Counterexamples concerning sectorial operators

Abstract.

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L ^p-space, with $1 \le p \neq 2 \le \infty$, and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L ^p-space; one acts on L ^p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on $L^2 ({\Bbb R}^2;X)$ admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property $(\alpha )$ (geometric property introduced by G. Pisier).