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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Operators from Banach spaces to complex interpolation spaces


Author: Vernon Williams
Journal: Proc. Amer. Math. Soc. 26 (1970), 248-254
MSC: Primary 47.30; Secondary 46.00
MathSciNet review: 0265971
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Abstract: Given a closed linear operator $ A$ with dense domain in a Banach space $ X$, M. Schechter [4] utilized the Lebesgue integral to construct a family of bounded linear operators from $ X$ to the Calderón complex interpolation space $ {(X,D(A))_8}$ [2], where $ D(A)$, the domain of $ A$ in $ X$, is a Banach space under the norm

$\displaystyle \vert\vert x\vert{\vert _{D(A)}} = \vert\vert x\vert\vert + \vert\vert Ax\vert\vert.$

In this paper we utilize the complex functional calculus, which provides a more natural setting, to construct a similar family of operators. At the same time we achieve a strengthening of the Schechter result, for in the proof of our theorem we make no use of the adjoint $ {A^ \ast }$ of $ A$ and consequently do not require the domain of $ A$ to be dense in $ X$. A completely analogous procedure would permit the removal from the Schechter theorem, referred to above, of the hypothesis that the domain of $ A$ is dense in $ X$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0265971-6
PII: S 0002-9939(1970)0265971-6
Keywords: Calderón complex interpolation space, functional calculus, closed linear operator with spectrum contained in a Cauchy domain
Article copyright: © Copyright 1970 American Mathematical Society