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

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.