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Too long for a comment. Actually I do not think that it is written anywhere but these kind of counteraxamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by \Delta+Bx \cdot \nabla, where B is a matrix. Assuming that all eigenvalues of B have negative real parts, then an invariant measure \mu exists (and is given by a Gaussian density). It turns out that the angle of analiticity in L^p of the invariant measure can be computed exactly and can be smaller that \pi/2, even for p=2. This can be found in a paper by Chill, Fasangova, Pallara and myself. To obtain similar examples in unweighted spaces one has only (but patiently) to compensate the weight thus obtaining an operator with a linear drift and a quadratic potential. This works however only from dimension 2 on; in the one dimensional case the operator is always self-adjont but still the angle of analitycity is different from \pi/2 in L^p, for p different from 2. There is also a paper by E. Priola and myself dealing with non-analytic Markov semigropus where one finds other examples. It would be nice to have a direct approach, avoiding the detour.