Analyticity of Determinants of Operators on a Banach Space
Analyticity of Determinants of Operators on a Banach Space
If $F(z)$ is an analytic family of operators on a Banach space which is of finite rank for each $z$, then rank $F(z)$ is constant except for isolated points, and det $(I + F(z))$ and tr $F(z)$ are analytic. Similarly if $F(z)$ is meromorphic.