10-64 Tepper L. Gill, Francis Mensah and Woodford W. Zachary

Adjoint Operators on Banach Spaces (91K, TeX) Apr 20, 10

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Abstract. In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications provide an extension of the Poincar\'{e} inequality and the Stone-von Neumann version of the spectral theorem for a large class of $C_0$-generators of contraction semigroups on separable Banach spaces. Our third application provides a natural extension of the Schatten-class of operators to all separable Banach spaces. As a part of this program, we introduce a new class of separable Banach spaces. In another direction, these spaces provide a natural framework for the (rigorous) construction of the path integral as envisioned by Feynman.

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