A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications (original) (raw)

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

Given a self-adjoint operator A:D(A)⊆→ and a continuous linear operator τ:D(A)→ with Range τ'∩' =0, a Banach space, we explicitly construct a family A^τ_Θ of self-adjoint operators such that any A^τ_Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭn-like formula where the role of the deficiency spaces is played by the dual pair (,'); the parameter Θ belongs to the space of symmetric operators from ' to . When = one recovers the "_-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 =L^2(^n) 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.