ACTA issues

## On power bounded operators with holomorphic eigenvectors. II

Maria F. Gamal'

Acta Sci. Math. (Szeged) 86:3-4(2020), 549-562
33/2020

 Abstract. In [uch] (among other results), M. Uchiyama gave necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [gam], 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 $2\leq N\leq \infty$, 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~$T^*$. DOI: 10.14232/actasm-020-283-1 AMS Subject Classification (1991): 47A05; 47B99, 47B32, 30H10 Keyword(s): power bounded operator, unilateral shift, similarity, quasisimilarity, quasiaffine transform, analytic family of eigenvectors received 3.3.2020, revised 1.10.2020, accepted 2.10.2020. (Registered under 33/2020.)