  Motivated by a variety of representations of fractional powers of operators,
we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ 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.
