Inequalities for the normalized determinant of positive operators in Hilbert spaces via Tominaga and Furuichi results

Authors

  • Silvestru Dragomir Victoria University, College of Engineering & Science, Melbourne City, MC 8001, Australia, University of the Witwatersrand, School of Computer Science & Applied Mathematics, Private Bag 3, Johannesburg 2050, South Africa https://orcid.org/0000-0003-2902-6805

Keywords:

Positive operators, normalized determinants, inequalities

Abstract

For positive invertible operators \(A\) on a Hilbert space \(H\) and a fixed unit vector \(x\in H,\) define the normalized determinant by \(\Delta_{x}(A):=\exp \left\langle \ln Ax,x\right\rangle\). In this paper, we prove among others that, if \(0<mI\leq A\leq MI,\) then


\begin{aligned}1&\leq \exp \left\langle \ln S\left( \left( \frac{M}{m}\right) ^{\frac{1}{2} I-\frac{1}{M-m}\left\vert A-\frac{1}{2}\left( m+M\right) I\right\vert }\right) x,x\right\rangle \\ &\leq \frac{\Delta _{x}(A)}{m^{\frac{M-\left\langle Ax,x\right\rangle }{M-m}
}M^{\frac{\left\langle Ax,x\right\rangle -m}{M-m}}}\leq S\left( \frac{M}{m} \right)\end{aligned}

for \(x\in H,\) \(\left\Vert x\right\Vert =1,\) where \(S\left( \cdot \right)\) is Specht's ratio.

References

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Published

01-01-2024

How to Cite

Dragomir, S. (2024). Inequalities for the normalized determinant of positive operators in Hilbert spaces via Tominaga and Furuichi results. Modern Mathematical Methods, 2(1), 1–9. Retrieved from https://modernmathmeth.com/index.php/pub/article/view/1