Andrea Posilicano
A Krein-like formula for singular perturbations of self-adjoint
operators and applications
(79K, AMS-TeX)
ABSTRACT. Given a self-adjoint operator $A:D(A)\subset H\to H$ and a continuous
linear operator $\tau:D(A)\to X$ with Range$\tau'\cap H'=\{0\}$, $X$ a
Banach space, we explicitly construct a family $A^\tau_\Theta$ of
self-adjoint operators such that any $A^\tau_\Theta$ coincides with the
original $A$ on the kernel of $\tau$. Such a family is obtained
by giving a Krein-like formula where the role of the deficiency
spaces is played by the dual pair $(X,X')$. The parameter
$\Theta$ belongs to the space of symmetric operators from $X'$ to
$X$. In the case $X$ is one dimensional one recovers the
``$H_{-2}$-construction'' of Kiselev and Simon and so, to some extent,
our results can be considered as an extension
of it to the infinite rank case. Various applications to singular
perturbations of non necessarily elliptic pseudo-differential operators are given,
thus unifying and extending previously known results.