On fractional powers of generators of fractional resolvent families

TL;DR

This paper investigates how fractional powers of generators of fractional resolvent families produce new resolvent families with different properties, extending the understanding of fractional calculus in operator theory.

Contribution

It establishes conditions under which fractional powers of generators produce analytic resolvent families and derives a generalized subordination principle, advancing fractional operator theory.

Findings

Fractional powers of generators produce analytic resolvent families under specific conditions.

A generalized subordination principle is established for fractional resolvent families.

Applications to fractional order Cauchy problems are demonstrated.

Abstract

We show that if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in (0,2]$, then $-A^{\beta}$ generates an analytic $\gamma$-times resolvent family for $\beta \in(0,\frac{2\pi-\pi\gamma}{2\pi-\pi\alpha})$ and $\gamma \in (0,2)$. And a generalized subordination principle is derived. In particular, if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in (1,2]$, then $-A^{1/\alpha}$ generates an analytic $C_0$-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.