Vol. 42, No. 2, 1972

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Complete non-selfadjointness of almost selfadjoint operators

Thomas Latimer Kriete, III

Vol. 42 (1972), No. 2, 413–437

Suppose that α is a real-valued measurable function defined on the unit interval [0,1] and that c is a function in the Lebesgue space L2(0,1). Let A be the (not necessarily bounded) operator on L2(0,1) associated with the pair (α,c) by

(Af )(x) = α (x)f(x)+ ic(x ) 0c(t)f (t)dt.

A has the domain

                   ∫ 1
𝒟 (A ) = {f ∈ L2(0,1) : |α(x)f(x )|2dx < ∞ }

which is dense in L2(0,1). One easily verifies that the imaginary part (2i)1(AA) extends to the bounded operator f 12f,cc. Thus A is almost selfadjoint in the sense that it differs from its reaI part by an operator of rank one. The operators A are more general than they appear. Livsic showed that every bounded operator B with real spectrum, no selfadjoint part, and with nonnegative imaginary part of rank one is unitarily equivalent to the completely non-selfadjoint part of such an operator A acting on L2(0,a) for some positive a. This raises the question of when (in terms of α and c) A is completely non-selfadjoint. The main result of this paper answers this question when the pair (α,c) is subject to a mild restriction that is always satisfied when A is bounded.

Mathematical Subject Classification 2000
Primary: 47A45
Received: 12 November 1970
Revised: 19 December 1971
Published: 1 August 1972
Thomas Latimer Kriete, III