An inverse spectral theory for finite CMV matrices

Abstract

For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV operators with different "boundary conditions", and the problem of reconstructing a CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by truncation. Bibliography : 24 references.2000 Mathematics Subject Classification. Primary 15A29; Secondary 42C05, 15A57.

Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators

“…The Daepp et al proof is essentially a POPUC analog of a standard proof of Wendroff for OPRL. Our proof (below) is new, albeit close to the earlier OPUC proof of Golinskii-Kudryavtsev [35], and we feel illuminating.…”

“…(1) (proven earlier by Cantero et al [8]; see also [35]). If λ, µ ∈ ∂D are different, then the zeros of Φ n+1 (z; λ) and Φ n+1 (z; µ) strictly interlace.…”

Rational interpolation and mixed inverse spectral problem for finite CMV matrices

“…For the definitions, notations and basic properties of finite CMV matrices see, for example, [14,9,6]. We will add some more to the list.…”

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

“…These matrices have been used by researchers in various contexts, see e.g. [6][7][8]16,24,25,27,28,32].…”