On typical properties of Hilbert space operators

TL;DR

This paper explores the common characteristics of bounded linear operators on infinite-dimensional Hilbert spaces across various topologies, focusing on spectral properties, unitary equivalence, and semigroup embeddability.

Contribution

It provides new insights into the typical spectral and structural properties of Hilbert space operators using Baire category methods.

Findings

Most operators have a dense spectrum in the complex plane.

Typical operators are not unitarily equivalent to each other.

A large class of operators can be embedded into C_0-semigroups.

Abstract

We study the typical behavior of bounded linear operators on infinite dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C_0-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.