Symmetric Operators on Hilbert C^*-Modules
by
Dan Popovici
Department of Mathematics, University of the West of Timisoara, B-dul V.Pârvan nr. 4, 1900 Timisoara, ROMANIA
Coauthors: Adriana Popovici (Department of Mathematics, University of the West Timisoara, B-dul V.Pârvan nr. 4, 1900 Timisoara, ROMANIA)

A Hilbert module obeys the same axioms as an ordinary Hilbert space except that the inner product, from which the geometry emerges, takes values in a more general C^*-algebra A then C. The study of these objects became important in many fields of mathematics as prediction theory, KK-theory, induced representation theory and Morita equivalence, index theory for operator-valued conditional expectations or Tomita-Takesaki theory for AW^*-algebras. The generalization of the unbounded operator notion has been pointed out by S.L.Woronowicz and K.Napiórkovsky in the early nineties to be useful in the C^*-algebraic approach to quantum group theory.

One of the most important well-known properties of a densely defined Hilbert space operator T with densely defined adjoint T^* is the boundedness of (I+T^*T)^-1. In order to carry out this condition in our more general framework we must introduce the so-called regular operators. The properties of a symmetric and regular operator on a Hilbert C^*-module can be deduced by using the more amenable (bounded operator with adjoint) Cayley transform. There is a bijective correspondence between regular symmetric operators and their Cayley transforms as partial isometries c such that (I-c)c^* has dense range. We finally obtain some sufficient conditions on a regular symmetric operator in order to posses a self-adjoint extension. One can prove that a regular symmetric operator T has a self-adjoint extension iff ker(T^*+iI) and ker(T^*-iI) are equivalent.

http://www.math.uvt.ro/~popovici

Date received: February 25, 2000