Certain class of unconditional bases in Hilbert space and its applications to functional model and scattering theory (Q756081)

Let H be a separable infinite dimensional Hilbert space and \(\{g_ n\}^{\infty}_{n=1}\) some Schauder basis for H. If there is an invertible bounded linear operator U on H for which \(\{Ug_ n\}^{\infty}_{n=1}\) is an orthonormal basis for H then U is called an orthogonalizer for \(\{g_ n\}^{\infty}_{n=1}\). The basis \(\{g_ n\}^{\infty}_{n=1}\) is called a \(V_ p\)-basis (for some \(0<p\leq \infty)\) if it has an orthogonalizer U of the form \(U=I+V\), where V is a compact operator in the Schatten trace-class \(C_ p\). Bases of this type were inroduced by \textit{V. A. Prigorskij} [Usp. Math. Nauk 20, No.5(125), 231-236 (1965; Zbl 0151.177)] and rediscovered by the author in the course of earlier investigations of scattering of acoustical waves. In the current paper characterizations and properties of \(V_ p\)-bases developed in Section 1 are applied in succeeding sections to obtaining a generalization of the Carleson-Bari criterion for \(V_ 2\)-basicity of the eigenfunctions of the restriction of the adjoint of the shift operator on the upper half-plane, and to the system of resonance states for a problem of scattering of acoustical waves by spherically symmetric inhomogeneities of the density of the medium.