Counterexamples concerning sectorial operators

In this paper we give two counterexamples to the closedness of the sum of t wo sectorial operators with commuting resolvents. In the first example the operators are defined on an LP-space, with 1 < p not equal 2 < infinity, an d one of them admits bounded imaginary powers The second example is concern ed with operators defined on a Hilbert valued LP-space; one acts on LP 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(R-2;X) admit a so-called bounded joint functional calculus if and only i f X is a UMD Banach space with property (alpha) (geometric property introdu ced by G. Pisier).