Vol. 80, No. 2, 1979

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Commutants and the operator equations AX = λXA

Carl Claudius Cowen

Vol. 80 (1979), No. 2, 337–340

Suppose A is a bounded operator on the Banach space such that A or A is one-to-one. In this note, we point out a relation between the commutant of A, the commutants of its powers, and operators which intertwine A and λA, where λ is a root of unity. A consequence of this relation is that the commutants of A and An are different if and only if there is an operator Y , not zero, that satisfies AY = λY A, where λn = 1, λ1. Combining this with Rosenblum’s theorem, we see that if the spectra of A and XA are disjoint, the commutant of A is the same as that of A2. We also use the theorem to give a counterexample to a conjecture of Deddens concerning intertwining analytic Toeplitz operators.

Mathematical Subject Classification 2000
Primary: 47A62
Secondary: 47B35
Received: 12 June 1978
Published: 1 February 1979
Carl Claudius Cowen