2014/09/30 by René Gebhardt, Konrad Schmüdgen, Gebhardt, René +1 · 1 citation
Mathematics · #47L05 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 46L08 #Secondary 47D40 #math.FA #math.OA #msc:46L08 #msc:47D40 #msc:47L05
paper · pdf · doi:10.48550/arxiv.1409.8523
arxiv created 2015/07/08 · arxiv updated 2015/07/09
Let E and F be Hilbert C^*-modules over a C^*-algebra \CAlgA. New classes of (possibly unbounded) operators t:E→ F are introduced and investigated. Instead of the density of the domain \Def(t) we only assume that t is essentially defined, that is, \Def(t)^\bot=\0\. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph \Graph(t) is orthogonally complemented in E⊕ F and orthogonally closed if \Graph(t)\bot\bot=\Graph(t). A theory of these operators is developed. Various characterizations of graph regular operators are given. A number of examples of graph regular operators are presented (E=C0(X), a fraction algebra related to the Weyl algebra, Toeplitz algebra, Heisenberg group). A new characterization of affiliated operators with a C^*-algebra in terms of resolvents is given.