Given a semi-group $U(t)$ of bounded linear operators with bounded self-adjoint generator $A$ we estimate the logarithm of the section determinants of $U(t)$ in terms of $A$. When $A$ is subject to an additional condition, which is related to so-called F\o lner sequences of orthogonal projections, this estimate implies a Szeg\H o type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when $A$ is a Toeplitz operator or a compact operator.

Szegö theorem, semi-group of linear operators, Følner sequences

47D06 (47B10 47B35)