1999/11/11 by Palle E. T. Jorgensen
Mathematics · Physics and Astronomy · #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:47A05 #msc:47A66 #msc:47B15
paper · pdf · doi:10.1063/1.533242
published as J. Math. Phys. 41 (2000), 2337--2349. · 12 pages; REVTeX; PACS numbers 02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db
arxiv created 1999/11/11 · openalex publication_date 2000/04/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper we provide a quantitative comparison of two obstructions for a given symmetric operator S with dense domain in Hilbert space ℋ to be self-adjoint. The first one is the pair of deficiency spaces of von Neumann, and the second one is of more recent vintage; Let P be a projection in ℋ. We say that it is smooth relative to S if its range is contained in the domain of S. We say that smooth projections Pii=1∞ diagonalize S if (a) (I−Pi)SPi=0 for all i, and (b) supi Pi=I. If such projections exist, then S has a self-adjoint closure (i.e., S̄ has a spectral resolution), and so our second obstruction to self-adjointness is defined from smooth projections Pi with (I−Pi)SPi≠0. We prove results both in the case of a single operator S and a system of operators.