#### PhD Thesis colloquium

##### Venue: Lecture Hall I, Dept of Mathematics

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 $% $ 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.

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Last updated: 17 Aug 2019