Tepper L. Gill and W. W. Zachary
The Linear Theory of S*-Algebras and Their Applications
(8933K, pdf)
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.