Operator Theory on Symmetrized Bidisc and Tetrablock-some Explicit Constructions

Abstract

A pair of commuting bounded operators (S; P ) acting on a Hilbert space, is called a -contraction, if it has the symmetrised bides = f(z1 + z2; z1z2) : jz1j 1; jz2j 1g C2 as a spectral set. For every -contraction (S; P ), the operator equation S S P = DP F DP has a unique solution F 2 B(DP ) with numerical radius, denoted by w(F ), no greater than one, where DP is the positive square root of (I P P ) and DP = RanDP . This unique operator is called the fundamental operator of (S; P ). This thesis constructs an explicit normal boundary dilation for -contractions. A triple of commuting bounded operators (A; B; P ) acting on a Hilbert space with the tetra block E = f(a11; a22; detA) : A = a11 a12 with kAk 1g C 3 a21 a22 as a spectral set, is called a tetra block contraction. Every tetra block contraction possesses two fundamental operators and these are the unique solutions of A B P = DP F1DP ; and B A P = DP F2DP : Moreover, w(F1) and w(F2) are no greater than one. This thesis also constructs an explicit normal boundary dilation for tetra block contractions. In these constructions, the fundamental operators play a pivotal role. Both the dilations in the symmetrised bidisc and in the tetra block are proved to be minimal. But unlike the one variable case, uniqueness of minimal dilations fails in general in several variables, e.g., Ando's dilation is not unique, see [44]. 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 tetra block. We also find a set of complete unitary invariants for this special class. Along the way, we find a Burling-Lax-Halmos type of result for a triple of multiplication operators acting on vector-valued Hardy 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 -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 tetra block setting.