Vol. 53, No. 2, 1974

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Linear operators for which TT and TT commute. II

Stephen LaVern Campbell

Vol. 53 (1974), No. 2, 355–361

Let (BN) denote the class of all bounded linear operators on a Hilbert space such that τ τ and TT commute. Let (BN)+ be those τ (BN) which are hyponorma]. Embry has observed that if T (BN), then 0 W(T) or T is normal. This is used to show that if τ (BN), then (T + λI)(BN) unless T is normal. It is also shown that if τ (BN)+, then Tn is hyponormal for n 1. An example of a T (BN)+ such that T2(BN) is given. Paranormality of operators in (BN) is shown to be equivalent to hyponormality. The relationship between T being in (BN) and T being centered is discussed. Finally, all 3 ×3 matrices in (BN) are characterized.

Mathematical Subject Classification 2000
Primary: 47B20
Received: 11 April 1973
Published: 1 August 1974
Stephen LaVern Campbell