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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0265971-6
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$.


References [Enhancements On Off] (What's this?)

  • [1] A. P. Calderón, Intermediate spaces and interpolation, Studia Math. Special Series 1 (1963), 31-34. MR 26 #5409. MR 0147896 (26:5409)
  • [2] -, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. MR 29 #5097. MR 0167830 (29:5097)
  • [3] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958, pp. 601-602. MR 22 #8302. MR 0117523 (22:8302)
  • [4] M. Schechter, Complex interpolation, Compositio Math. 18 (1967), 117-147. MR 36 #6927. MR 0223880 (36:6927)

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Additional Information

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

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