Abstract
Given a self-adjoint operator A:D(A)โHโH and a continuous linear operator ฯ:D(A)โX with Range ฯโฒโฉHโฒ={0}, X a Banach space, we explicitly construct a family Aฯฮ of self-adjoint operators such that any Aฯฮ coincides with the original A on the kernel of ฯ. Such a family is obtained by giving a Kreฤฑn-like formula where the role of the deficiency spaces is played by the dual pair (X, Xโฒ); the parameter ฮ belongs to the space of symmetric operators from Xโฒ to X. When X=C one recovers the โHโ2-constructionโ of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which H=L2(Rn) and ฯ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.
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