Examples of pseudo-differential operators in \(L^ p\) spaces with unbounded imaginary powers (Q1924591)

Let \(A\) be a closed, densely-defined Banach space operator which is injective and has dense range. Then \(A\) is said to be sectorial if its resolvent contains the negative reals and \(|\lambda(\lambda+A)^{-1}|\) is uniformly bounded for \(\lambda>0\). For \(1<p<\infty\) other than \(2\), the author exhibits a pseudo--differential operator acting on \(L^p(\mathbf R)\) which is sectorial, but does not admit bounded imaginary powers. The construction is based on the theory of Fourier multipliers.