Vol. 23, No. 3, 1967

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Minimal range theorems for operators with thin spectra

Joseph Gail Stampfli

Vol. 23 (1967), No. 3, 601–612

Let T be a bounded linear operator on a Hilbert space H. Let W(T) = {(Tx,x) : x H and x= 1} denote the numerical range of T, and let Σ(T) designate the convex hull of σ(T), the spectrum of T. It is well known that for an arbitrary operator T,Σ(T) W(T). Moreover, if T is normal, then W(T) = Σ(T). In general, if W(T) = Σ(T), one can not expect T to be normal. However, if the spectrum of T is sufficiently thin, then relations of this sort do imply something about the operator.

First it is shown that, for operators with spectrum on certain “flat” convex curves, one can infer from the relations W(T±1) Σ(T±1) alone that T is normal. Examples are presented which show that this inference can not be made for arbitrary convex curves. However, the second result states that if σ(T) lies on a smooth convex curve, and

(a) W(T) Σ(T)

(b) W[(T zI)1] Σ[(T zI)1] for zσ(T), then T is normal.

Many conditions on T, short of normality, are known to imply (a) or (b), and corollaries are stated to cover these situations.

Mathematical Subject Classification
Primary: 47.30
Received: 17 October 1966
Published: 1 December 1967
Joseph Gail Stampfli