2000/05/09 by Andrea Posilicano, Posilicano, Andrea · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP
paper · pdf · doi:10.48550/arxiv.math/0005082
Proposition 2.1 revised. Remarks 2.15 and 2.16 added. 38 pages. To appear in Journal of Functional Analysis
openalex publication_date 2000/05/09 · arxiv created 2000/12/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a self-adjoint operator A:D(A)⊆\calH→\calH and a continuous linear operator τ:D(A)→\X with Range τ'∩\calH' =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 Kre\uın-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 ``\calH-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 \calH=L2(\REn) 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.