Some inequalities for continuous functions of selfadjoint operators in hilbert spaces

Abstract

If $\{ E_{\lambda} \} _{\lambda\in\mathbb{R}}$ is the spectral family of a bounded selfadjoint operator A on a Hilbert space H and m=minSp(A) and M=maxSp(A), we show that for any continuous function φ: $[ m,M ] \rightarrow \mathbb{C}$ , we have the inequality $$\begin{aligned} \bigl\vert \bigl\langle \varphi ( A ) x,y \bigr\rangle \bigr\vert ^{2} \leq& \Biggl( \int_{m-0}^{M}\bigl\vert \varphi ( t ) \bigr\vert \,d \Biggl( \bigvee_{m-0}^{t} \bigl( \langle E_{ ( \cdot ) }x,y \rangle \bigr) \Biggr) \Biggr) ^{2} \\ \leq& \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert x,x \bigr\rangle \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert y,y \bigr\rangle \end{aligned}$$ for any vectors x and y from H. Some related results and applications are also given.