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.

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