Reflexivity of Isometries

Wing-Suet Li ^{∗} John E. McCarthy ^{†} Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A. Washington University, St. Louis, Missouri 63130, U.S.A.

July 24 1994

Abstract We prove that any set of commuting isometries on a separable Hilbert space is

reflexive.

Let C be a set of bounded operators on a Hilbert space H, Lat(C) be the lattice of all closed subspaces of H left invariant by every element of C, and AlgLat(C) be the algebra of all bounded operators that leave invariant every element of Lat(C). Any polynomial in elements of C will be in AlgLat(C), and so will anything in the closure, in the weak operator topology, of this set of polynomials: denote this set by W (C). If W (C) is all of AlgLat(C), then C is called reflexive. In this note we prove that any set of commuting isometries on a separable space is reflexive.

The idea of reflexivity was first introduced by D. Sarason, who proved that any set of commuting normal operators is reflexive, and that so is any set of analytic Toeplitz operators [10]. In [4], J. Deddens proved that every isometry is reflexive, and later it was shown that a pair of commuting isometries T_{1} and T_{2} with the additional property that T_{1} commuted with T_{2}^{⋆} is reflexive [7] and [8, 9]. The first author and H. Bercovici have shown that a pair of commuting isometries, one of which is a shift of finite multiplicity, is reflexive [2]. Recently, K. Horák and V. Müller showed that if C is any set of commuting isometries, then AlgLat(C) is contained in both the commutant of C and in its double commutant [6], and we use their result to prove that if C is any collection of commuting isometries on a separable Hilbert space, then it is reflexive. Independently, H. Bercovici has also shown that any collection of commuting isometries is reflexive [1].

∗Partially supported by National Science Foundation grant DMS 9303702.

†Partially supported by National Science Foundation grant DMS 9301508.

Mathematics Subject Classification 47A15

Theorem 1 Let C = {T_{α}}_{α∈I} be a collection of commuting isometries on a separable Hilbert space. Then C is reflexive.

Proof: The isometries {T_{α}} have commuting unitary extensions {U_{α}}, and so are a (possibly infinite) subnormal tuple. (A subnormal tuple is, by definition, a set of commuting subnormal operators that have simultaneous extensions to commuting normal operators; commuting subnormal operators need not, in general, have commuting normal extensions, but commuting isometries do. See [7] for basic facts about subnormal tuples). By Lemma 2, we can assume that {U_{α}} acts on a separable space.

It was observed in [7, Section 2] for finite subnormal tuples, but is true for all subnormal tuples with separably acting minimal normal extensions, that they are reflexive if and only if their restriction to every cyclic subspace is reflexive (the proof is an easy modification of the proof for a single subnormal operator given in the book [3, Theorem VII.8.5] ). Moreover, a cyclic tuple of isometries {T_{α}} can be represented as the tuple {M_{z_{α}}} on some space P^{2}(μ), where μ is a regular Borel measure on a torus, a product of unit circles, P^{2}(μ) denotes the closure in L^{2}(μ) of the polynomials, and M_{z_{α}} denotes multiplication by the α^{th} coordinate function.

By a result of Horák and Müller [6], any operator in AlgLat(C) is in the commutant of C. The commutant of {M_{z_{α}}} is the set of all multiplication operators M_{ϕ}, where ϕ is in P^{2}(μ) ∩ L^{∞}(μ), and the weak-star topology on the set of multiplication operators, as a subset of the bounded linear operators on P^{2}(μ), coincides with the weak-star topology on P^{2}(μ) ∩ L^{∞}(μ) considered as a subset of L^{∞}(μ). We claim (Lemma 3 below) that the algebra P^{2}(μ) ∩ L^{∞}(μ) is elementary (also known as having property 𝔸_{1}), which means that for every weak-star continuous linear functional Λ on {M_{ϕ} : ϕ ∈ P^{2}(μ)∩L^{∞}(μ)}, there exist vectors f and g in P^{2}(μ), such that

Λ(M_{ϕ}) = (M_{ϕ}f, g) = ∫ ϕfḡdμ.

Given the claim, it follows that any operator M_{ϕ} in AlgLat(C) cannot be separated from the weak-star closure of the polynomials in M_{z_{α}} by a weak-star continuous linear functional. Indeed, if there were a Λ that vanished on the polynomials of M_{z_{1}}, M_{z_{2}}, …, the vector g would have to be orthogonal to the invariant subspace generated by f, and this invariant subspace would be left invariant by M_{ϕ}, and so Λ would vanish on M_{ϕ} also (this argument first appeared in [5]). Therefore AlgLat(C) coincides with the weak-star closed algebra generated by T_{1}, T_{2}, …, and so C is reflexive. □

Lemma 2 Let [S] be a subnormal tuple on a separable space H. Then the minimal normal extension of [S] acts also on a separable space.

Proof: As H is the closure of a countable union of cyclic subspaces, it is sufficient to prove the lemma for cyclic tuples. Assume, therefore, that H = P^{2}(μ) for some finite Borel measure μ. As P^{2}(μ) is separable, the set of monomials {z_{α_{1}}^{i_{1}}z_{α_{2}}^{i_{2}} … z_{α_{k}}^{i_{k}}} has a countable dense subset {z^{β_{1}}, z^{β_{2}}, …}, for some multi-indices β_{1}, β_{2}, . . ..

The set of all rational linear combinations of {z^{β_{m}}} and {z̄^{β_{n}}} is then a countable dense subset of L^{2}(μ). □

Lemma 3 Let μ be a finite Borel measure on a torus. For every weak-star continuous linear functional Λ on P^{2}(μ) ∩ L^{∞}(μ), and every ϵ > 0, there exist f, g ∈ P^{2}(μ) such that

Λ(ϕ) = ∫ ϕfḡdμ

for every ϕ in P^{2}(μ) ∩ L^{∞}(μ), and ∥f∥, ∥g∥ ≤ (1 + ϵ)∥Λ∥^{1/2}.

Proof: First, let us assume that μ is a finite measure on 𝕋^{2}. Let Λ be any weak-star continuous linear functional on P^{2}(μ) ∩ L^{∞}(μ). We will assume that Λ is non-zero. Let δ be a small positive number that we shall specify later. Fix a function F in L^{1}(μ) so that

Λ(ϕ) = ∫ ϕF dμ, and ∥F ∥_{L^{1}(μ)} ≤ (1 + δ)∥Λ∥.

We wish to write F as fḡ for some f, g in P^{2}(μ).

(i) First we do it approximately. Observe that F can be approximated within δ′ = δ^{2}/12 in L^{1}(μ) by a finite linear combination of characteristic functions of disjoint sets that are the product of open intervals in the unit circle:

∫ |F − ∑_{k=1}^{n} a_{k}χ_{I_{k}×J_{k}}|dμ < δ′.

Moreover, this can be done in such a way that for any open rectangle I_{k} × J_{k}, there are larger intervals I′_{k} ⊃ Ī_{k} and J′_{k} ⊃ J̄_{k} such that

μ(I′_{k} × J′_{k} \ I_{k} × J_{k}) < \frac{δ′}{nM} ,

where M = ∥μ∥(1 + max{|a_{k}| : 1 ≤ k ≤ n}). Let ξ_{k} (resp. η_{k}) be functions in the disk algebra (the closure of the polynomials in the uniform norm on the unit disk) that have norm 1, modulus 1 on I_{k} (resp. J_{k}) and modulus δ′/nM off I′_{k} (resp. J′_{k}). As ξ_{k} and η_{k} are approximable in sup-norm by polynomials, the map (z_{1}, z_{2}) → ξ_{k}(z_{1})η_{k}(z_{2}) is in P^{2}(μ). Let

f_{0}(z_{1}, z_{2}) = ∑_{k=1}^{n} \frac{a_{k}}{\sqrt{|a_{k}|}}ξ_{k}(z_{1})η_{k}(z_{2}),
g_{0}(z_{1}, z_{2}) = ∑_{k=1}^{n} \sqrt{|a_{k}|}ξ_{k}(z_{1})η_{k}(z_{2}).

Both f_{0} and g_{0} are in P^{2}(μ), ∥f_{0}∥, ∥g_{0}∥ ≤ (∥F ∥_{L^{1}(μ)} + 3δ′)^{1/2}, and

∫ |F − f_{0}ḡ_{0}|dμ < 3δ′ = δ^{2}/4.

(ii) Now we try to improve our approximation in the standard way. Suppose we are given f_{1} in P^{2}(μ), g_{1} in L^{2}(μ), ϵ_{1}, ϵ_{2} > 0, such that

∫ |F − f_{1}ḡ_{1}|dμ < ϵ_{1}, and ∥f_{1}∥, ∥g_{1}∥ ≤ (∥F ∥_{L^{1}(μ)} + ϵ_{1})^{1/2}.

We claim we can find f_{2} in P^{2}(μ), g_{2} in L^{2}(μ), such that

∥f_{1} − f_{2}∥ ≤ 2\sqrt{ϵ_{1}}, ∥g_{2}∥ ≤ ∥g_{1}∥ + 2\sqrt{ϵ_{1}}  (4)

and

∫ |F − f_{2}ḡ_{2}|dμ < ϵ_{2}.  (5)

To prove the claim, fix a very small ϵ′ > 0. By part (i), we can find ξ in P^{2}(μ), η in L^{2}(μ), with

∥ξ∥^{2} = ∥η∥^{2} ≤ (1 + ϵ′)∥F − f_{1}ḡ_{1}∥_{L^{1}(μ)},

and

∥F − f_{1}ḡ_{1} − ξη̄∥_{L^{1}(μ)} ≤ ϵ′.

Let E = {z ∈ 𝕋^{2} : |f_{1}(z)| < |ξ(z)|}. We can find two open sets B_{1} and B_{2} in 𝕋^{2}, such that E ⊂ B_{2}, B̄_{1} ⊂ B_{2}, μ(B_{1} \ E) + μ(E \ B_{1}) < ϵ′, μ(B_{2} \ B_{1}) < ϵ′. There exists u in the poly-disk algebra of norm 2, |u(z)| = 2 if z ∈ B̄_{1}, and |u(z)| = ϵ′ if z ∉ B_{2}. Now define f_{2} = f_{1} + uξ and

g_{2}(z) = \{ \begin{matrix}\frac{f̄_{1}g_{1}+ξ̄η}{f̄_{1}+ūξ̄} & z ∈ (B_{1} ∩ E) ∪ (𝕋^{2} \ B_{2}) \\ 0 & z ∈ B_{2} \ B_{1}. \end{matrix}

Thus f_{2}, g_{2} will satisfy (4) and (5) if ϵ′ is sufficiently small (to get (5), note that ∥g_{2}∥ ≤ \frac{1}{1−ϵ′} (∥g_{1}∥ + ∥η∥) ).

One can now continue inductively, to get f_{n} in P^{2}(μ) and g_{n} in L^{2}(μ) with

∫ |F − f_{n}ḡ_{n}|dμ ≤ 4^{−n−1}δ^{2}
∥f_{n} − f_{n+1}∥ ≤ 2^{−n}δ
∥g_{n+1}∥ ≤ ∥g_{n}∥ + 2^{−n}δ

Let f be the norm limit of {f_{n}}, and h be any weak cluster-point of {g_{n}}. Finally let g be the projection of h from L^{2}(μ) onto P^{2}(μ). Then for any ϕ in P^{2}(μ) ∩ L^{∞}(μ),

Λ(ϕ) = ∫ ϕF dμ = ∫ ϕfh̄dμ = ∫ ϕfḡdμ,
∥f∥ ≤ ∥f_{0}∥ + δ ≤ ((∥F ∥_{L^{1}(μ)} + 3δ′)^{1/2} + δ) ≤ (∥Λ∥ + δ)^{1/2} + δ,

and

∥g∥ ≤ ∥g_{0}∥ + δ ≤ (∥Λ∥ + δ)^{1/2} + δ.

Thus f and g are as required if δ is small enough.

The modifications for finitely many isometries are obvious. If there are an infinite number, we use the fact that any finite Borel measure on a torus is regular, and again approximate L^{1}(μ) functions by linear combinations of characteristic functions of products of intervals, where all but finitely many of the factors are the whole circle. □

References

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[3] J.B. Conway. The Theory of Subnormal Operators. American Mathematical Society, Providence, 1991.

[4] J.A. Deddens. Every isometry is reflexive. Proc. Amer. Math. Soc., 28:509–512, 1971.

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