Opuscula Math. 33, no. 2 (2013), 283-291
http://dx.doi.org/10.7494/OpMath.2013.33.2.283

Opuscula Mathematica

# Inequalities for regularized determinants of operators with the Nakano type modulars

Michael Gil'

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

Keywords: Hilbert space, compact operators, regularized determinant, Nakano type modular.

Mathematics Subject Classification: 47B10, 47A55.

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