Constructing Finite Frames of a given Spectrum and Quasianalytic Spectral Sets of Cyclic Contractions

Abstract

We show the equivalence relations, distances between Hilbert frames, ellipsoidal tight frames completions with prescribed norms and projection decompositions of operators. We characterize the generalization of Gram–Schmidt orthogonalization generating all Parseval frames and verify the Schur-Horn theorem for operators and frames. We study the spectra of contractions belonging to spectral classes and the hyperinvariant subspace problem for asymptotically nonvanishing contractions, with invariant subspaces for power-bounded operator of class C_1. We discuss the equal-norm Parseval frames and constructing finite frames of a given spectrum and set of lengths. We show the shift-type invariant subspaces of contractions quasianalytic contractions, function algebras and the compression of quasianalytic spectral sets of cyclic contractions.