Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces | ||||
| Electronic Journal of Mathematical Analysis and Applications | ||||
| Volume 11, Issue 2, July 2023, Page 1-15 PDF (521.71 K) | ||||
| Document Type: Regular research papers | ||||
| DOI: 10.21608/ejmaa.2023.199881.1014 | ||||
 View on SCiNiTO | ||||
| Author | ||||
Vuk Stojiljkovic 
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| Stevana Mokranjca 8 | ||||
| Abstract | ||||
| In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert space have been obtained. The recent progression of the Hilbert space inequalities following the definition of the convex operator inequality has lead researchers to explore the concept of Hilbert space inequalities even further. The motivation for this paper stems from the recent development in the theory of tensorial and Hilbert space inequalities. Multiple inequalities are obtained with variations due to the convexity properties of the mapping $f$ $$\bigg|\bigg|\frac{1}{6}\left(\operatorname{exp}(A)\otimes 1+4\operatorname{exp}\left(\frac{A\otimes 1+1\otimes B}{2}\right)+1\otimes \operatorname{exp}(B)\right)$$ $$-\frac{1}{4}\bigg(\int_{0}^{1}\operatorname{exp}\left(\left(\frac{1-k}{2}\right)A\otimes 1+\left(\frac{1+k}{2}\right)1\otimes B\right)k^{-\frac{1}{2}}dk$$ $$+\int_{0}^{1}\operatorname{exp}\left(\left(1-\frac{k}{2}\right)A\otimes 1+\frac{k}{2}1\otimes B\right)(1-k)^{-\frac{1}{2}}dk\bigg)\bigg|\bigg|$$ $$\leq \frac{47}{360}\norm{1\otimes B-A\otimes 1}^{2}(\norm{\operatorname{exp}(A)}+\norm{\operatorname{exp}(B)}).$$ Tensorial version of a Lemma given by Hezenci is derived and utilized to obtain the desired inequalities. In the introduction section is given a brief history of the inequalities, while in the preliminary section we give necessary Lemmas and results in order to understand the paper. Structure and novelty of the paper are discussed at the end of the introduction section. | ||||
| Keywords | ||||
| Tensorial product; Selfadjoint operators; convex functions; Spectra | ||||
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