An alternative to Plemelj-Smithies formulas on infinite determinants

Abstract

An alternative to Plemelj - Smithies formulas for the p -regularized quantities $d^{(p)}(K)$ and $D^{(p)}(K)$ is presented which generalizes previous expressions with $p=1$ due to Grothendieck and Fredholm. It is also presented global upper bounds for these quantities. In particular we prove that $$ |d^{(p)}(K)| \le e^{κ\n K \n_p^p} $$ holds with $κ= κ(p) \le κ(\infty) = \exp \bigl\{-{1 \over 4(1+ e^{2 π})} \bigr\}$ for $ p\ge 3$ which improves previous estimate yielding $κ(p) = e (2 + \ln(p-1))$.