On the weighted geometric mean of accretive matrices
Tóm tắt
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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