Results on the Spectral Schwartz Distribution

TL;DR

This paper introduces the concept of spectral Schwartz distributions as a generalization of the resolvent function, providing a spectral decomposition framework for various operators, including non-self-adjoint and perturbed cases.

Contribution

It defines spectral Schwartz distributions, extending resolvent theory to distributions, and computes these distributions for matrices, unitary, and non-self-adjoint operators.

Findings

Spectral Schwartz distribution generalizes the resolvent to distributions.

Provides spectral decomposition for non-self-adjoint and perturbed operators.

Calculates spectral distributions for specific operators like multiplication and rank-one perturbations.

Abstract

The resolvent of an operator in a Banach space is defined on an open subset of the complex plane and is holomorphic. It obeys the resolvent equation. A generalization of this equation to Schwartz distributions is defined and a Schwartz distribution, which satisfies that equation is called a resolvent distribution. Its restriction to the subset, where it is continuous, is the usual resolvent function. Its complex conjugate derivative is,but a factor, the spectral Schwartz distribution, which is carried by a subset of the spectral set of the operator. The spectral distribution yields a spectral decomposition. The spectral distribution of a matrix and a unitary operator are given. If the the operator is a self-adjoint operator on a Hilbert space, the spectral distribution is the derivative of the spectral family. We calculate the spectral distribution of the multiplication operator and some rank one perturbations. These operators are not necessarily self adjoint and may have discrete real or imaginary eigenvalues or a nontrivial Jordan decomposition.