Relations between two inequalities \((B^{\frac r2} A^p B^{\frac r2})^{\frac r{p+r}}\geq B^r\) and \(A^p\geq(A^{\frac p2} B^r A^{\frac p2})^{\frac p{p+r}}\) and their applications. (English) Zbl 1028.47013
Let \(A\) and \(B\) be positive bounded linear operators on a complex Hilbert space and let \(p\) and \(r\) be non-negative real numbers. In this paper, the authors give relations between the two inequalities \((B^{r/2}A^p B^{r/2})^{r/p+r}\geq B^r\) and \(A^p\geq(A^{p/2} B^rA^{p/2})^{p/p+r}\) as follows:
(i) If \((B^{r/2}A^pB^{r/2})^{r/p+r}\geq B^r\), then \(A^p\geq(A^{p/2} B^rA^{p /2})^{p/p+r}\).
(ii) If \(A^p\geq(A^{p/2}B^rA^{r/2})^{p/p+r}\) and \(N(A) \subset N(B)\), then \((B^{r/2}A^pB)^{r/p+r}\geq B^r\), where \(N(X)\) denotes the null space of the operator \(X\).
(i) If \((B^{r/2}A^pB^{r/2})^{r/p+r}\geq B^r\), then \(A^p\geq(A^{p/2} B^rA^{p /2})^{p/p+r}\).
(ii) If \(A^p\geq(A^{p/2}B^rA^{r/2})^{p/p+r}\) and \(N(A) \subset N(B)\), then \((B^{r/2}A^pB)^{r/p+r}\geq B^r\), where \(N(X)\) denotes the null space of the operator \(X\).
Reviewer: Omar Hirzallah (Zarqa)
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