A Fredholm determinant formula for section determinants of bounded operators (Q1882521)

The author proves a Fredholm determinant formula for section determinants of bounded operators. To be more precise, let \(A\) be a linear bounded operator action on a Hilbert space \(H\) and let \(U:=e^A\) be the exponential function of \(A\). Splitting \(H=H_1\oplus H_2\), where \(H_1\) is finite-dimensional and \(H_2\) is separable, \(U\) can be represented by a \(2\times 2\)-matrix corresponding to this decomposition \(U=\left({ U_{11} \,\,\,U_{12} \atop U_{21} \,\,\, U_{22}}\right)\). Under some weak condition, the following formula is shown: det\(\,U_{11}= e^{\text{ tr}\, A_{11}}\text{ det}\,(I+\hat{K_0})\). Here we have decomposed \(A\) analogously to \(U\) and \(\hat{K_0}\) is an integral operator on \(L^2([0,1];H_1)\) of trace class. A formula for the kernel of \(\hat{K_0}\) is given as well.