Friday, 14 December 2007

soft question - Dimension Leaps

Here is a closely related pair of examples from operator theory, von Neumann's inequality and the theory of unitary dilations of contractions on Hilbert space, where things work for 1 or 2 variables but not for 3 or more.



In one variable, von Neumann's inequality says that if TT is an operator on a (complex) Hilbert space HH with |T|leq1|T|leq1 and pp is in mathbbC[z]mathbbC[z], then |p(T)|leqsup|p(z)|:|z|=1|p(T)|leqsup|p(z)|:|z|=1. Szőkefalvi-Nagy's dilation theorem says that (with the same assumptions on TT) there is a unitary operator UU on a Hilbert space KK containing HH such that if P:KtoHP:KtoH denotes orthogonal projection of KK onto HH, then Tn=PUn|HTn=PUn|H for each positive integer nn.



These results extend to two commuting variables, as Ando proved in 1963. If T1T1 and T2T2 are commuting contractions on HH, Ando's theorem says that there are commuting unitary operators U1U1 and U2U2 on a Hilbert space KK containing HH such that if P:KtoHP:KtoH denotes orthogonal projection of KK onto HH, then Tn11Tn22=PUn11Un22|HTn11Tn22=PUn11Un22|H for each pair of nonnegative integers n1n1 and n2n2. This extension of Sz.-Nagy's theorem has the extension of von Neumann's inequality as a corollary: If T1T1 and T2T2 are commuting contractions on a Hilbert space and pp is in mathbbC[z1,z2]mathbbC[z1,z2], then |p(T1,T2)|leqsup|p(z1,z2)|:|z1|=|z2|=1|p(T1,T2)|leqsup|p(z1,z2)|:|z1|=|z2|=1.



Things aren't so nice in 3 (or more) variables. Parrott showed in 1970 that 3 or more commuting contractions need not have commuting unitary dilations. Even worse, the analogues of von Neumann's inequality don't hold for nn-tuples of commuting contractions when ngeq3ngeq3. Some have considered the problem of quantifying how badly the inequalities can fail. Let KnKn denote the infimum of the set of those positive constants KK such that if T1,ldots,TnT1,ldots,Tn are commuting contractions and pp is in mathbbC[z1,ldots,zn]mathbbC[z1,ldots,zn], then |p(T1,ldots,Tn)|leqKcdotsup|p(z1,ldots,zn)|:|z1|=cdots=|zn|=1|p(T1,ldots,Tn)|leqKcdotsup|p(z1,ldots,zn)|:|z1|=cdots=|zn|=1. So von Neumann's inequality says that K1=1K1=1, and Ando's Theorem yields K2=1K2=1. It is known in general that Kngeqfracsqrtn11Kngeqfracsqrtn11. When n>2n>2, it is not known whether KnltinftyKnltinfty.



See Paulsen's book (2002) for more. On page 69 he writes:




The fact that von Neumann’s inequality holds for two commuting contractions
but not three or more is still the source of many surprising results and
intriguing questions. Many deep results about analytic functions come
from this dichotomy. For example, Agler [used] Ando’s theorem to deduce an
analogue of the classical Nevanlinna–Pick interpolation formula
for analytic functions on the bidisk. Because of the failure of a von
Neumann inequality for three or more commuting contractions, the analogous
formula for the tridisk is known to be false, and the problem of finding the
correct analogue of the Nevanlinna–Pick formula for polydisks
in three or more variables remains open.


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