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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.

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Acknowledgements

The author would like to thank the anonymous referee for reading carefully the paper and providing some useful suggestions to improve it.

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Correspondence to S. S. Dragomir.

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Dragomir, S.S. Some inequalities for continuous functions of selfadjoint operators in hilbert spaces. Acta Math Vietnam 39, 287–303 (2014). https://doi.org/10.1007/s40306-014-0061-4

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Keywords

  • Selfadjoint operators
  • Functions of selfadjoint operators
  • Spectral representation
  • Inequalities for selfadjoint operators

Mathematics Subject Classification (2010)

  • 47A63
  • 47A64
  • 47A99