On curvature and similarity
Let \(B_ 1(\Omega)\) denote the class of bounded Hilbert space operators T such that \(T-\lambda I\) is onto and has one dimensional kernel for \(\lambda \in \Omega\) and such that \(\bigvee_{\lambda \in \Omega}\ker(T- \lambda I)\) is dense in the Hilbert space. Define the curvature \({\mathcal K}_ T\), for \(T\in B_ 1(\Omega)\), by \({\mathcal K}_ T(\lambda)=- \partial^ 2/\partial {\bar \lambda}\partial \lambda \log \| k_{\lambda}\|^ 2,\) where \(\{k_{\lambda}\}\) is an analytic determination of the eigenvectors of T. In [Acta Math. 141, 187-261 (1978; Zbl 0427.47016)], \textit{M. J. Cowen} and \textit{R. G. Douglas} conjectured that two operators \(S,T\in B_ 1({\mathbb{D}})\), \({\mathbb{D}}\) the unit disk, are similar if and only if \({\mathcal K}_ T(\lambda)/{\mathcal K}_ S(\lambda)\to 1\), as \(\lambda \to \lambda_ 0\in \partial {\mathbb{D}}\). In the present paper, a counterexample is given to the ''only if'' half of the conjecture. The example consists of a (function of a) restriction of a Toeplitz operator and uses the similarity theory for Toeplitz operators obtained by the first author [J. Oper. Theory 4, 37-54 (1980; Zbl 0468.47014)]. The appropriate curvature estimates also give k-spectral set information about te operator involved. \(\{\) In a later paper [London J. Math., to appear], the authors have found a counterexample to the ''if'' half of the above conjecture and some results on a revised condition for similarity.\(\}\)
similarity theory for Toeplitz operators
scientific article; zbMATH DE number 3866004