Essentially hyponormal operators with essential spectrum contained in a circle

Abstract

In this paper two results are given . It is proved that if the essential spectrum σ(π(T)) of the bounded hyponormal operator T is contained in a circle, then T is essentially normal operator. Based on this result it is proved that if T∈ L(H) with ind T = 0 then T = λU + K (where λ ∈ R^+, U is a unitary operator and K is a compact operator) if and only if TT^∗ is quasi-diagonal with respect to any sequence {P_n } in PF(H) such that Pn → I, strongly.

Keywords

• Essentiall spectrum
• Quasidiagonal operator