- Konuralp Journal of Mathematics
- Cilt: 11 Sayı: 2
- Numerical Radius and $p$-Schatten Norm Inequalities for Analytic Functions of Operators in Hilbert S...
Numerical Radius and $p$-Schatten Norm Inequalities for Analytic Functions of Operators in Hilbert Spaces
Authors : Sever Dragomir
Pages : 109-126
View : 24 | Download : 77
Publication Date : 2023-10-31
Article Type : Research
Abstract :Let $H$ be a complex Hilbert space, $f:G\\subset \\mathbb{C}\\rightarrow \\mathbb{C}$ an analytic function on the domain $G$ and $A\\in \\mathcal{B} \\left( H\\right) $ with $\\mbox{Sp}\\left( A\\right) \\subset G$ and $\\gamma $ a closed rectifiable path in $G$ and such that $\\mbox{Sp}\\left( A\\right) \\subset \\mbox{ins}\\left( \\gamma \\right) .$ If we denote \\begin{equation*} B\\left( f,\\gamma ;A\\right) :=\\frac{1}{2\\pi }\\int_{\\gamma }\\left\\vert f\\left( \\xi \\right) \\right\\vert \\left( \\left\\vert \\xi \\right\\vert -\\left\\Vert A\\right\\Vert \\right) ^{-1}\\left\\vert d\\xi \\right\\vert , \\end{equation*} then for $B,$ $C\\in \\mathcal{B}\\left( H\\right) $ we have \\begin{equation*} \\left\\vert \\left\\langle C^{\\ast }Af\\left( A\\right) Bx,y\\right\\rangle \\right\\vert \\leq B\\left( f,\\gamma ;A\\right) \\left\\langle \\left\\vert \\left\\vert A\\right\\vert ^{\\alpha }B\\right\\vert ^{2}x,x\\right\\rangle ^{1/2}\\left\\langle \\left\\vert \\left\\vert A^{\\ast }\\right\\vert ^{1-\\alpha }C\\right\\vert ^{2}y,y\\right\\rangle ^{1/2} \\end{equation*} for $\\alpha \\in \\left[ 0,1\\right] $ and $x,$ $y\\in H.$ Some natural applications for \\textit{numerical radius} and $p$-\\textit{Schatten norm } are also provided.Keywords : Vector inequalities, Numerical Radius, Norm inequalities