Inequalities for regularized determinants of operators with the Nakano type modulars

Abstract

Let $\{p_k\}$ be a nondecreasing sequence of integers, and $A$ be a compact operator in a Hilbert space whose eigenvalues and singular values are $\lambda_k(A)$ and $s_k(A)$ $(k=1, 2, .... )$, respectively. We establish upper and lower bounds for the regularized determinant $\prod_{k=1}^\infty (1-\lambda_k(A)){\rm exp}\;[\sum_{m=1}^{p_k-1} \frac{\lambda_k^m(A)}{m}],\mbox{ assuming that } \sum_{j=1}^{\infty} \frac{s_j^{p_j}(A/c)}{p_j}\lt \infty$ for a constant $c\in (0,1)$.