A generalization of the concept of sectorial operator

Let be a Banach space and a non-increasing function such that as and is a Lipschitz function on .A linear operator is said to be -sectorial if there exist constants and such that the spectrum of  lies in the set and  there exists     0$ SRC=http://ej.iop.org/images/1064-5616/197/7/A02/tex_sm_3785_img13.gif/>    such that         for  where is the resolvent of the operator .The properties of the operator exponential and fractional powers of a -sectorial operator are analysed alongside the question of the unique solubility of the Cauchy problem for the linear differential operator with -sectorial operator-valued coefficient.