 00215 Andrea Posilicano
 A Kreinlike formula for singular perturbations of selfadjoint
operators and applications
(79K, AMSTeX)
May 9, 00

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Abstract. Given a selfadjoint 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
selfadjoint 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 Kreinlike 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 pseudodifferential operators are given,
thus unifying and extending previously known results.
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