On normaloid operators
I. H. Sheth · Pacific Journal of Mathematics · 1969
The purpose of the present paper is to extend an earlier theorem of the author's on hyponormal operators to the following, on normaloid operators. THEOREM.Let N be an operator such that Nzl is normaloid for all complex values of z.If AN = N*A, for an arbitrary operator A, for which OgCl(TF(A)), then N= N*.2* Notations.We consider bounded linear operators defined on a Hubert space H.As usual, the symbols s(T), Σ(T), W(T) and Cl(W(T)) stand for the spectrum of an operator T, the closed convex hull of s(T), the numerical range of T and the closure of W(T) respectively.An operator T is said to be normaloid if || T || = sup {| z |; z e s(T)} and hyponormal, if Γ*T -TT* ^ 0. It is known that if T is hyponormal, then T is normaloid and T -zl is also hyponormal for all complex numbers z.When the original version of this paper was submitted, the referee told me of [3] then existing as a preprint and this makes possible the following shorter proof.Proof of Theorem.Since AN = JV*A and 0 g Cl (W(A)), s(N) is real [3].Also Σ(N) = Cl (W(N)) for such a normaloid operator N [1].Hence Cl(T^(iV)) is real.This completes the proof of theorem.The corresponding result for hyponormal operators now follows as corollary from this theorem and the remark made above.