An alternative to Plemelj-Smithies formulas on infinite determinants

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 $$|d^{(p)}(K)| \le e^{\kappa \n K \n_p^p}$$ holds with $\kappa = \kappa(p) \le \kappa(\infty) = \exp \bigl\{-{1 \over 4(1+ e^{2 \pi})} \bigr\}$ for $p\ge 3$ which improves previous estimate yielding $\kappa(p) = e (2 + \ln(p-1))$.