We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

Let $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$, where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

We present an abstract approach to the construction of holomorphic functional calculi for unbounded operators and apply it to the special case of sectorial operators. In effect, we obtain a calculus for a much larger class of functions than was known before, including certain meromorphic functions. We discuss the role of topology. Then we prove in detail a composition rule $(f\circ g)(A)=f(g(A))$ which is the main result of the paper. This is done in such a way that the proof can easily be transferred to functional calculi for other classes of operators.