NORM INEQUALITIES FOR THE NONCOMMUTATIVE ČEBYŠEV FUNCTIONAL IN BANACH ALGEBRAS

Abstract

Let $ℬ$ be a complex Banach algebra. For two continuous functions $x$, $y:\left[a,b\right]\to ℬ$ we define the noncommutative Čebyšev functional

$$D\left(x,y\right):=\left(b-a\right){\displaystyle {\int }_a^{b}x}\left(t\right)y\left(t\right)dt-{\displaystyle {\int }_a^{b}x}\left(t\right)dt{\displaystyle {\int }_a^{b}y}\left(t\right)dt.$$

In this paper we show among other that if $x$, $y$ are strongly differentiable, then

$$\Vert D\left(x,y\right)\Vert \le \frac{1}{4}{\left(b-a\right)}^2\times \{\begin{array}{l}{\Vert x^{\prime }\Vert }_{\left[a,b\right],1}{\Vert y^{\prime }\Vert }_{\left[a,b\right],1},\hfill \\ \frac{1}{3}{\left(b-a\right)}^2{\Vert x^{\prime }\Vert }_{\left[a,b\right],\infty }{\Vert y^{\prime }\Vert }_{\left[a,b\right],\infty }\hfill \\ \frac{1}{2}\left(b-a\right){\Vert x^{\prime }\Vert }_{\left[a,b\right],1}{\Vert y^{\prime }\Vert }_{\left[a,b\right],\infty },\hfill \end{array}$$

where

$${\Vert z^{\prime }\Vert }_{\left[a,b\right],1}:={\displaystyle {\int }_a^b\Vert z^{\prime }\left(t\right)\Vert }dt\text{ and }{\Vert z^{\prime }\Vert }_{\left[a,b\right],\infty }:=\underset{t\in \left(a,b\right)}{\sup }\Vert z^{\prime }\left(t\right)\Vert$$

for a strongly differentiable function $z$ on $\left(a,b\right)$. Some applications for analytic functions of elements in Banach algebras with examples for exponential function are also given.