DILATION FOR PAIRS OF CONTRACTIONS

Abstract

Dilation theory is a paradigm for understanding operators in which one operator is represented as a compression of another operator that is well behaved in some way. Every contraction, for example, can be dilated to (and so is a compression of) a unitary operator. In this work we begin with unitary dilation of a single contraction due to Sz.-Nagy and Foias. After that we discuss here a generaliza tion to a pair of commuting contractions given by T. Ando [1]. He demonstrated a pair like this has a simultaneous commuting dilation. Then there was a shocking revelation from , Parrott [2] that there is no way to generalise this event any further. They demonstrated triples of commuting contractions that lacked any commuting isometric di latation. So here, We study the concept of Dilation of tuples in a large Hilbert Space (K) that contains a smaller Hilbert space (H). The main results that we want to study are: determination of the tuples of com muting operators (V1,V2) in a larger Hilbert space that dilates tuples (T1,T2) in a smaller Hilbert Space and to establish the example which shows that this result does not exist for 3 and 4 pairs of operators .We will discuss about existence of minimal isometric dilation, extension of isometry to unitary, von Neumann’s inequality, Sz.-Nagy existence theorem. Using these concepts we will investigate the existence for a pair of commuting unitary operators on some Hilbert space K ⊃ H for the pair of commuting contractions on the Hilbert space H.