Inhaltszusammenfassung

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 infinit...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» weiterlesen» einklappen

Autoren

Klassifikation

DDC Sachgruppe:
Mathematik