Generalized Bebiano-Lemos-Providência inequalities and their reverses
Let \(A,B \geq 0\) be bounded operators acting on a Hilbert space. The Furuta inequality states that \[ A^{1+r}\geq\left(A^{\frac{r}{2}}B^pA^{\frac{r}{2}}\right)^{\frac{1+r}{p+r}} \] holds if \(A\geq B \geq 0\), \(r\geq 0\) and \(p \geq 1\) [cf.\ \textit{T.\,Furuta}, Proc.\ Am.\ Math.\ Soc.\ 101, No.\,1, 85--88 (1987; Zbl 0721.47023)]. Utilizing this inequality, the authors generalize the Bebiano--Lemos--Providência norm inequality [see \textit{N.\,Bebiano}, \textit{R.\,Lemos} and \textit{J.\,da Providência}, Linear Algebra Appl.\ 401, 159--172 (2005; Zbl 1076.15019)] \[ \left\| A^{\frac{1+t}{2}}B^tA^{\frac{1+t}{2}}\right\| \leq \left\| A^{\frac{1}{2}}\left(A^{\frac{s}{2}}B^sA^{\frac{s}{2}}\right)^{\frac{t}{s}} A^{\frac{1}{2}}\right\|,\quad s \geq t \geq 0, \] by proving the inequality \[ \left\| A^{\frac{1+s}{2}}B^{1+s}A^{\frac{1+s}{2}}\right\| ^{\frac{p+s}{p(1+s)}} \leq \left\| A^{\frac{1}{2}}\left(A^{\frac{s}{2}}B^{p+s}A^{\frac{s}{2}}\right)^{\frac{1}{p}} A^{\frac{1}{2}}\right\|,\quad p \geq 1,\quad s \geq 0. \] Under certain assumptions, the authors also investigate some reverse Bebiano--Lemos--Providên\-cia inequalities.
scientific article; zbMATH DE number 5190758