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 | References | Similar Articles | Additional Information
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 \[ ||x|{|_{D(A)}} = ||x|| + ||Ax||.\] 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 $X$.
- A.-P. Calderón, Intermediate spaces and interpolation, Studia Math. (Ser. Specjalna) Zeszyt 1 (1963), 31–34. MR 0147896
- A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI https://doi.org/10.4064/sm-24-2-113-190
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Martin Schechter, Complex interpolation, Compositio Math. 18 (1967), 117–147 (1967). MR 223880
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Additional Information
Keywords:
Calderón complex interpolation space,
functional calculus,
closed linear operator with spectrum contained in a Cauchy domain
Article copyright:
© Copyright 1970
American Mathematical Society