Abstract
In [U] (among other results), M. Uchiyama gave necessary and sufficient conditions for contractions to be similar to the unilateral shift S of multi-plicity 1 in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [Gam2], a cyclic power bounded operator is constructed which has the requested norm-estimates, is a quasiaffine transform of S, but is not quasisimilar to S. In this paper, a power bounded operator is constructed which has the requested norm-estimates, is quasisimilar to S, but is not similar to S. The question whether the criterion for contractions to be similar to S can be generalized to polynomially bounded operators remains open.
Also, for every cardinal number 2 ≤ N ≤ ∞, a power bounded operator T is constructed such that T is a quasiaffine transform of S and dim ker T* = N. This is impossible for polynomially bounded operators. Moreover, the constructed operators T have the requested norm-estimates of complete analytic families of eigenvectors of T*.
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Gamal’, M.F. On power bounded operators with holomorphic eigenvectors. II. ActaSci.Math. 86, 549–562 (2020). https://doi.org/10.14232/actasm-020-283-1
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DOI: https://doi.org/10.14232/actasm-020-283-1