Vol. 34, No. 2, 1970

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Unitary dilations for commuting contractions

Stephen Parrott

Vol. 34 (1970), No. 2, 481–490

Let S1,S2,,,Sn be a set of commuting contraction operators on a Hilbert space H, let U1,U2,,Un be a set of commuting unitary operators on a Hilbert space K containing H, and let P be the projection from K to H. The set U1,,Un is called a set of commuting unitary dilations for S1,,Sn provided that

P Um11U m22⋅⋅⋅Umnnx = lSm11 Sm22Smnn x

for all x in H and for all nonnegative integers m1,m2,,mn. Sz.-Nagy proved that a single contraction has a unitary dilation, and Ando showed that any two commuting contractions possess a pair of commuting unitary dilations. This note presents several counterexamples which disprove the corresponding conjecture for three or more contractions.

In §3, three commuting contractions, R,S,T are constructed which do not have commuting unitary dilations. The operators R and S each have norm one, while the operator T may be chosen to have any norm between zero and one. However, the proof yielding the counterexample fails completely if the operators R,S,T are replaced by λR,λS,T with 0 < λ < 1, and this raises another question.

It is known that a finite or infinite set of commuting contractions S1,S2, which satisfies Sk2 1 possesses a set of commuting unitary dilations. Thus it appears that the “size” of a set of contractions may be relevant to the existence of commuting unitary dilations; and since two of the contractions in §3 have norm one, it is conceivable that this example might be only a peculiar “boundary” phenomenon. In §4 this notion is dispelled by a more complicated example showing that three commuting contractions, each of norm strictly less than one, can fail to have commuting unitary dilations. Although the example of §4 is in most (but not all) respects more powerful than that of §3, the latter is presented separately because of its simplicity.

Section 3 also observes that a recent result of Sz.-Nagy and Foias is equivalent to Ando’s theorem. Section 5 shows that the counterexamples constructed in this paper to the unitary dilation conjecture cannot be used as counterexamples to another well-known conjecture concerning spectral sets.

Mathematical Subject Classification
Primary: 47.40
Received: 9 June 1969
Revised: 5 January 1970
Published: 1 August 1970
Stephen Parrott