## Advances in Operator Theory

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

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

#### 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