On unitary dilations of contractions

Abstract

Using some relatively deep facts about complex functions and spectral measures, B. Sz.-Nagy [2 ] has recently proved that to every contraction A on a Hilbert space H there corresponds a unitary operator on a larger Hilbert space K so that U is a dilation of An for every positive integer n. In other words, if |A|l 0, and U(i, j) =0 in all other cases. An elementary computation shows that is unitary once it is proved that SA =A T. This was proved by Halmos [1 ] by noting that S2A =AT2 and approximating the square root function in the unit interval by polynomials. The fact that Un is a dilation of An is evident since, except for U(0, 0), all the nonzero entries of are above the main diagonal. The possibility of such an explicit proof was suggested by P. R. Halmos who also, contributed several useful comments.