Eigenvalues of seminormal operators, examples

Richard Carey and Joel David Pincus

Vol. 52 (1974), No. 2, 347–357

DOI: 10.2140/pjm.1974.52.347

Abstract

If τ∗ is completely hyponormal and [T,T^∗] has one dimensional range, a necessary and sufficient condition for a point z to belong to the point spectrum of T is known. Using this criterion two examples are constructed.

In the first example the point spectrum of T is empty, in the second example the spectrum of T is nowhere dense but almost every point of it is an eigenvalue.

The construction of both examples uses results about trigonometric series and t he so-called principal function map T ↔ g which associates with every bounded operator T with TT^∗− T^∗T ≡ 2∕πC trace class a Lebesgue summable function g(ν,μ) defined on σ(T), the spectrum of T.

Mathematical Subject Classification 2000

Primary: 47B20

Milestones

Received: 7 August 1973

Published: 1 June 1974