Vol. 41, No. 1, 1972

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A class of operators on Hilbert space

Glenn Richard Luecke

Vol. 41 (1972), No. 1, 153–156

If T is an operator (bounded endormorphism) on the complex Hilbert space H, then T ∈ℛ if and only if (T zI)1= 1∕d(z,W(T)) for all z Cl W(T), where Cl W(T) is the closure of the numerical range of T and d(z,W(T)) = inf{|z u| : u W(T)}. The main results of this paper are: (1) T ∈ℛ if and only if the boundary of the numerical range of T is a subset of σ(T), the spectrum of T; and (2) is an arc-wise connected, closed nowhere dense subset of the set of all operators on H (norm topology) when dimH 2.

Mathematical Subject Classification 2000
Primary: 47B20
Received: 23 October 1970
Published: 1 April 1972
Glenn Richard Luecke