Samuel multiplicity and the structure of semi-Fredholm operators
In this paper, the shift Samual multiplicity, denoted \(s_{mul}(T)\), and the backward shift Samual multiplicity, denoted \(bs_{mul}(T)\), are introduced for any semi-Fredholm operator \(T\) on a separable complex Hilbert space \(H\): \[ s_{mul}(T)=\text{ lim}_{t\to \infty} ({\text{dim}(H/T^k(H))/k)}, \] \[ bs_{mul}(T)=\text{ lim}_{t\to \infty} (\text{dim} (\text{ker}(T^k)))/k). \] These numerical invariants refining the Fredholm index can be regarded as stabilized dimensions of the cokernel \(H/T(H)\) and kernel \(\ker (T)\), respectively. A geometric interpretation of \(s_{mul}(T)\) and \(bs_{mul}(T)\) leads to a local version of Apostol's triangular representation theorem and to Gohberg's punctured neighborhood theorem. Banach space operators are also considered.
scientific article; zbMATH DE number 2093017
scientific article