An alternative to Plemelj-Smithies formulas on infinite determinants (Q1310809)

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. Global upper bounds for these quantities are also presented. In particular, it is proved that \[ | d^{(p)}(K)|\leq e^{\kappa\| K\|^ p_ p} \] holds with \(\kappa=\kappa(p)\leq\kappa(\infty)=e^{-1/4(1+e^{2\pi})}\) for \(p\geq 3\) which improves a previous estimate yielding \(\kappa(p)=e(2+ln(p-1))\).