Multi-variable Operator Theory: Rational Dilation, Realization Formula for Bounded Analytic Functions and its Applications

Abstract: Ever since Sz.-Nagy proved his dilation theorem for contractive Hilbert space operators in 1953, the *ration dilation problem* - a natural generalization of Sz.-Nagy's theorem to the multi-variable setting formulated by W. B. Arveson in 1972 - has been in the forefront of research in operator theory. In this talk, we shall discuss the rational dilation problem associated to several domains in both 2- and 3-dimensional complex spaces.

Associated to every bounded analytic function in the unit disk, there is a 2x2 block operator unitary matrix such that the function can be realized in terms of the entries of the 2x2 unitary matrix. Conversely, for every 2x2 block operator unitary matrix, there is a bounded analytic function in the unit disk (possibly operator-valued) that can be expressed in terms of the entries of the unitary matrix. This is known as the *realization formula*. We shall discuss an analogue of this formula for two domains in 2-dimensional complex space. An application of this formula will also be discussed.