On the weighted geometric mean of accretive matrices

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 $$r\in (0,1).$$