Hyponormal Operators and Spectral Density

Abstract

Introduction.We say an operator T on a Hubert space H is hyponormal if || Tx || St I T*x || for xeH, or equivalently T*T-TT* = 0.In this paperwe will first examine some general properties of hyponormal operators.Then we restrictour interest to hyponormal operators with "thin" spectra.The importance ofthe topological nature ofthe spectrum is evident in our main result (Theorem 4) which states that a hyponormal operator whose spectrum lies on a smooth Jordan arc is normal.We continue with a general discussion of a certain growth condition on the resolvent which obtains for hyponormal operators.We conclude with a counterexample to a relation between hyponormal and subnormal operators.The reader is advised that additional facts about hyponormal operators may be found in [11].We shall denote the spectrum and the resolvent set of an operator by cr(T) and piT), respectively.The spectral radius Rsp(T) = sup {| z | : zeo(T)}.The numerical range = closure {z: z = ( Tx.x) \\ x \\ = 1} is designated by W(T).Throughout the paper the underlying vector space is always a separable Hubert space H. I. Lemma 1.If T is hyponormal and (T -zI)~i exists (as a bounded operator) then (T -zl)'1 is hyponormal.Proof.Since hyponormality is preserved under translation (see [11, Lemma 1]), we may assume z = 0. Thus T*T -TT* = 0 and hence 0 < T~1(T*T -TT*) T*~1 = T~iT*TT*~1 -I Now since A _ I implies A'1 g I we have / -T*T_1 T*_1 T St 0, and hence (T*_1 T_1 -T"1 T*_1) = T*_1(7 -T*T~l T*'1 T)T~l ^ 0 which completes the proof.Theorem 1.Let T be hyponormal with zepiT).Then || (T -ziylx\\^lldiz,aiT)) or, equivalently, || (T -z/)x || St d(z, o-(T)) where \ x | = 1, xeH and diz, <t(T)) = min { | z -w \ : w e oCT)} .