Vol. 38, No. 1, 1971

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Classifying special operators by means of subsets associated with the numerical range

Mary Rodriguez Embry

Vol. 38 (1971), No. 1, 61–65

Let A be a continuous linear operator on a complex Hilbert space X, with inner product <,> and associated norm ∥∥. For each complex number z let Mz(A) = {x : Ax,x= zx2}. The following classifications of special operators are obtained: (i) A is a scalar multiple of an isometry if and only if AMz(A) Mz(A) for each complex z; (ii) A is a nonzero scalar multiple of a unitary operator if and only if AMz(A) = Mz(A) for each complex z; and (iii) A is normal if and only if for each complex z{x|Ax Mz(A)} = {x|Ax Mz(A)}.

Mathematical Subject Classification 2000
Primary: 47B15
Received: 16 June 1970
Revised: 5 October 1970
Published: 1 July 1971
Mary Rodriguez Embry