## A Lewent type determinantal inequality.(English)Zbl 1300.47028

Let $$A_i$$, $$i=1,\dots ,n$$, be strictly contractive trace class operators over a separable Hilbert space. The author proves the determinant inequality $\left | \det \left (\frac{I+\sum _{i=1}^{n}\lambda _iA_i}{I-\sum _{i=1}^{n}\lambda _iA_i}\right )\right |\leq \prod _{i=1}^n\det \left (\frac{I+|A_i|}{I-|A_i|}\right )^{\lambda _i},$ where $$\sum _{i=1}^n \lambda _i=1$$, $$\lambda _i\geq 0$$, $$i=1,\dots ,n$$. This generalizes the following numerical inequality due to L. Lewent [Sitzungsber. Berl. Math. Ges. 7, 95–101 (1908; JFM 39.0466.02)]: $\frac{1+\sum _{i=1}^{n}\lambda _ix_i}{1-\sum _{i=1}^{n}\lambda _ix_i} |\leq \prod _{i=1}^n\det \left (\frac{1+|x_i|}{1-|x_i|}\right )^{\lambda _i},$ where $$x_i\in [0,1), \sum _{i=1}^n \lambda _i=1$$, $$\lambda _i\geq 0$$, $$i=1,\dots ,n$$.

### MSC:

 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 15A45 Miscellaneous inequalities involving matrices

JFM 39.0466.02
Full Text: