\(p\)-Schatten Norm Inequalities for Cebysev's Functional Via a Biernacki, Pidek and Ryll-Nardzewski Result

Abstract

An operator \(A\in \mathcal{B}\left( H\right)\) is said to belong to the von Neumann-Schatten class \(\mathcal{B}_{p}\left( H\right),\) \(1\leq p<\infty \) if the \(p\)-Schatten norm is finite

\[
\left\Vert A\right\Vert _{p}:=\left[ tr \left( \left\vert A\right\vert ^{p}\right) \right] ^{1/p}<\infty .
\]

For \(\mathbf{A=}\left( A_{1},...,A_{n}\right) \ , \ \mathbf{B=}\left(B_{1},...,B_{n}\right) \in B\left( H\right)\) we define the noncommutative

operator Cebysev functional

\[
D_{n}\left( \mathbf{A},\mathbf{B}\right) :=\frac{1}{n}\sum \limits_{i=1}^{n}A_{i}B_{i}-\frac{1}{n}\sum\limits_{i=1}^{n}A_{i}\frac{1}{n}\sum\limits_{i=1}^{n}B_{i}.
\]

and we provide several bounds for the \(p\)-Schatten norm \(\left\Vert D_{n}\left( \mathbf{A},\mathbf{B}\right) \right\Vert _{p}\) with \(p\geq 1\) in terms of forward difference defined by \(\Delta A_{k}:=A_{k+1}-A_{k}, \ 1\leq k\leq n-1, \ n\geq 2.\)