Variant versions of the Lewent type determinantal inequality

Ali Morassaei

Abstract

In this paper, we present a refinement of the Lewent determinantal inequality and show that the following inequality holds $$\det\frac{I_{\mathcal{H}}+A_1}{I_{\mathcal{H}}-A_1}+\det\frac{I_{\mathcal{H}}+A_n}{I_{\mathcal{H}}-A_n}-\sum_{j=1}^n\lambda_j \det\left(\frac{I_{\mathcal{H}}+A_j}{I_{\mathcal{H}}-A_j}\right)$$ $$\ge \det\left[\left(\frac{I_{\mathcal{H}}+A_1}{I_{\mathcal{H}}-A_1}\right)\left(\frac{I_{\mathcal{H}}+A_n}{I_{\mathcal{H}}-A_n}\right)\prod_{j=1}^n \left(\frac{I_{\mathcal{H}}+A_j}{I_{\mathcal{H}}-A_j}\right)^{-\lambda_j}\right],$$ where $A_j\in\mathbb{B}(\mathcal{H})$‎, ‎$0\le A_j < I_\mathcal{H}$‎, ‎$A_j$'s are trace class operators and $A_1 \le A_j \le A_n~(j=1,\ldots,n)$ and $\sum_{j=1}^n\lambda_j=1,‎~ ‎\lambda_j \ge 0‎~ ‎(j=1,\ldots,n)$‎. ‎In addition, we present some new versions of the Lewent type determinantal inequality.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 3 (2018), 632-638.

Dates
Accepted: 25 February 2018
First available in Project Euclid: 4 April 2018

https://projecteuclid.org/euclid.aot/1522807280

Digital Object Identifier
doi:10.15352/aot.1711-1259

Mathematical Reviews number (MathSciNet)
MR3795104

Zentralblatt MATH identifier
06902456

Subjects
Primary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
Secondary: 15A45‎ ‎47A63‎ ‎47A64

Citation

Morassaei, Ali. Variant versions of the Lewent type determinantal inequality. Adv. Oper. Theory 3 (2018), no. 3, 632--638. doi:10.15352/aot.1711-1259. https://projecteuclid.org/euclid.aot/1522807280

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