Annals of Functional Analysis

On compactness in‎ ‎complex interpolation

Jürgen Voigt

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Abstract

‎We show that‎, ‎in complex interpolation‎, ‎an operator function that is compact‎ ‎on one side of the interpolation scale will be compact for all proper‎ ‎interpolating‎ ‎spaces if the right hand side $(Y^0,Y^1)$ is reduced to a single space‎. ‎A corresponding result‎, ‎in restricted generality‎, ‎is shown if the left hand side‎ ‎$(X^0,X^1)$ is reduced to a single space‎. ‎These results are derived from the fact that a holomorphic operator valued‎ ‎function on an open subset of $\mathbb{C}$ which‎ ‎takes values in the compact operators on part of the boundary is in fact compact‎ ‎operator valued‎.

Article information

Source
Ann. Funct. Anal., Volume 3, Number 1 (2012), 121-127.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900029

Digital Object Identifier
doi:10.15352/afa/1399900029

Mathematical Reviews number (MathSciNet)
MR2903273

Zentralblatt MATH identifier
1271.46025

Subjects
Primary: 46B70: Interpolation between normed linear spaces [See also 46M35]
Secondary: 47B07: Operators defined by compactness properties

Keywords
Complex interpolation ‎compact operator function

Citation

Voigt, Jürgen. On compactness in‎ ‎complex interpolation. Ann. Funct. Anal. 3 (2012), no. 1, 121--127. doi:10.15352/afa/1399900029. https://projecteuclid.org/euclid.afa/1399900029


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