Localization and Compactness in Bergman and Fock spaces

Joshua Isralowitz; Mishko Mitkovski; Brett D. Wick

Home/Archive/arXiv

arXiv · 1306.0316

Localization and Compactness in Bergman and Fock spaces

Joshua Isralowitz·Mishko Mitkovski·Brett D. Wick

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < \infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < \infty$ case.

Explore related subjects

math.CV math.FA

Keep this discovery

BibTeX RIS

Joshua Isralowitz, Mishko Mitkovski, Brett D. Wick. 2013-06-03. Localization and Compactness in Bergman and Fock spaces. https://arxiv.org/abs/1306.0316

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗

arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.

math.CV↗

arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗