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 $$\mathrm{\mid }{d}^{(p)}(K)\mathrm{\mid }\le e^{\kappa \text{n}K{\text{n}}_p^p}$$ holds with $\kappa = \kappa(p) \le \kappa(\infty) = \exp \bigl{-{1 \over 4(1+ e^{2 \pi})} \bigr}$for$ p\ge 3$whichimprovespreviousestimateyielding$\kappa(p) = e (2 + \ln(p-1))$.