Adjoint for operators in Banach spaces

Gill, Tepper; Basu, Sudeshna; Zachary, Woodford; Steadman, V.

doi:10.1090/S0002-9939-03-07204-6

Adjoint for operators in Banach spaces
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by Tepper L. Gill, Sudeshna Basu, Woodford W. Zachary and V. Steadman PDF

Proc. Amer. Math. Soc. 132 (2004), 1429-1434 Request permission

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.

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