Functional calculus for sectorial operators via the entire function with the growth regularity

In this paper, having analyzed the previously obtained results devoted to the root vectors series expansion in the Abel-Lidskii sense, we come to the conclusion that the concept can be formulated in the classical terms of the spectral theorem. Though, the spectral theorem for a sectorial operator has not been formulated even in the m-sectorial case, we can consider from this point of view a most simplified case related to the sectorial operator with a discrete spectrum. Thus, in accordance with the terms of the spectral theorem, we naturally arrive at the functional calculus for sectorial operators which is the main focus of this paper. Due to the functional calculus methods, we construct the operator class with the asymptotics more subtle then one of the power type.