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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 z∉σ(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.
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