Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result

Authors : Sever Dragomır

Pages : 56-70

Doi:10.33434/cams.1362711

View : 65 | Download : 76

Publication Date : 2024-03-04

Article Type : Research Paper

Abstract :Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\\leq A,$ $B\\leq M,$ for some constants $m,$ $M,$ then \\begin{align*} 0& \\leq \\frac{m}{M^{2}}\\nu \\left( 1-\\nu \\right) \\left( \\frac{A^{2}\\otimes 1+1\\otimes B^{2}}{2}-A\\otimes B\\right) \\\\ & \\leq \\left( 1-\\nu \\right) A\\otimes 1+\\nu 1\\otimes B-A^{1-\\nu }\\otimes B^{\\nu } \\\\ & \\leq \\frac{M}{m^{2}}\\nu \\left( 1-\\nu \\right) \\left( \\frac{A^{2}\\otimes 1+1\\otimes B^{2}}{2}-A\\otimes B\\right) \\end{align*} for all $\\nu \\in \\left[ 0,1\\right] .$ We also have the inequalities for Hadamard product \\begin{align*} 0& \\leq \\frac{m}{M^{2}}\\nu \\left( 1-\\nu \\right) \\left( \\frac{A^{2}+B^{2}}{2}% \\circ 1-A\\circ B\\right) \\\\ & \\leq \\left[ \\left( 1-\\nu \\right) A+\\nu B\\right] \\circ 1-A^{1-\\nu }\\circ B^{\\nu } \\\\ & \\leq \\frac{M}{m^{2}}\\nu \\left( 1-\\nu \\right) \\left( \\frac{A^{2}+B^{2}}{2}% \\circ 1-A\\circ B\\right) \\end{align*} for all $\\nu \\in \\left[ 0,1\\right] .$
Keywords : Tensorial product, Hadamard Product, Selfadjoint operators, Convex functions

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