Similarity of analytic Toeplitz operators on the Bergman spaces

Abstract

In this paper we give a function theoretic similarity classification for Toeplitz operators on weighted Bergman spaces with symbol analytic on the closure of the unit disk.

Localization and the Toeplitz algebra on the Bergman space

“…Given what we know about Toeplitz operators (see, e.g., [1][2][3][4][5]7,9,12,15]), the C * -algebra T is certainly much better understood than C * (A s ). It is known, for example, that T coincides with its commutator ideal [11,8].…”

On quasi-similarity of multiplication operator on the weighted Bergman space in the unit ball

“…Next, Li (see [10]) in 2009 proved that multiplication operator M z n is similar to ⊕ n 1 M z on the weighted Bergman space. And then, Jiang and Zheng in [9] extended the main result in [8] to the weighted Bergman space. In 2011, Douglas and Kim in [6] investigated the reducing subspaces for an analytic multiplication operator M z n on the Bergman space A 2 α (A r ) of the annulus A r .…”

News from P.R.China

“…The question was answered in the affirmative (see [17] or [9]). Later, the question was also answered in the affirmative on many other analytic function Hilbert spaces, such as weighted Bergman spaces A 2 α (see [18]), Sobolev disk algebra R(D) (see [20] or [14]), and Dirichlet space D (see [19]). Recently in [11], Hou and Jiang proved that the question still holds on the weighted Hardy space of polynomial growth, which covers the weighted Bergman space, the weighted Dirichlet space, and many weighted Hardy spaces defined without measures.…”

“…An application of Douglas's question, together with the technology of strongly irreducible operator and K 0 -group, is the similarity of analytic Toeplitz operators (e.g. Theorem 1.1 in [18]), which is analogous to the result on Hardy space (see [3]).…”