2009/06/30 by S. Hassi, Seppo Hassi, H. S. V. de Snoo +5
Computer Science · Mathematics · #47A05 #47A06 (Primary) #47A12 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math.FA #msc:47A05 #msc:47A06 #msc:47A12
paper · pdf · doi:10.48550/arxiv.0906.5406
arxiv created 2009/06/30 · openalex publication_date 2009/06/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a, not necessarily closed, linear relation in a Hilbert space \sH with a multivalued part \mul A. An operator B in \sH with \ran B⊥\mul A** is said to be an operator part of A when A=B \hplus (\0\× \mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of A. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation A is said to have a Cartesian decomposition if A=U+\I V, where U and V are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of A and the real and imaginary parts of A is investigated.