Advances in Operator Theory

Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals

Stefan Milošević

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Abstract

If $A,B \in \mathcal {B}(\mathcal {H})$ are normal contractions, then for every $X \in \mathcal {C}_{||| \cdot |||}(\mathcal {H})$ and $\alpha > 0$ holds $$\Big|\Big|\Big| (I - A^{*}A)^{\frac{\alpha}{2}} X(I - B^{*}B)^{\frac{\alpha}{2}} \Big|\Big|\Big| \leqslant \Big|\Big|\Big| \sum_{n=0}^\infty (-1)^{n} \binom{\alpha}{n}A^{n} X B^{n} \Big|\Big|\Big|,$$ which generalizes a result of D.R. Jocić [Proc. Amer. Math. Soc. 126 (1998), no. 9, 2705-2713] for $\alpha$ not being an integer. Similar inequalities in the Schatten $p$-norms, for non-normal $A,B$ and in the $Q$-norms if one of $A$ or $B$ is normal, are also given.

Article information

Source
Adv. Oper. Theory, Volume 1, Number 2 (2016), 147-159.

Dates
Received: 29 September 2016
Accepted: 1 December 2016
First available in Project Euclid: 4 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512431392

Digital Object Identifier
doi:10.22034/aot.1609.1019

Mathematical Reviews number (MathSciNet)
MR3723616

Zentralblatt MATH identifier
1355.47006

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 47B49: Transformers, preservers (operators on spaces of operators) 47A30: Norms (inequalities, more than one norm, etc.) 47A63: Operator inequalities 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

Keywords
norm inequality elementary operator $Q$-norm

Citation

Milošević, Stefan. Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals. Adv. Oper. Theory 1 (2016), no. 2, 147--159. doi:10.22034/aot.1609.1019. https://projecteuclid.org/euclid.aot/1512431392


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References

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