The spectra of compact operators in Hilbert spaces

Abstract

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

On properties of the operator equation TT*=T+T*

“…, then it is a compact and normal operator. However, σ(T) = { 1 2 , 1 3 } = σ(T * ) is not contained in the unit circle with centre 1, hence T S(C 2 ) from Theorem 4.2 of T. T. West (see [12]). On the other hand, T ∈ S(C n ) if and only if σ(T) lies on the unit circle with centre 1.…”

“…In [12], T. T. West observed that if T ∈ B(H) is compact and normal, then σ(T) is contained in the unit circle with centre at 1 if and only if the operator equation (1) holds. From this paper we are interested in what spectral properties hold for nonnormal solutions of the operator equation (1).…”