Composition operators on Hilbert spaces of entire functions with analytic symbols

TL;DR

This paper investigates composition operators with analytic symbols on certain Hilbert spaces of entire functions, establishing conditions for boundedness and providing explicit models and formulas for these operators.

Contribution

It proves that bounded composition operators have affine symbols, develops a Fock-type model for linear symbols, and generalizes existing theorems to infinite-dimensional Segal-Bargmann spaces.

Findings

Bounded composition operators have affine symbols of degree at most 1.

Explicit formulas for polar decomposition, Aluthge transform, and powers of these operators.

Generalization of a theorem to infinite-dimensional Segal-Bargmann spaces.

Abstract

Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Carswell, MacCluer and Schuster is generalized to the case of Segal-Bargmann spaces of infinite order. Some related questions are also discussed.