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Unbounded Operators on Hilbert -Modules
Let and be Hilbert -modules over a -algebra \CAlg{A}. New
classes of (possibly unbounded) operators are introduced and
investigated. Instead of the density of the domain \Def(t) we only assume
that is essentially defined, that is, \Def(t)^\bot=\{0\}. Then has a
well-defined adjoint. We call an essentially defined operator graph regular
if its graph \Graph(t) is orthogonally complemented in 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
(, a fraction algebra related to the Weyl algebra, Toeplitz algebra,
Heisenberg group). A new characterization of affiliated operators with a
-algebra in terms of resolvents is given
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