Analytic equivalence and similarity of operators.
Two bounded linear operators \(T,R\) on a complex separable Hilbert space \(H\) are said to be analytically equivalent on an open set \(G\subset{\mathbb C}\) if there exist holomorphic maps \(A,B\) from \(G\) to the set of invertible operators such that \(T-\lambda I=A(\lambda)(R-\lambda I)B(\lambda)\) for all \(\lambda\in G\). If \(T\) and \(R\) are similar, then this can be achieved with constant \(A\) and \(B\). It then becomes natural to ask conditions under which analytic equivalence implies similarity. \textit{C. Apostol} [Oper. Theory Adv. Appl. 32, 15--35 (1988; Zbl 0655.47003)] conjectured that if \(G\supset \sigma_g(T)\cup \sigma_g(R)\), where \(\sigma_g\) denotes the generalised spectrum, then analytic equivalence of \(T\) and \(R\) on \(G\) implies similarity. In the paper under review, the author proves the conjecture in the special case where \(T\) is an isometry with \(\dim \ker(T^*)\) finite. One ingredient of the proof is the following result which is of independent interest: an operator \(R\) is similar to an isometry if and only if its eigenvalues, if any, are on the unit circle and there exist an operator \(K\) with closed range and finite-dimensional kernel and two constants \(M,m>0\) such that \(m\| Kx\| \leq\| R^nx\| \leq M\| x\| \) for all \(x\in H\) and \(n\geq 0\).
scientific article; zbMATH DE number 1890564
scientific article