Journal of Operator Theory

Volume 57, Issue 2, Spring 2007 pp. 409-427.

Interpolation classes and matrix monotone functions

Authors: Yacin Ameur (1), Sten Kaijser (2), and Sergei Silvestrov (3)
Author institution: (1) Royal Institute of Technology, SE-100 44 Stockholm, Sweden
(2) Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden
(3) Department of Mathematics, Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden

Summary: An interpolation function of order $n$ is a positive function $f$ on $(0,\infty)$ such that $\|f(A)^{1/2}Tf(A)^{-1/2}\|\leqslant\max(\|T\|,\|A^{1/2}TA^{-1/2}\|)$ for all $n\times n$ matrices $T$ and $A$ such that $A$ is positive definite. By a theorem of Donoghue, the class $C_n$ of interpolation functions of order $n$ coincides with the class of functions $f$ such that for each $n$-subset $S=\{\lambda_i\}_{i=1}^n$ of $(0, \infty)$ there exists a positive Pick function $h$ on $(0,\infty)$ interpolating $f$ at $S$. This note comprises a study of the classes $C_n$ and their relations to matrix monotone functions of finite order. We also consider interpolation functions on general unital $C^*$-algebras.

Keywords: Interpolation function, matrix monotone function, Pick function

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