Numerical Radius and $p$-Schatten Norm Inequalities for Analytic Functions of Operators in Hilbert Spaces

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

References

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