00-215 Andrea Posilicano

A Krein-like formula for singular perturbations of self-adjoint operators and applications (79K, AMS-TeX) May 9, 00

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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.

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