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Adjoint for operators in Banach spaces
Authors:Tepper L. Gill   Sudeshna Basu   Woodford W. Zachary   V. Steadman
Affiliation:Department of Electrical Engineering, Howard University, Washington, DC 20059 ; Department of Mathematics, Howard University, Washington, DC 20059 ; Department of Electrical Engineering, Howard University, Washington, DC 20059 ; Department of Mathematics, University of the District of Columbia, Washington, DC 20058
Abstract:In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

Keywords:Adjoints   Banach space embeddings   Hilbert spaces
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