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
A has the domain
which is dense in L2(0,1). One easily verifies that the imaginary part (2i)−1(A−A∗)
extends to the bounded operator f → 1∕2⟨f,c⟩c. 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.
