A Krein-like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Given a self-adjoint operator A: D(A)⊆H→H and a continuous linear operator τ: D(A)→X with Range τ′∩H′={0}, X a Banach space, we explicitly construct a family AτΘ of self-adjoint operators such that any AτΘ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreın-like formula where the role of the deficiency spaces is played by the dual pair (X, X′); the parameter Θ 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=L2(Rn) and τ 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.