces.

COMPACT AND HILBERT–SCHMIDT WEIGHTED COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES

AbstractLet u and $\varphi $ be two analytic functions on the unit disk D such that $\varphi (D) \subset D$ . A weighted composition operator $uC_{\varphi }$ induced by u and $\varphi $ is defined on $A^2_{\alpha }$ , the weighted Bergman space of D, by $uC_{\varphi }f := u \cdot f \circ \varphi $ for every $f \in A^2_{\alpha }$ . We obtain sufficient conditions for the compactness of $uC_{\varphi }$ in terms of function-theoretic properties of u and $\varphi $ . We also characterize when $uC_{\varphi }$ on $A^2_{\alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $\alpha $ when $\varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{\alpha }$ .

Some function theoretic properties of nonlinear resistive networks

We consider nonlinear resistive networks. The equivalence of the Liouville property, the Khas'minskii condition and the weak maximum principle for operators of Laplacian with potential is proved, and a number of criteria for these properties are given. The parabolicity of networks is also discussed.

Difference of quaternionic weighted composition operators on slice regular Fock spaces

Quaternionic Fock space is a useful generalization of the Fock space in the complex plane, which plays an important role in quantum mechanics. In view of quaternionic operator theory this topic attains more diversity and complexity. In this paper we first explore the connection between the properties and about the difference of quaternionic weighted composition operators acting on slice regular Fock space with the function theoretic properties of the symbols. It can further imply some topological information on the set of quaternionic weighted composition operators.

Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces

Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

Cyclic row contractions and rigidity of invariant subspaces

It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard d-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be “transformed” (in a precise technical sense) into compressions of the standard d-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multiplicities. We show that such decompositions face multivariate obstacles of an algebraic nature. Nevertheless, decompositions are obtained for nilpotent commuting row contractions by analyzing function theoretic rigidity properties of their invariant subspaces.

Function Theoretic Properties of Symmetric Powers of Complex Manifolds

In the paper we study properties of symmetric powers of complex manifolds. We investigate a number of function theoretic properties [e. g. (quasi) c-finite compactness, existence of peak functions] that are preserved by taking the symmetric power. The case of symmetric products of planar domains is studied in a more detailed way. In particular, a complete description of the Carathéodory and Kobayashi hyperbolicity and Kobayashi completeness in that class of domains is presented.

The Mittag Leffler reproducing kernel Hilbert spaces of entire and analytic functions

This paper investigates the function theoretic properties of two reproducing kernel functions based on the Mittag-Leffler function that are related through a composition. Both spaces provide one parameter generalizations of the traditional Bargmann–Fock space. In particular, the Mittag-Leffler space of entire functions yields many similar properties to the Bargmann–Fock space, and several results are demonstrated involving zero sets and growth rates. The second generalization, the Mittag-Leffler space of the slitted plane, is a reproducing kernel Hilbert space (RKHS) of functions for which Caputo fractional differentiation and multiplication by zq (for q>0) are densely defined adjoints of one another.

Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

We discuss the value distribution of Borel measurable functions which are subharmonic or meromorphic along leaves on laminations. They are called leafwise subharmonic functions or meromorphic functions respectively. We consider cases that each leaf is a negatively curved Riemannian manifold or Kahler manifold. We first consider the case when leaves are Riemannian with a harmonic measure in L.Garnett sense. We show some Liouville type theorem holds for leafwise subharmonic functions in this case. In the case of laminations whose leaves are Kahler manifolds with some curvature condition we consider the value distribution of leafwise meromorphic functions. If a lamination has an ergodic harmonic measure, a variant of defect relation in Nevanlinna theory is obtained for almost all leaves. It gives a bound of the number of omitted points by those functions. Consequently we have a Picard type theorem for leafwise meromorphic functions.

On the Spectrum of Volterra-Type Integral Operators on Fock–Sobolev Spaces

We determine the spectrum of the Voltterra-type integral operators \(V_g\) on the growth type Fock–Sobolev spaces \(\mathcal {F}_{\psi _m}^\infty \). We also characterized the bounded and compact spectral properties of the operators in terms of function-theoretic properties of the inducing map g. As a means to prove our main results, we first described the spaces in terms of Littlewood–Paley type formula which is interest of its own.

Restricted interpolation by meromorphic inner functions

AbstractMeromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

AbstractWe elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Generalized Volterra Companion Operators on Fock Spaces

We characterize the bounded and compact generalized Volterra companion integral operators acting between the standard Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain Berezin type integral transforms which involve some function theoretic properties of symbols inducing the operators. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces.

Topological Structure of the Space of Weighted Composition Operators Between Different Hardy Spaces

We consider properties related to weighted composition operators boundedly acting from the classical Hardy space H p to H q for $${1 \leq q < p < \infty}$$ . Especially, we shall completely determine path connected components in the set of weighted composition operators and explicitly characterize by function-theoretic properties of analytic self-maps.

Formula omitted]-Liouville theorems on complete smooth metric measure spaces

We study some function-theoretic properties on a complete smooth metric measure space (M,g,e−fdv) with Bakry–Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the f-heat equation, which leads to upper and lower Gaussian bounds on the f-heat kernel. We also prove Lp-Liouville theorems in terms of the lower bound of Bakry–Émery Ricci curvature and the bound of function f, which generalize the classical Ricci curvature case and the N-Bakry–Émery Ricci curvature case.

Weighted Composition Operators from Hardy to Zygmund Type Spaces

This paper aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator from the Hardy spaces to the Zygmund type spaces and the little Zygmund type spaces in terms of function theoretic properties of the symbols and .

Commutator of composition operators with adjoints of composition operators

We characterize the compactness of the linear fractionally induced commutator in terms of the function theoretic properties of ϕ and ψ. We show that in the automorphic case the commutator is compact if and only if ϕ and ψ are simple rotations of the unit disc. On the other hand, when one of the inducing maps is not an automorphism of the disc, we show that the commutator is non-trivially compact if and only if the inducing maps are both parabolic with the same boundary fixed point or they are both hyperbolic with the same boundary fixed point and their other fixed points are conjugate reciprocals.

On the Boundary Behaviour of Polymorphic Functions

Let Φ be a Fuchsian group acting on the unit disk D. This group may be infinitely generated and may have elliptic elements. A non-constant function f meromorphic in D is called Φ-polymorphic if, for every ϕ ∈Φ there is a Moebius transformation γ such that f ∘ φ = γ ∘ f. This induces a homomorphism f* of Φ into PSL(2, ℂ). The image group Г:= f *(Φ) need not be discrete. In this paper we study function-theoretic properties of a polymorphic function f and its relation to the limit sets and fixed points of Φ and the image group Г. This has some consequences for degenerate Kleinian groups.

Weighted composition operators from Bergman-type spaces into Bloch spaces

Let ϕ be an analytic self-map and u be a fixed analytic function on the open unit disk D in the complex plane ℂ. The weighted composition operator is defined by $$ uC_\phi f = u \cdot (f \circ \phi ), f \in H(D). $$ Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.

Some non-linear function theoretic properties of Riemannian manifolds

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p -Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.