Backward intertwiners and a transitive commutant

1 Introduction

Let H be a complex Hilbert space and let \B(H) denote its bounded linear operators. The commutant of S\in\B(H) is

A closed subspace is hyperinvariant for S if every operator in \{S\}' leaves it invariant. An operator algebra is transitive if its only closed common invariant subspaces are \{0\} and H. Thus S has no nonzero proper closed hyperinvariant subspace precisely when its commutant is transitive.

The hyperinvariant-subspace problem asks whether every bounded nonscalar operator on an infinite-dimensional separable complex Hilbert space has a nonzero proper closed hyperinvariant subspace. The construction below gives a negative answer. Its purpose is to exhibit the mechanism explicitly: the ordinary invariant subspaces of each weighted-shift fiber are coordinate tails, and a second, commuting operator moves those tails backward. Invariance under the commutant therefore imposes a strict descent that cannot persist on a probability space.

1.1 The construction theorem

Put K=\ell^2(\Z), with orthonormal basis (e_j)_{j\in\Z}, and define

Let X=\Z_2 be the compact additive group of 2-adic integers, the inverse limit of the groups \Z/2^m\Z. Equip it with completed Haar probability \mu, put \theta x=x+1, and set H_0=L^2(X,\mu;K). For g\in L^\infty(X), the scalar multiplier is (M_gf)(x)=g(x)f(x).

All generators in the last assertion commute with S. The explicit power bound implies norm-quasinilpotence, \lim_{n\to\infty}\|S^n\|^{1/n}=0, while the first assertion will give many ordinary invariant subspaces of the resulting operator.

Here the strong operator topology is pointwise norm convergence on H. Consequently the commutant in the corollary also gives a negative answer to the question whether every unital transitive operator algebra is strongly dense in \B(H).

1.2 Context and relation to the companion

Douglas and Pearcy studied the connection between hyperinvariant subspaces and transitive algebras [4]. A major positive result is Arveson’s density theorem: a transitive algebra on a separable Hilbert space which contains a maximal abelian self-adjoint subalgebra of \B(H) is strongly dense in \B(H) [1, Theorem 3.3]. The scalar multiplier algebra used here has infinite multiplicity. Section 6 gives an explicit operator commuting with every such multiplier that is not itself a scalar multiplier, so the maximality hypothesis is absent.

Domar’s work on invariant subspaces of weighted translation spaces [3] and Grabiner’s work on unicellular shifts [6] provide the classical lattice ideas behind the fibers. Section 2 proves the precise quadratic weighted-translation criterion we need. Its convex-envelope and support-separation argument follows Domar’s method. The proof includes both normalized convolution-tail estimates; no general criterion for arbitrary weights is assumed. Our additional task is to link the fibers by a bounded commuting operator that moves their invariant tails backward. Opposite logarithmic changes at the two ends of the bilateral sequence make this possible, and one measurable normalization controls the operator over the whole base.

Atzmon and Sodin constructed completely indecomposable bilateral weighted shifts with spectrum the unit circle [2]; their 1999 paper left open whether those shifts have proper hyperinvariant subspaces. Here the extra commutant action is supplied by transport between different fibers. For a study of operators with shift-type invariant subspaces, see Kérchy [7]. The invariant- and hyperinvariant-subspace problems must remain distinct: the Kérchy–Pearcy preprint claiming an equivalence was withdrawn after a counterexample to its Theorem 3 [8]. The operator constructed here preserves every fixed tail L^2(X;\Ktail{h}); its hyperinvariant conclusion is proved directly.

The companion paper [9] constructs, for every prescribed irrational rotation, a continuous circle weight whose operator in the rotation factor has no nontrivial invariant projection in that factor. It derives the existence conclusion of Corollary 1.2 by the usual projection argument; see also Dykema [5, Proposition 2.1]. The present construction has a different purpose: it exhibits the ordinary invariant tails, the bounded operator that moves them backward, and the explicit estimate (1). The two proofs use different constructions. Our only proof imports from the companion are the two elementary facts stated below.

1.3 Proof overview and elementary inputs

The proof first makes the possible fiber subspaces completely explicit. Section 2 classifies invariant subspaces of a weighted translation whose logarithmic weights have opposite quadratic growth at the two ends. Section 3 constructs weights over \Z_2 satisfying that criterion in every fiber. Their decay also gives the uniform power estimate, which proves norm-quasinilpotence of the operator on H_0 rather than merely of its individual fibers.

To couple the fibers, we prescribe a map from e_j at x to a positive multiple of e_{j-1} at \theta x. Commutation with the forward shifts determines a recurrence for those multiples. Section 4 solves the recurrence in both directions. The logarithmic changes of the weights under \theta have opposite signs at the two ends; the coefficients therefore decay in both directions. Dividing all coefficients in each fiber by their common supremum yields the contraction V without changing commutation.

Finally, a subspace invariant under every scalar multiplier has an orthogonal projection represented by a measurable projection field. Invariance under S makes each fiber a tail \Ktail{s(x)}. Section 5 uses the dense images under V_x to obtain

Probability preservation excludes finite labels, and ergodicity excludes a nonconstant mixture of full and zero fibers. Section 6 then proves the corollary and records the structure of the example.

We use the following two elementary inputs, with the stated scope. Their complete proofs are in the companion [9].

1. 1.

   Fields from scalar multipliers [9, Lemma 3.1]. For a probability space (Y,\nu), K=\ell^2(\Z), and A\in\B(L^2(Y;K)) commuting with every scalar multiplier, there is an essentially bounded field A_y\in\B(K), unique almost everywhere, such that

   (Af)(y)=A_yf(y),\qquad \|A_y\|\le\|A\|\quad\text{a.e.}

   For every v\in K, the map y\mapsto A_yv is measurable. Products and adjoints act fiberwise; an orthogonal projection has orthogonal projection fibers outside one null set.

2. 2.

   The commutant algebra [9, Lemma 1.2]. For any bounded nonscalar operator on a complex Hilbert space, its commutant is a proper strongly closed unital complex algebra. If its only closed common invariant subspaces are zero and the whole space, that algebra is transitive.

The first input requires no base transformation, covariance, trace or factor assumption. We apply it only after proving reduction by all scalar multipliers. The second requires neither separability nor quasinilpotence; the constructed operator will satisfy its nonscalarity hypothesis. The field input is used in Section 5, and the commutant input in Section 6.

2 The quadratic weighted-translation criterion

We first prove a criterion that makes every invariant fiber subspace a coordinate tail. The proof concerns one weighted sequence and does not involve the base transformation.

For a positive sequence w=(w_j)_{j\in\Z}, set

If w is decreasing, right translation (Rz)_j=z_{j-1} is a contraction on this space. For a sequence q_j on an integer interval, convexity means that q_{j+1}-q_j is nondecreasing wherever the differences are defined; concavity means the reverse.

The convex-envelope and support-separation argument below follows the method of Domar’s weighted-translation work [3, Section 3]. We prove the stated quadratic specialization in full, including the normalized estimates for both convolution tails. The signed exponent for a negative index is essential: the corresponding summand is w_{-n}^{-1/n} for n>0.

3 The 2-adic weights

We now construct the measurable family of shifts. Their logarithmic weights must satisfy the criterion of Section 2 while changing in opposite directions at the two ends when the base moves by one.

3.1 The base transformation

For the inverse-limit base X=\Z_2 fixed in the introduction and m\ge1, let

Thus each residue is uniformly distributed. The map \theta x=x+1 is an invertible probability-preserving transformation, and

The finite residue classes generate the Borel sigma-algebra of X.

3.2 The weights

Fix

The integer blocks I_m=\{k:\ell_{m-1}\le k<\ell_m\} partition the positive integers. Set J_1=0 and J_{m+1}=J_m+DN_m; equivalently J_m=D(N_m-2). Define a_0^+(x)=a_0^-(x)=c, and for k\in I_m put

On the full integer axis define

The two central weights satisfy \beta_{-1}(x)=\beta_0(x)=e^{-c} for every x. The linear terms in a_k^\pm give quadratic growth of the cumulative logarithms. The smaller residue corrections change by +D and -D under \theta unless the residue wraps around; the increments of J_m preserve monotonicity across block boundaries. We first check the invariant-subspace criterion, then use those opposite changes to construct the backward intertwiner.

We may now define S\in\B(H_0) by (Sf)(x)=S_xf(x). It has norm e^{-c}. For each g\in L^\infty(X), the scalar multiplier (M_gf)(x)=g(x)f(x) commutes with S. Before constructing the operator that links different fibers and moves their tails backward, we record the uniform power bound for S.

4 A bounded backward intertwiner

We seek an operator which sends e_j in the fiber at x to a positive multiple of e_{j-1} in the fiber at \theta x. Commutation with the forward shift determines a recurrence for those multiples. The opposite logarithmic drifts at the two ends of the fiber make its solutions bounded after normalization.

4.1 Two-ended drift

Write \Delta a_j(x)=a_j(\theta x)-a_j(x). A block I_m has an exceptional residue at x when T_m(x)=N_m-1. Since

the Borel–Cantelli Lemma shows that almost every x has only finitely many such blocks. Intersect this conull set over all iterates of \theta to obtain a measurable \theta-invariant conull set X_*. No independence of the exceptional events is needed.

The constant C(x) may be unbounded and nonintegrable as a function of x. Only its pointwise finiteness is used below.

4.2 Solving and normalizing the recurrence

For positive coefficients b_j(x), sending e_j at x to b_j(x)e_{j-1} at \theta x would commute with S precisely when

Figure 1 shows the two paths in this coefficient identity. The left side is obtained by moving backward and then forward, and the right side in the opposite order.

We first solve this equation with initial value one, then divide all coefficients in each fiber by a common bound. For every x\in X set \widetilde b_0(x)=1 and define positive coefficients at all integer indices by

Every coefficient is a finite product of positive measurable functions and their reciprocals, so it is finite and measurable.

Define a backward shift V_x:K\to K by V_xe_j=b_j(x)e_{j-1}. Every V_x has norm at most one, and x\mapsto V_xv is measurable for every v\in K by finite-coordinate approximation. It defines V\in\B(H_0) by

Invariance of \mu gives \norm{Vf}^2=\int_X\norm{V_xf(x)}^2\,d\mu(x)\le\norm{f}^2.

The density conclusion, rather than surjectivity, is exactly what is needed for closed invariant subspaces. No bounded inverse of V_x or V is asserted.

5 All common invariant subspaces

We now show that the scalar multipliers, S, and V have no proper nonzero closed common invariant subspace. The first step accounts for every closed subspace invariant under the multipliers, without assuming in advance that it is defined fiber by fiber.

5.1 Multipliers force a measurable field of projections

Let M\subseteq H_0 be a closed subspace invariant under S, V, and every scalar multiplier. Since M_{\bar g}=M_g^* belongs to the same multiplier family, M reduces each multiplier. Its orthogonal projection P therefore commutes with all of them. The field lemma [9, Lemma 3.1], applied with the probability space (X,\mu) and K=\ell^2(\Z), gives

where the P_x are orthogonal projections outside one null set and x\mapsto P_xv is measurable for each v\in K. These are exactly the projection and measurability conclusions needed below; no covariance property is imposed on this field.

5.2 Fiber invariance and measurable tail labels

For this fixed subspace M and its projection field P_x, the global invariance identities (I-P)SP=0 and (I-P)VP=0 imply

almost everywhere. To justify the passage, apply each global identity to the constant field e_j. In the second identity evaluate the output at \theta x and use probability preservation. The resulting equalities hold for every j on a common conull set and extend to all of K by boundedness. Intersect this set, the projection set, and X_* over all integer iterates of \theta. We can henceforth use all these identities on a single invariant conull set. This set is chosen after fixing M; no intersection over all candidate subspaces is required.

By Proposition 3.2 and (31), there is a label

It is measurable: for every j\in\Z,

on the conull set, and the label can be set to +\infty elsewhere. Equation (32) and Proposition 4.3 give two implications:

The first uses \overline{V_xK}=K. The second uses \overline{V_x\Ktail{h}}=\Ktail{h-1} and the inclusion \Ktail{h-1}\subseteq\ran P_{\theta x}. When the input fiber is zero, there is no restriction on the output label.

5.3 Strict descent on a probability space

The final measure argument is useful independently of the construction.

Applying Lemma 5.1 shows that P_x=I almost everywhere or P_x=0 almost everywhere. We have therefore proved that the unital complex algebra generated by S, V, and all scalar multipliers is transitive.

6 The commutant and structural consequences

6.1 Ordinary invariant subspaces and compactness

For every h\in\Z, the subspace L^2(X;\Ktail{h}) is closed, nonzero, proper, and S-invariant. It is not invariant under V: the image of the constant field e_h has its nonzero coordinate at h-1 almost everywhere. Thus the construction separates ordinary invariance from hyperinvariance in an explicit way.

Each fiber S_x is compact. Indeed a_k^\pm\ge c(k+1) implies \beta_j(x)\to0 as |j|\to\infty, and cutting off all but finitely many shift weights approximates S_x in operator norm. The operator S on H_0 is not compact. There is an infinite orthonormal sequence (\varphi_n) in L^2(X), since the N_m disjoint residue classes give N_m orthogonal indicators for every m. The orthonormal vectors \varphi_n\otimes e_0 have images

These images are orthogonal with a common positive norm and have no norm-convergent subsequence.

6.2 The multiplier algebra has infinite multiplicity

Let Q_0 be the projection onto L^2(X)\otimes\C e_0. This gives a concrete witness to the properness of the commutant: on 1\otimes e_0,

so Q_0\notin\{S\}'.

The same projection commutes with every scalar multiplier, but is not itself a scalar multiplier. If Q_0=M_g, its action on 1\otimes e_0 would give g=1 almost everywhere, whereas its action on 1\otimes e_1 is zero. The abelian self-adjoint algebra \{M_g:g\in L^\infty(X)\} is therefore not maximal abelian in \B(H_0). Arveson’s maximal-abelian inclusion theorem [1, Theorem 3.3] does not apply to this subalgebra of the transitive commutant.

6.3 Atomic finite families and a diffuse joint algebra

We finish with a spectral property of the constructed weights. This subsection is independent of the transitivity proof.

Let |S|=(S^*S)^{1/2}. Identify H_0 with L^2(X\times\Z,\mu\times\#), where \# is counting measure, and let Ue_j=e_{j+1} on every fiber. Then

All weights are positive, so |S| is injective with dense range and U is the unitary polar part of S. Define

For a family of operators, W^* denotes the smallest strongly closed unital self-adjoint algebra containing that family. A nonzero projection in an abelian von Neumann algebra is minimal if it contains no smaller nonzero projection in that algebra. Such an algebra is atomic when its identity is the strong sum of its minimal projections, and diffuse when it has no minimal projection. We will show that W^*(A_n:n\in F) is atomic for every finite F\subset\Z, whereas W^*(A_n:n\in\Z) is diffuse.

For any finite F\subset\Z and fixed j, the tuple (\beta_{j+n}(x))_{n\in F} depends on finitely many residues T_m(x). Because the residues are nested, it depends on one residue T_M(x) and takes finitely many values. Allowing the countably many values of j gives a countable range for the joint tuple on X\times\Z. Its nonempty level sets have positive measure: on every fixed coordinate slice they are finite unions of residue classes. These level sets give the minimal projections of the generated abelian algebra, and their sum is the identity. Thus W^*(A_n:n\in F) is atomic. In particular, |S| has a pure-point spectral resolution.

For the full family, the two equal central weights determine the coordinate slice. They are the only weights equal to e^{-c}, since a_k^\pm\ge c(k+1)>c for every k\ge1. At (x,j) the tuple indices n for which \beta_{j+n}(x)=e^{-c} are therefore exactly -j-1 and -j. For every r\in\Z, the projection onto the slice j=r, denoted by P_r, belongs to W^*(A_n:n\in\Z): it is

Next fix m\ge1 and choose k_m=\ell_{m-1}\in I_m. On the slice j=r, the multiplier A_{k_m-r} supplies \beta_{k_m}(x). For 0\le t<N_m, put

Equation (17) shows that the multiplication projection of the residue rectangle \{T_m=t\}\times\{r\} is

Thus every residue-class projection on every coordinate slice belongs to the generated algebra. Such rectangles generate the product sigma-algebra. Conversely all A_n are multiplication operators, so, modulo completion and null sets,

in its multiplication representation. Haar measure on \Z_2 has no atoms, and consequently this algebra is diffuse.

The pure-point assertion does not assert discreteness of the spectrum or compactness of S. Nor is this full multiplier algebra the scalar base algebra used in the transitivity proof. In fact |S| does not commute with S, because the adjacent weights \beta_0=e^{-c} and \beta_1\le e^{-2c} differ. The polar-conjugate algebra is therefore not contained in \{S\}'.