The principal purpose of this thesis is to develop and study several functional calculi
for commuting unbounded operators. This is achieved by combining and generalising the
works of Taylor [41,42], M c Intosh [30,31], and Cowling et al. . However, we also
obtain, under special circumstances, an explicit expression for Taylor's functional calculus;
and utilise various functional calculi to give a different outlook to the work of Dore and
Let us now briefly look at Taylor [41,42], M c Intosh [30,31], and Cowling et al. .
We will then mention what we have achieved, before giving an outline of the thesis.
Consider Taylor [41,42]. In these papers Taylor introduced a notion of a joint spectrum
and defined a functional calculus for a commuting tuple of bounded operators on a Banach
space. This work generalised the spectrum and functional calculus for a single operator,
and the work by Shilov , Arens , Arens--Calderon , and Waelbrock . Later,
Putinar  showed the uniqueness of Taylor's functional calculus. Vasilescu  showed
how to extend the notion of joint spectrum to Dcommuting systems, and Vasilescu 
and Janas  obtained explicit formulae for the functional calculus.
Now, consider M c Intosh . In this paper M c Intosh developed a functional calculus
for type ! operators, which we will call sector type ! operators, in a Hilbert space. This
functional calculus provided a simple means to derive the basic properties of semigroups
and fractional powers, of these operators. It also provides, by the consideration of quadratic