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Operator means and positivity of block operators

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Abstract

The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators. It is shown that for positive operators A, B, C and for every non-negative operator monotone function f on \( (0,\infty )\), the block matrix

$$\begin{aligned} \left( \begin{array}{cc} f(A) &{} f(C) \\ f(C) &{} f(B) \\ \end{array} \right) \end{aligned}$$

is positive if and only if \(C \le A!B\). In particular, if \(C \le A!B\) then

$$\begin{aligned} \left( \begin{array}{cc} A &{} C \\ C &{} B \\ \end{array} \right) , \end{aligned}$$

is positive.

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Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    Hamed Najafi

  2. Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, 91775, Mashhad, Iran

    Hamed Najafi

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  1. Hamed Najafi
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Correspondence to Hamed Najafi.

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Najafi, H. Operator means and positivity of block operators. Math. Z. 289, 445–454 (2018). https://doi.org/10.1007/s00209-017-1958-0

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  • DOI: https://doi.org/10.1007/s00209-017-1958-0

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