Quasinilpotent operators and non-Euclidean metrics
For a quasinilpotent operator \(A\) on a Hilbert space \(\mathcal H\), denote \( k_{x}=\operatorname{lim\,sup}_{|z|\to 0} \frac{\log\|(A - z I)^{-1}x\|}{\log\|(A - z I)^{-1}\|} \), \(x\in \mathcal H\), \(x\neq 0\). The power set of this operator is defined by the equality \(\Lambda(A)=\{ k_{x}:x\in {\mathcal H}\), \(x\neq 0\}\). The properties of this set are closely connected with the lattice of \(A\)-hyperinvariant subspaces. In the paper under review, it is proved that, for the Volterra integral operator on the Hardy space \(H^{2}(\mathbb D)\), the equality \(\Lambda(A)=\{ 1\}\) holds true. In the second part of the paper, ther authors analyze the properties of some notion of the length of curves. Let \(\gamma = \{ z(t): 0\leq t \leq 1\}\) be a piece-wise smooth curve lying inside the resolvent set of a linear bounded operator \(A\) on \(\mathcal H\). The length of \(\gamma\) is defined by \(L_{x}(\gamma)=\int_{0}^{1}\| (A - z(t)I)^{-1}x\|\,|z'(t)|\,dt\), \(x\in \mathcal H\). The authors obtain estimates and extremal values of this object.
scientific article; zbMATH DE number 6951113