The Absolute Continuity of Phase Operators

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Let H be a separable Hilbert space with orthonormal basis {φn}, V the unilateral shift operator on H so that Vφn = φn+1, and A be defined by Aφn = an-1φn with a0 = 0. Consider the operator C = 1/2(V*A + AV). If the sequence {an} converges monotonically to 1 and is so chosen that the spectrum of C is exactly the interval [-1,1], the operator C is called a phase operator. It was previously known that certain phase operators were absolutely continuous and that all phase operators had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous. (Received November 5, 1974.)


First published in Notices of the American Mathematical Society 22.1 (1975), published by the American Mathematical Society.

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