A pair of commuting bounded operators $(S,P)$ acting on a Hilbert space, is
called a $\Gamma$-contraction, if it has the symmetrised bidisc $$\Gamma=\{
(z_1+z_2,z_1z_2):|z_1| \leq 1,|z_2| \leq 1 \}\subseteq \mathbb{C}^2$$ as a spectral
set. For
every $\Gamma$-contraction $(S,P)$, the operator equation $$S-S^*P=D_PFD_P$$ has a
unique solution $F$ with numerical radius, $w(F)$ no greater than one, where $D_P$
is the
positive square root of $(I-P^*P)$. This unique operator is called the fundamental
operator of
$(S,P)$. This thesis constructs an explicit normal boundary dilation for a
$\Gamma$-contraction. A triple of commuting bounded operators $(A,B,P)$ acting on a
Hilbert space with the closure of the tetrablock $E=\{(a_{11},a_{22},\text{det}A):
A=\begin{pmatrix} a_{11} & a_{12} \\
a_{21} & a_{22} \end{pmatrix}\text{ with }\lVert A \rVert < 1\}\subseteq\mathbb{C}^3
$$ as a spectral set, is called a tetrablock contraction. Every tetrablock contraction
possesses two fundamental operators and these are the unique solutions of $$
A-B^*P=D_PF_1D_P, \text{ and }  B-A^*P=D_PF_2D_P. $$ Moreover, $w(F_1)$ and
$w(F_2)$ are no greater than one. This thesisalso constructs an explicit normal
boundary
dilation for a tetrablock contraction. In these constructions, the fundamental
operators play a
pivotal role. Both the dilations in the symmetrized bidisc and in the tetrablock are
proved to
be minimal. But unlike the one variable case, uniqueness of minimal dilations
usually does
not hold good in several variables, e.g., Ando's dilation is not unique. However, we
show that
the dilations are unique under a certain natural condition. In view of the abundance of
operators and their complicated structure, a basic problem in operator theory is to
find nice
functional models and complete sets of unitary invariants. We develop a functional
model
theory for a special class of triples of commuting bounded operators associated with
the
tetrablock. We also find a set of complete sort of unitary invariants for this
special class.
Along the way, we find a Beurling-Lax-Halmos type of result for a triple of
multiplication
operators acting on vector-valuedHardy spaces. In both the model theory and unitary
invariance,fundamental operators play a fundamental role. This thesis answers the
question
when two operators $F$ and $G$ with $w(F)$ and $w(G)$ no greater than one, are
admissible as fundamental operators, in other words, when there exists a
$\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and
$G$ is the fundamental operator of $(S^*,P^*)$. This thesis also answers a similar
question
in the tetrablock setting.