The orbit of a bounded operator under the M\"{o}bius group modulo similarity equivalence

DOI10.1007/S11856-020-2016-XarXiv1811.05428MaRDI QIDQ6309661

Author name not available (Why is that?)

Publication date: 13 November 2018

Abstract: Let M"{o}b denote the group of biholomorphic automorphisms of the unit disc and

be the orbit of a Hilbert space operator

T

under the action of M"{o}b. If the quotient

, where

s i m

is the similarity between two operators is a singleton, then the operator

T

is said to be weakly homogeneous. In this paper, we obtain a criterion to determine if the operator

M_{z}

of multiplication by the coordinate function

z

on a reproducing kernel Hilbert space is weakly homogeneous. We use this to show that there exists a M"{o}bius bounded weakly homogeneous operator which is not similar to any homogeneous operator, answering a question of Bagchi and Misra in the negative. Some necessary conditions for the M"{o}bius boundedness of a weighted shift are also obtained. As a consequence, it is shown that the Dirichlet shift is not M"{o}bius bounded.

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