- Communications in Advanced Mathematical Sciences
- Cilt: 7 Sayı: 1
- Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in...
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 : 19 | Download : 69
Publication Date : 2024-03-04
Article Type : Research
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