Meromorphic resolvents and power bounded operators (Q1923874)

The aim of the paper is to derive power boundedness \[ \| A^n\|\leq C, \hbox{ for all } n\geq 0 \] from the resolvent condition \[ \|(I-zA)^{-1}\|\leq {K\over(1-|z|)} \hbox { for } |z|<1 \] of a bounded linear operator \(A\) in a complex separable Hilbert space. The main result of the paper concerns an operator \(A\) which can be decomposed as \(A=B+N\) with \(N\) nuclear and \(B\) bounded, \(\| B\|\leq\eta\). The author proves that for any \(\eta\) there exists a number \(c(\eta)<\infty\) such that \[ C\leq c(\eta)(\| N\|_1+1)K. \] The proof uses results on rational approximation of meromorphic functions.