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One step extension to positive definite map on Z2
by
Ion Suciu
Institute of Mathematics, Romanian Academy, Bucharest
Given a pair [T, S]) of two commuting contractions on the Hilbert space H we can find a positive definite map g --> Tg, g in Z2 , of Z2 into B(H) such that T[1, 0] = T, T[0, 1] = S. According to Naimark dilation theorem any such extension is the compression to H of a uniquely determined unitary representation of the group Z2 on a larger Hilbert space K.
If we look for an extension whose unitary dilation leaves the initial space H semi-invariant then the existence is assured by the celebrated Ando dilation Theorem.
The problem of recurrent construction, by a system of free parameters, of such type of solution is very complicate. In a series of preceding papers we proposed some methods based on the successive dilations and lifting of the commutant methods from one contraction case.
Here we define a two variable one step extension and describe, in a nalogy with the first step in construction of the choice sequence in one variable case, its free parameter.
Date received: June 13, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-62.