Spectral theory of finite rank perturbations of normal operators

Let $N$ be a bounded normal operator on a separable Hilbert space
and let $\mu$ stand for its scalar spectral measure. The spectral
nature of the perturbation $T=N+K$, where $K$ is  a sufficiently
``smooth'' compact  operator, will be discussed.  We are
interested in the existence of invariant subspaces,
decomposability and other questions.


We introduce the perturbation operator-valued function of $T$,
defined in the whole complex plane, except for a certain thin set,
and explain its role. We discuss the dependence of the answers on
geometric properties of $\mu$.


The case when $\mu$ is absolutely continuous with respect to the
area measure has been considered in \cite{Yakubovich-1} in 1993;
this is the case of $\mu$ of ``dimension'' two. A quotient model
for $T$, constructed in terms of certain vector-valued Sobolev
classes of functions, was established in this work. The case of a
discrete measure $\mu$ (that is, of a diagonalizable operator $N$)
has been studied more recently in a series of papers by C. Foias
and his coauthors (see \cite{Foias-1}). This can be seen as the
case of zero dimension.
In this work, we extend the model from \cite{Yakubovich-1} to a wide class of measures, which we
call dissectible.
The case when the ``dimension'' $\mu$ is close to one 
(that is, when $\mu$ behaves roughly like the arc length measure)
is not covered by our techniques.

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The talk is based on the joint work \cite{Putinar-Yak} of the speaker with Mihai Putinar.