Examples of peseudo-differential operators in L~p spaces with unbounded imaginary powers

摘要

1. Introduction. In 1987, Dore and Venni [5] introduced an operator-theoretical concept in order to obtain results on "maximal regularity" for linear elliptic and parabolic equations. The theory of bounded imaginary powers of operators plays a central role in their approach. Besides applications to interpolation theory, this is a main reason why the class of operators which admit bounded imaginary powers has been investigated in detail in recent years (see [5], [18], [17], [4], [11], [1] and the references given therein). In particular, it is known that the Lp-realization of certain linear differential operators have bounded imaginary powers (see [20], [8], [19], [2], [10], [6]). However, a characterization of the class of operators which admit bounded imaginary powers seems to be unknown. This is not at least due to the fact that only a few examples of operators of positive type, also called sectorial operators, are known which do not possess bounded imaginary powers. In order to find such a characterization, it might be useful to find and to investigate different kinds of examples of operators which do not admit bounded imag-inary powers.