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Upper and lower bounds for regularized determinants. (English) Zbl 1162.47019

Let \(p\geq2\) be an integer and \(S_{p}\) be the von Neumann-Schatten ideal of compact operators \(A\) in a separable Hilbert space with the finite norm \(N_{p}(A)= [\text{Trace}(AA*)^{p/2}]^{1/p}\), where \(A^{\ast}\) denotes the adjoint of \(A\). If \(A\in S_{p}\), the regularized determinant of \(A\) is defined as \(\det{}_{p}(A)=\prod_{j=1}^{\infty}(1-\lambda_{j}(A))\exp[\,\sum_{m=1}^{p-1}\frac{\lambda_{j}^{m}(A)}{m}]\), where \(\lambda_{j}(A)\) are the eigenvalues of \(A\) with their multiplicities arranged in decreasing order. The main goal of the paper is to find bounds for the constant \(q_{p}\) (\(p>2\)) that appears in the inequality \(\det{}_{p}(A)\leq\exp[ q_{p}N_{p} ^{p}(A)]\). The paper also offers lower bounds for \(\det{}_{p}(A)\).

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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