07-6 Tepper L. Gill and W. W. Zachary

The Linear Theory of S*-Algebras and Their Applications (8933K, pdf) Jan 4, 07

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Abstract. In this paper, we provide an introduction to the theory of isotopes in infinite dimensional spaces. Although we consider this to be an introduction, most of the results a new, and have never appeared in print. We restrict ourselves to Hilbert spaces and develop the linear theory, providing detailed proofs for all major results. After a few examples, in the first section we consider an isotope as a change in operator multiplication on the space of bounded linear operators over a fixed Hilbert space in the second section. The basic theory is developed leading to the notion of an S*-algebra (in honor of R. M. Santilli), which is a natural generalization of C*-algebras. The basic theory is then used in the third section to develop a complete theory of one-parameter linear isosemigroups of operators, which extend the theory of one-parameter semigroups of operators, which have played, and still play an important role in applied analysis. In the fourth section we apply our theory of iso-semigroups of operators to unify and simply two different approaches to the important class of Sobolev-Galpern equations. We close with a discussion of the general nonlinear case, where the operators may be nonlinear, singular and/or multivalued.

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