Besov spaces associated with non-negative operators on Banach spaces

Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B_{q,X}^{s,A}$ for non-negative operators $A$ on Banach spaces $X$ with a full range of indices $s \in $\mathbb{R}$$and$0 < q \leq \infty$. The approach we use is the dyadic decomposition of resolvents for non-negative operators, an analogue of the Littlewood-Paley decomposition in the construction of the classical Besov spaces. In particular, by using the reproducing formulas for fractional powers of operators and explicit quasi-norms estimates for Besov spaces we discuss the connections between the smoothness of Besov spaces associated with operators and the boundedness of fractional powers of the underlying operators.