Tensorial and Hadamard Product Inequalities for Synchronous Functions

Authors : Sever Dragomir

Pages : 177-187

Doi:10.33434/cams.1362694

View : 119 | Download : 79

Publication Date : 2023-12-25

Article Type : Research Paper

Abstract :Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\\left( A\\right) ,$ ${Sp}\\left( B\\right) \\subset I,$ then% \\begin{equation*} \\left( f\\left( A\\right) g\\left( A\\right) \\right) \\otimes 1+1\\otimes \\left( f\\left( B\\right) g\\left( B\\right) \\right) \\geq f\\left( A\\right) \\otimes g\\left( B\\right) +g\\left( A\\right) \\otimes f\\left( B\\right) \\end{equation*}% and the inequality for Hadamard product% \\begin{equation*} \\left( f\\left( A\\right) g\\left( A\\right) +f\\left( B\\right) g\\left( B\\right) \\right) \\circ 1\\geq f\\left( A\\right) \\circ g\\left( B\\right) +f\\left( B\\right) \\circ g\\left( A\\right) . \\end{equation*}% Let either $p,q\\in \\left( 0,\\infty \\right) $ or $p,q\\in \\left( -\\infty ,0\\right) $. If $A,$ $B>0,$ then \\begin{equation*} A^{p+q}\\otimes 1+1\\otimes B^{p+q}\\geq A^{p}\\otimes B^{q}+A^{q}\\otimes B^{p}, \\end{equation*}% and% \\begin{equation*} \\left( A^{p+q}+B^{p+q}\\right) \\circ 1\\geq A^{p}\\circ B^{q}+A^{q}\\circ B^{p}. \\end{equation*}
Keywords : Convex functions, Hadamard Product, Selfadjoint operators, Tensorial product