Operator Theory on Symmetrized Bidisc

A commuting pair of operators (S, P) on a Hilbert space H is said to be a Gamma-contraction if the symmetrized bidisc is a spectral set of the tuple (S, P). In this paper we develop some operator theory inspired by Agler and Young's results on a model theory for Gamma-contractions. We prove a Beurling-Lax-Halmos type theorem for Gamma-isometries. Along the way we solve a problem in the classical one-variable operator theory. We use a "pull back" technique to prove that a completely non-unitary Gamma-contraction (S, P) can be dilated to a direct sum of a Gamma-isometry and a Gamma-unitary on the Sz.-Nagy and Foias functional model of P, and that (S, P) can be realized as a compression of the above pair in the functional model of P. Moreover, we show that the representation is unique. We prove that a commuting tuple (S, P) with |S| \leq 2 and |P \leq 1 is a Gamma-contraction if and only if there exists a compressed scalar operator X with the decompressed numerical radius not greater than one such that S = X + P X^*. In the commutant lifting set up, we obtain a unique and explicit solution to the lifting of S where (S, P) is a completely non-unitary Gamma-contraction. Our results concerning the Beurling-Lax-Halmos theorem of Gamma-isometries and the functional model of Gamma-contractions answers a pair of questions of J. Agler and N. J. Young.Comment: 26 pages, revised and final version. To appear in Indiana University Mathematics Journa

Cornerstones of Sampling of Operator Theory

This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains