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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

Citations:

JFM 39.0466.02
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