A contractive homomorphism of the algebra $\mathcal A(\mathbb D)$(respectively, $\mathcal A(\mathbb D^2)$) dilates to a * - homomorphism of the algebra of the continuous functions $\mathcal C(\mathbb T) $(respectively, $\mathcal C(\mathbb T^2)$) on the boundary $\mathbb T$ as shown by Sz.-Nagy(respectively, Ando). Parrott produced examples to show that contractive homomorphisms of the algebra $\mathcal A(\mathbb D^n) $ need not dilate if n is strictly greater than 2. It was shown by G. Misra that the Parrott examples are not even 2-contractive. It was shown that the study of Parrott examples is equivalent to studying a class of contractive linear maps on a normed linear space. As a consequence, it is natural to ask if contractive linear maps on a (finite dimensional) normed linear space are necessarily completely contractive. Paulsen and E. Ricard have shown that there is always a contractive linear map on any normed linear space of dimension n strictly greater than 2; whichisnot completely contractive. However, Andos theorem shows that every contractive linear map of $\mathbb C^2$ equipped with the the $ l_{\infty}$ norm is completely contractive. By duality, the same is true of the $l_1$ norm on $\mathbb C^2$ as well. We settled the problem of contractive linear map which is not completely contractive for a class of domain in $\mathbb C^2.$ So, the question of which contractive linear maps are completely contractive for the remaining domain in $\mathbb C^2$ remains open. In future, we want to explore this problem. I think that this results may lead to new directions in multi-variable operator theory. On the other hand, we dealt with the relationship of the homomorphisms $\rho_V $ to to the well known theory of Cowen-Douglas class of operators using the concepts from complex geometry. We obtained an interesting relationship, expressing the curvature matrix induced by the reproducing kernel $K(z,w)$ in terms of the localization matrices $V_i$ of size $1 \times m.$ This leads to the study of more general curvature matrices of the type $((\partial_i\bar{\partial}_jK(z;w)^\lambda))$ for $\lambda $ is strictly greater than $0.$ In this setting contractivity is related to curvature inequalities. We showed, using the Bergman kernel of the matrix unit ball, that the curvature inequalities do not imply a von Neumann type inequality. In future, we want to compute the Bergman Kernel for domain in $\mathbb C^2$ which is of the form $\Omega_\mathbfA:=\{(z_1,z_2):\|z_1A_1+z_2A_2\|_{\rm op}\leq 1\},$ where $\mathbf A=(A_1, A_2)$ in $\mathbb C^2 \otimes \mathcal M_2(\mathbb C).$ As we see, this is a challenging topic in which plenty is to be done, requiring significant work.