Spectral analysis of finite section normal integral operators

The Finite Section Method for Dissipative Operators

We show that for self-adjoint Jacobi matrices and Schrödinger operators, peturbed by dissipative potentials in l(N) and L(0,∞) respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for a.c. spectrum. This is the a...

Inverse spectral analysis for finite matrix-valued Jacobi operators

Consider the Jacobi operators J given by (J y)n = anyn+1+bnyn+a∗n−1yn−1, yn ∈ C (here y0 = yp+1 = 0), where bn = b ∗ n and an : det an 6= 0 are the sequences of m × m matrices, n = 1, .., p. We study two cases: (i) an = a∗n > 0; (ii) an is a lower triangular matrix with real positive entries on the diagonal (the matrix J is (2m+1)-band mp×mp matrix with positive entries on the first and the las...

Integral Formulas for Section Determinants of Bounded Operators

Diagonals of normal operators with finite spectrum.

Let X={lambda1, ..., lambdaN} be a finite set of complex numbers, and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e1, e2, ... for H, A gives rise to a matrix whose diagonal is a sequence d=(d1, d2, ...) with the property that each of its terms dn belongs to the convex hull of X. Not all sequences with that property c...

Spectral Theorem for Normal Operators: Applications

In these notes, we present a number of interesting and diverse applications of the Spectral Theorem for Normal Operators. These include a Spectral Mapping Theorem for Normals Operators, a Spectral Characterization of Algebraic Operators, von Neumann’s Mean Ergodic Theorem, Pathconnectivity of the Group of Invertible Operators, and some Polynomial Approximation Results for Operators.