Vol. 18, No. 3, 1966

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Conditions implying normality in Hilbert space

Mary Rodriguez Embry

Vol. 18 (1966), No. 3, 457–460

The problem with which this paper is concerned is that of finding new conditions which imply the normality of an operator on a complete inner product space S. Each such condition, presented in this paper, involves the commutativity of certain operators, associated with a given operator A. Theorem 1 states the equivalence of the following conditions: (i) A is normal, (ii) each of AA and AA commutes with Re A, (iii) AA commutes with Re A and AA commutes with Im A. Theorem 2 states that A is normal if AA and AA commute and Re A is nonnegative definite. Finally, Theorem 3 states that if AA commutes with each of AA and Re A, then AA commutes with A. In this case, if A is reversible, then A is normal.

Mathematical Subject Classification
Primary: 47.40
Received: 4 June 1965
Published: 1 September 1966
Mary Rodriguez Embry