Numerical radius and p-Schatten norm inequalities for power series of operators in Hilbert spaces

Authors : Sever Dragomır

Pages : 365-390

Doi:10.31801/cfsuasmas.1341138

View : 42 | Download : 85

Publication Date : 2024-06-21

Article Type : Research Paper

Abstract :Let $H$ be a complex Hilbert space. Assume that the power series with complex coefficients $f(z):=\\sum\\nolimits_{k=0}^{\\infty }a_{k}z^{k}$ is convergent on the open disk $D(0,R),~f_{a}(z):=\\sum\\nolimits_{k=0}^{\\infty}\\left\\vert a_{k}\\right\\vert z^{k}$ that has the same radius of convergence $R$ and $A,~B,~C\\in B(H)$ with $\\left\\Vert A\\right\\Vert $ <$R$, then we have the following Schwarz type inequality $ \\left\\vert \\left\\langle C^{\\ast }Af(A)Bx,y\\right\\rangle \\right\\vert \\leq f_{a}(\\left\\Vert A\\right\\Vert )\\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} $ for $\\alpha \\in \\lbrack 0,1]$ and $x,y\\in H.$ Some natural applications for numerical radius and p-Schatten norm are also provided.
Keywords : Vector inequality, bounded operators, Aluthge transform, Dougal transform, partial isometry, numerical radius, p Schatten norm