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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
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caet-02.