Unbounded operators on Hilbert C*-modules: graph regular operators

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sightings 1. Unitary *-module spaces Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13 a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13 b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15 c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16 2. Operators on unitary *-module spaces Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. Graph regularity Pragmatism between weak and (strong) regularity . . . . . . . . . 27 a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 b) The case C(X) . . . . . . . . . . . . . . . . . . . . .