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On the weighted geometric mean of accretive matrices

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Abstract

In this paper, we discuss new inequalities for accretive matrices through non-standard domains. In particular, we present several relations for \(A^r\) and \(A\sharp _rB\), when A, B are accretive and \(r\in (-1,0)\cup (1,2).\) This complements the well-established discussion of such quantities for accretive matrices when \(r\in [0,1],\) and provides accretive versions of known results for positive matrices. Among many other results, we show that the accretive matrices A, B satisfy

$$\begin{aligned} \mathfrak {R}(A\sharp _rB)\le \mathfrak {R}A\sharp _r \mathfrak {R}B, r\in (-1,0)\cup (1,2). \end{aligned}$$

This, and other results, gain their significance due to the fact that they are reversed when \(r\in (0,1).\)

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Authors and Affiliations

  1. Department of Mathematics, The University of Jordan, Amman, Jordan

    Yassine Bedrani & Fuad Kittaneh

  2. Department of Basic Sciences, Princess Sumaya University for Technology, Amman, 11941, Jordan

    Mohammed Sababheh

Authors
  1. Yassine Bedrani
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  2. Fuad Kittaneh
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  3. Mohammed Sababheh
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Correspondence to Mohammed Sababheh.

Additional information

Communicated by Jean-Christophe Bourin.

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Bedrani, Y., Kittaneh, F. & Sababheh, M. On the weighted geometric mean of accretive matrices. Ann. Funct. Anal. 12, 2 (2021). https://doi.org/10.1007/s43034-020-00094-6

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  • DOI: https://doi.org/10.1007/s43034-020-00094-6

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