On the difference of spectral projections
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Abstract
The aim of the present Ph.D. thesis is to investigate the relationship between differences of spectral projections and Hankel integral operators. This leads us to the following question: Is the difference of two spectral projections
$E_{(-infty, lambda)}(A+B)$ and $E_{(-infty, lambda)}(A)$ associated with an open interval $(-infty, lambda)$ unitarily equivalent to a Hankel integral operator, provided that $A$ and $B$ are self-adjoint operators on a complex separable Hilbert space of infinite dimension, where $A$ is semibounded and $B$ is of rank 1?
We show that, roughly speaking, the answer to this question is positive for all but at most countably many $lambda in R$. Further, we prove a similar result in the more general case when $B$ is compact.
The above question is motivated by the following classical example given by M. Krein: The difference of the resolvents of the Neumann and Dirichlet Laplacians on the semi-axis at the spectral point -1 is a rank one operator, but the difference of the spectral projections of these resolvents associated with $(-infty, lambda)$ is not even Hilbert Schmidt, for all $0