  In this monograph, we formulated the sufficient conditions of the
Abel-Lidskii basis property for a sectorial operator. Having studied such an
operator class, we strengthened the conditions regarding the semi-angle of the
sector and weakened a great deal conditions regarding the involved parameters.
Thus, we clarified the results by Lidskii V.B. devoted to the decomposition on
the root vector system of the non-selfadjoint operator. We used a technique of
the entire function theory and introduced the so-called Schatten-von Neumann
class of the convergence exponent. Having considered strictly accretive
operators satisfying special conditions formulated in terms of the norm and
used a sequence of contours of the power type, we invented a peculiar method
how to calculate a contour integral involved in the problem. Finally, we
consider evolution equations in the abstract Hilbert space.
