Annals of Functional Analysis

On compactness in‎ ‎complex interpolation

Jürgen Voigt

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‎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

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

First available in Project Euclid: 12 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Complex interpolation ‎compact operator function


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

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