Unbounded Operators on Hilbert $C^*$-Modules

Abstract

Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra \CAlg{A}. New classes of (possibly unbounded) operators $t:E\to F$ 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 $E\oplus F$ 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 ($E=C_0(X)$, 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