A Krein-like formula for singular perturbations of self-adjoint operators and applications

Given a self-adjoint operator A:D(A) subset of or equal to H --> H and a continuous linear operator tau: D(A) --> X with Range tau ' boolean AND H ' = {0}, H a Banach space, we explicitly construct a family A(Theta)(tau) of self-adjoint operators such that any A(Theta)(tau) 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. When X = C one recovers the "H-2-construction" 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 = L-2(R-n) and tau 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. (C) 2001 Academic Press.