Trace conditions for regular spectral behavior of vector-valued analytic functions
The authors consider the so-called logarithmic residue \[ \frac{1}{2\pi i} \int_\Delta F'(\lambda) F'(\lambda)^{-1} \] of a function \(F\) with values in a complex Banach algebra and \(\Delta\) a bounded Cauchy domain in \(\mathbb C\) such that \(F\) is invertible on \(\partial \Delta\). If \(F\) is invertible in every \(\lambda\in\Delta\), then the logarithmic residue is zero by Cauchy's theorem. A function is called spectrally regular if also the reverse holds, that is, if the vanishing of its logarithmic residue on every Cauchy domain where it makes sense, implies that \(F(\lambda)\) is invertible for every \(\lambda\) in the domain. The authors are interested in giving conditions when \(F\) is spectrally regular. To this end, they consider plain functions, that is, functions that are analytically equivalent to elementary polynomials, or more generally, to \(J\)-plain functions, where \(J\) is an ideal in the Banach algebra. As main results, the authors show that a \(J\)-plain function is spectrally regular if there exists a \(J\)-resolving family of traces.
scientific article; zbMATH DE number 5549253
scientific article