 
Summary: Summary
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. [11]. 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
Venni [14].
Let us now briefly look at Taylor [41,42], M c Intosh [30,31], and Cowling et al. [11].
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 [38], Arens [4], ArensCalderon [5], and Waelbrock [45]. Later,
Putinar [35] showed the uniqueness of Taylor's functional calculus. Vasilescu [44] showed
how to extend the notion of joint spectrum to Dcommuting systems, and Vasilescu [43]
and Janas [24] obtained explicit formulae for the functional calculus.
Now, consider M c Intosh [30]. 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
