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

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.