Invariant subspaces for commuting contractions
by
Marek Kosiek
Jagiellonian University, Institute of Mathematics, Krakow, Poland

Let T=(T₁, ..., T_N) be an N-tuple of commuting contractions acting on a complex, separable, infinite dimensional Hilbert space. By s_H(T) we denote its Harte spectrum. Our main result is as follows

Theorem If T has the unit polydisk D ^N as a spectral set, and s_H(T)ÇD ^N is dominating for H^¥(D ^N) then T has a common (nontrivial) invariant subspace.

In the case of N=2, we can avoid the hypothesis that D ^N is a spectral set for T, by Ando Dilation Theorem.

In the proof of the above result a version of the dual algebra technique is applied. An important role in this technique play so called vanishing properties which, roughly speaking, say that for any vectors x, y we can choose suitable orthonormal sequences {x_n}, {y_n} such that

sup |áh(T)x, ynñ|® 0,     sup |áh(T)xn, yñ|® 0,

where the supremum is taken over all h Î H^¥(D ^N) of norm one.

Since T is an N-tuple of commuting contractions, we have

0 £ T*nTn £ I

for all multiindices n. Hence the nonnegative operator B:=inf_n T^*nTⁿ exists, satisfies the same inequality, and has a spectral measure E supported on the interval [0, 1].

In our result we get vanishing properties without any kind of C_0· or C_·0 conditions, by proving the following älmost orthogonality" property and using it together with the domination assumption.

Theorem Let T=(T₁, ..., T_N) be a completely nonisometric N-tuple of commuting contractions and let d > 0 be given. Then for a fixed vector x and an arbitrary positive integer n we can find a finite sequence of real numbers 0 < t₁ < ... < t_n < t_n+1 < 1 and a corresponding sequence of multiintegers k₁ < ... < k_n such that

||E([0, 1]\[ti, ti+1))Tki(x)|| < d

for i=1, ..., n.

Date received: June 17, 2002