Spectral analysis of structured matrices with applications

Abstract

The thesis is aiming at several interrelated topics from spectral and operator theory with applications in other branches of mathematical analysis and physics. Its primary object is in spectral analysis of linear operators on Hilbert spaces that can be represented by in nite struc tured matrices. A signi cant part of the thesis is devoted to discrete operators of mathematicalphysics in the non-self-adjoint setting.Chapters from the rst part of this thesis are devoted to particular inequalities re ecting vari ous spectral properties of studied operators. Optimal bounds for discrete eigenvalues and spectral stability of one-dimensional discrete Schrödinger operators are analyzed. New proof for the op timal discrete Hardy inequality is found, new improved discrete Rellich inequality is discovered, and criticality as well as related Hardy-like inequalities for powers of the discrete Laplacian are further discussed. Open problems on Lieb Thirring inequalities for non-self-adjoint Jacobi and Schrödinger operators are solved. Inequality criteria for absence or existence of an eigenvalue at the bottom of the essential spectrum of lattice Schrödinger operators are established.Second part of the thesis is devoted to L-matrices, particularly, to comprehensive spectral analysis of the Hilbert L-matrix. Open problems concerning its operator norm are solved. Asymptotic spectral properties of the nite Hilbert L-matrix of large order are analyzed. As an applica tion, Wilf's asymptotic formula for the best constant in the Hardy inequality on nite dimensional spaces is improved