Inequalities for regularized determinants of operators with the Nakano type modulars

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 \(c\in (0,1)\)