Let $T\in {\mathcal{L}}(X)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f(z)\mapsto (z-T)f(z)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.
It is shown that the sum and the product of two commuting Banach space operators with Dunford’s property $\mathrm (C)$ have the single-valued extension property.