Bounded, compact and Schatten class weighted composition operators on the Bergman space are characterized by the use of generalized Berezin transforms. An estimate of the essential norms of weighted composition operators on the Bergman space is given. Most of the results remain true for Hardy spaces and weighted Bergman spaces.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0024610704005605

Weighted Composition Operators on the Bergman Space

http://www.math.chalmers.se/Math/Grundutb/CTH/mve021/1112/conference/slides/Banach-2013/Lindstrom.pdf

Spectra of weighted composition operators with automorphic symbols

https://people.clas.ufl.edu/sam/files/zhao.pdf

New estimates of Essential norms of weighted composition operators between Bloch type spaces

In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear). 2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α (0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy (1)

Isometries of weighted Bergman spaces

In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in ℂ n with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space H∞.

Weighted Composition Operators from H∞ to the Bloch Space of a Bounded Homogeneous Domain

1.1. The isometries of the Hardy spaces Hp (0 2 was then done by Forelli [3]; Rudin [8] removed the restriction p > 2 by proving a theorem on equimeasurability. Finally, Koranyi and Vagi [6] noted that the methods developed by Forelli, Rudin and Schneider applied to bounded symmetric domains. In this note it will be shown that their methods also apply to the Bergman spaces over bounded Runge domains. The isometries which are onto are completely characterized; the special cases of the ball and polydisc are particularly nice and are given separately.

Clinton J. Kolaski

Papers

Isometries of weighted Bergman spaces

Isometries of Bergman Spaces over Bounded Runge Domains