A VERSION OF LOMONOSOV’S THEOREM FOR COLLECTIONS OF POSITIVE OPERATORS

ALEXEY I. POPOV AND VLADIMIR G. TROITSKY

Abstract. It is known that for every Banach space X and every proper W OT -closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x ∈ X and f ∈ X^{∗} such that ⟨f, T x⟩ = 0 for all T ∈ A. In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this as follows: without having a compact operator in the algebra, one has |⟨f, T x⟩| ⩽ ∥T_{∗}∥_{e} for all T ∈ A. In this paper, we prove a similar extension of a result of R. Drnovšek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero x ∈ X_{+} and f ∈ X_{+}^{∗} such that ⟨f, T x⟩ ⩽ ∥T_{∗}∥_{e} for all T ∈ C.

In this paper we use techniques which were recently developed for transitive algebras to obtain analogous results for collections of positive operators on Banach lattices. Let us first briefly describe these two branches of the Invariant Subspace research.

Transitive algebras. Suppose that X is a Banach space. A subspace Z of X is said to be invariant under an operator T ∈ L(X) if {0} ≠ Z ≠ X and T (Z) ⊆ Z. The Invariant Subspace Problem deals with the question: “Which operators have invariant subspaces?”. Lomonosov proved in [7] that an operator which commutes with a compact operator has an invariant subspace. There is also an algebraic version of the problem: which subalgebras of L(X) have no (common) invariant subspaces? Such subalgebras are called transitive. The classical Burnside’s theorem asserts that if X is finite-dimensional then L(X) has no proper transitive subalgebras (clearly, L(X) itself is always transitive). Using Lomonosov’s technique, Burnside’s theorem can be extended to the infinite-dimensional case as follows:

Theorem 1 ([11, Theorem 8.23]). A proper W OT -closed subalgebra of L(X) containing a compact operator is not transitive.

A “quantitative” version of the latter theorem was obtained by Lomonosov in [8] for algebras of adjoint operators. Before we state it, we need to introduce some notation. It is easy to see that a subalgebra A of L(X) has an invariant subspace if and only if

Proc. AMS, 137 (2009), 1793-1800.

1 2 A. I. POPOV AND V. G. TROITSKY

2000 Mathematics Subject Classification. 47B65; 47A15.

Key words and phrases. Positive operator, adjoint operator, transitive algebra.

there exist non-zero x ∈ X and f ∈ X^{∗} such that ⟨f, T x⟩ = 0 for every T ∈ A. Now suppose that X is a dual space; that is, X = Y^{∗} for some Banach space Y. If T ∈ L(X) is a bounded adjoint operator on X then there is a unique operator S ∈ L(Y) such that S^{∗} = T. We will write S = T_{∗}; there will be no ambiguity as T_{∗} will always be taken with respect to Y. We will write ∥T ∥_{e} for the essential norm of T, i.e., the distance from T to the space of compact operators. Note that in general, for an adjoint operator T, one has ∥T ∥_{e} ⩽ ∥T_{∗}∥_{e}. See [2] for an example of T such that ∥T ∥_{e} < ∥T_{∗}∥_{e}.

Theorem 2 ([8]). Let X be a dual Banach space and A a proper W^{∗}OT -closed subalgebra of L(X) consisting of adjoint operators. Then there exist non-zero x ∈ X and f ∈ X^{∗} such that |⟨f, T x⟩| ⩽ ∥T_{∗}∥_{e} for all T ∈ A.

Invariant ideals of collections of positive operators. Suppose now that X is a Banach lattice. Recall that a linear (not necessarily closed) subspace J ⊆ X is called an order ideal if it is solid, i.e., y ∈ J implies x ∈ J whenever |x| ⩽ |y|. The following version of Lomonosov’s theorem for positive operators was proved by B. de Pagter [10]: a positive quasinilpotent compact operator on X has a closed invariant order ideal. There have been many extensions of this result, see, e.g., [1]. In particular, R. Drnovšek [3] showed that a collection of positive operators satisfying certain assumptions has a (common) invariant closed ideal. To state his result precisely, we need to introduce more notations.

As usual, we write X_{+}, X_{+}^{∗}, and L(X)_{+} for the cones of positive elements in X, X^{∗}, and L(X), respectively. Let C be a collection of positive operators on X. Following [1], we will denote by symbols ⟨C] and [C⟩ the super left and the super right commutants of C, respectively, i.e.,

⟨C] = {S ∈ L(X)_{+} : ST ⩽ T S for each T ∈ C},
[C⟩ = {S ∈ L(X)_{+} : ST ⩾ T S for each T ∈ C}.

If D is another collection of operators then we write CD = {T S: T ∈ C, S ∈ D}. The symbol C^{n} is defined as the product of n copies of C.

An operator T is locally quasinilpotent at x if lim sup_{n}∥T^{n}x∥^{\frac{1}{n}} = 0. If U is a subset of X then we write ∥U∥ = sup{∥x∥: x ∈ U}. We call a collection C of operators finitely quasinilpotent at a vector x ∈ X if lim sup_{n}∥F^{n}x∥^{\frac{1}{n}} = 0 for every finite subcollection F of C. Clearly, finite quasinilpotence at x implies local quasinilpotence at x of every operator in the collection.

If E is a Banach lattice then an operator T : E → E is called AM-compact if the image of every order interval under T is relatively compact. Since order intervals are norm bounded, every compact operator is AM-compact. An operator T is said to dominate an operator S if |Sx| ⩽ T |x| holds for all x ∈ E.

Theorem 3 ([3]). If C is a collection of positive operators on a Banach lattice X such that

(i) C is finitely quasinilpotent at some positive non-zero vector, and (ii) some operator in C dominates a non-zero AM-compact operator,

then C and [C⟩ have a common closed invariant order ideal.

Observe that if a collection C of positive operators has a (closed nontrivial) invariant ideal then there exist non-zero positive x and f such that ⟨f, T x⟩ = 0 for all T ∈ C. The converse is also true when C is a semigroup.

The goal of this paper is to “quantize” Theorem 3 in the same manner that Theorem 1 was “quantized” into Theorem 2. Our proofs use ideas from [6] and [9].

In the following lemma, we collect several standard facts that we will use later. See, e.g., [1] for the proofs.

Lemma 4. Let Z be a vector lattice, x ∈ Z_{+}. Then for each y, z ∈ Z one has (i) |x ∧ y − x ∧ z| ⩽ |y − z|; (ii) if |y| ⩽ z then |x − x ∧ z| ⩽ |x − x ∧ y|; (iii) |x − x ∧ y| ⩽ |x − y|.

From now on, X will be a real Banach lattice. We will also assume that X is a dual Banach space; that is, X = Y^{∗} for some (fixed) Banach space Y. We will start with a version of Theorem 2 for convex collections of positive operators.

Theorem 5. Let C be a convex collection of positive adjoint operators on X. If there is x_{0} > 0 such that every operator from C is locally quasinilpotent at x_{0} then there exist non-zero x ∈ X_{+} and f ∈ X_{+}^{∗} such that ⟨f, T x⟩ ⩽ ∥T_{∗}∥_{e} for all T ∈ C.

Remark 6. One might try to deduce Theorem 5 from Theorem 2 by considering the W^{∗}OT -closed algebra generated by C. However, the example in [5] shows that there exists an algebra of nilpotent operators on a Hilbert space H which is W OT -dense in L(H).

Proof of Theorem 5. Clearly, we may assume that ∥x_{0}∥ = 1. Also, without loss of generality, C is closed under taking positive multiples of its elements, otherwise we replace C with \{αT: T ∈ C, 0 < α ∈ ℝ\}. Fix 0 < ε < \frac{1}{10}. Define

C_{ε} = {T ∈ C: ∥T_{∗}∥_{e} < ε} and
H_{ε}(x) = {z ∈ X: |z| ⩽ T x for some T ∈ C_{ε}}, x ∈ X_{+}.

Then H_{ε}(x) is convex and solid for all x ∈ X_{+}.

Suppose that \overline{H_{ε}(x)} ≠ X for some nonzero x ∈ X_{+}. Since H_{ε}(x) is convex, there is a nonzero g ∈ X^{∗} such that g(y) ⩽ 1 for all y ∈ H_{ε}(x). Consider h = |g| ∈ X^{∗}. Then for any y ∈ H_{ε}(x) we have

h(y) ⩽ h(|y|) = sup{g(u): − |y| ⩽ u ⩽ |y|} ⩽ 1

since H_{ε}(x) is solid. In particular, ⟨h, T x⟩ ⩽ 1 for all T ∈ C_{ε}.

Put f = \frac{ε}{2}h. We claim that ⟨f, T x⟩ ⩽ ∥T_{∗}∥_{e} for each T ∈ C. Indeed, if T is compact, i.e., ∥T_{∗}∥_{e} = 0, then αT ∈ C_{ε} for all 0 < α ∈ ℝ. Therefore ⟨h, αT x⟩ ⩽ 1 for all 0 < α ∈ ℝ, so that ⟨f, T x⟩ = ⟨h, T x⟩ = 0. If T is not compact then \frac{εT}{2∥T_{∗}∥_{e}} ∈ C_{ε}, whence

⟨f, T x⟩ = ∥T_{∗}∥_{e}⟨h, \frac{εT}{2∥T_{∗}∥_{e}} x⟩ ⩽ ∥T_{∗}∥_{e}.

Suppose now that \overline{H_{ε}(x)} = X for all nonzero x ∈ X_{+}. Then, in particular, for each x ∈ X there is y_{x} ∈ H_{ε}(x) such that ∥x_{0} − y_{x}∥ < ε. Fix an operator T_{x} ∈ C_{ε} such that |y_{x}| ⩽ T_{x}x. Then (ii) and (iii) of Lemma 4 yield ∥x_{0} − x_{0} ∧ T_{x}x∥ < ε.

Let U_{0} = \{x ∈ X_{+} : ∥x − x_{0}∥ ⩽ \frac{1}{2}\}. Since ∥(T_{x})_{∗}∥_{e} < ε, there is an adjoint compact operator K_{x} ∈ K(X) such that ∥K_{x} − T_{x}∥ < ε. As compact adjoint operators are w^{∗}-∥·∥ continuous on norm bounded sets, it follows that there is a relative (to U_{0}) w^{∗}- open neigborhood W_{x} ⊆ U_{0} of x such that ∥K_{x}z − K_{x}x∥ < ε whenever z ∈ W_{x}. Then for every y ∈ W_{x} we have:

∥x_{0} − x_{0} ∧ T_{x}y∥ ⩽ ∥x_{0} − x_{0} ∧ T_{x}x∥ + ∥x_{0} ∧ T_{x}x − x_{0} ∧ K_{x}x∥

+ ∥x_{0} ∧ K_{x}x − x_{0} ∧ K_{x}y∥ + ∥x_{0} ∧ K_{x}y − x_{0} ∧ T_{x}y∥

⩽ ∥x_{0} − x_{0} ∧ T_{x}x∥ + ∥T_{x}x − K_{x}x∥ + ∥K_{x}x − K_{x}y∥ + ∥K_{x}y − T_{x}y∥

< ε + ε∥x∥ + ε + ε∥y∥ < 5ε < \frac{1}{2}.

Together with T_{x} ⩾ 0 this yields (x_{0} ∧ T_{x}y) ∈ U_{0} for each y ∈ W_{x}.

Note that U_{0} is w^{∗}- compact since U_{0} is the intersection of X_{+} with a closed ball. Hence, we can find x_{1}, …, x_{n} ∈ U_{0} such that U_{0} = ⋃_{k=1}^{n} W_{x_{k}}. Define T = T_{x_{1}} + ⋯ + T_{x_{n}} ∈ C. Then by Lemma 4(ii), we have x_{0} ∧ T x ∈ U_{0} for every x ∈ U_{0}.

A VERSION OF LOMONOSOV’S THEOREM 5

Define a sequence (y_{n}) ⊆ U_{0} by y_{0} = x_{0} and y_{n+1} = x_{0} ∧ T y_{n}. Clearly 0 ⩽ y_{n} for all n, and y_{n} ⩽ T y_{n−1} ⩽ … ⩽ T^{n}y_{0}, so that ∥y_{n}∥ ⩽ ∥T^{n}x_{0}∥. Thus y_{n} → 0 as n → ∞ by local quasinilpotence. This is a contradiction by the definition of U_{0}. □

The next theorem shows that the conclusion of Theorem 5 is also true for some collections of operators which are not necessarily convex. We will, however, use a more restrictive quasinilpotence condition. We will need some additional definitions.

Let C be a collection of positive operators. Following [1], define

D_{C} = {D ∈ L(X)_{+} : ∃T_{1}, …, T_{k} ∈ [C⟩ and

S_{1}, …, S_{k} ∈ ⋃_{n=1}^{∞} C^{n} such that D ⩽ ∑_{i=1}^{k} T_{i}S_{i}}

In other words, D_{C} is the smallest additive and multiplicative semigroup which contains the collection [C⟩ · C and such that T ∈ D_{C} and 0 ⩽ S ⩽ T imply S ∈ D_{C} (see [1]).

Let C be a collection of positive adjoint operators on X. Define

E_{C} = {T ∈ D_{C}: T = S^{∗} for some S ∈ L(Y)}.

Since adjoint operators are stable under addition and multiplication, E_{C} is an additive and multiplicative semigroup. It is also clear that C ⊆ E_{C}.

Theorem 7. Let C be a collection of positive adjoint operators on X. If C is finitely quasinilpotent at some x_{0} > 0 then there exist non-zero x ∈ X_{+} and f ∈ X_{+}^{∗} such that ⟨f, T x⟩ ⩽ ∥T_{∗}∥_{e} for all T ∈ E_{C}.

Proof. Clearly E_{C} is convex. Note that the finite quasinilpotence of C at x_{0} implies the finite quasinilpotence of D_{C} (and, therefore, of E_{C}) at x_{0} (see, e.g., [1, Lemma 10.43]). Finally, apply Theorem 5 to E_{C}. □

Now suppose, in addition, that Y is itself a Banach lattice. Then we can improve the conclusion of Theorem 5. For an operator T acting on Y, define

θ(T) = inf{∥T − K∥: K is AM-compact}.

Clearly, θ is a seminorm on L(Y) and θ(T) = 0 if and only if T is AM-compact (because the subspace of AM-compact operators in L(Y) is norm closed).

For ξ ∈ Y_{+}, define a seminorm ρ_{ξ} on X via ρ_{ξ}(x) = |x|(ξ).

Lemma 8. If ξ ∈ Y_{+} and K ∈ L(Y) is AM-compact, then K^{∗}: (B_{X}, w^{∗}) → (X, ρ_{ξ}) is continuous.

Proof. Let x_{α} ⟶^{w^{∗}} x, with x_{α}, x ∈ B_{X}. Write

ρ_{ξ}(K^{∗}x_{α} − K^{∗}x) = |K^{∗}x_{α} − K^{∗}x|(ξ) = sup_{−ξ⩽ζ⩽ξ}⟨x_{α} − x, Kζ⟩ = sup_{ν∈A}⟨x_{α} − x, ν⟩,

where A = K([−ξ, ξ]). By assumption, K is AM-compact, thus A is a ∥·∥-compact set.

For ν ∈ A, fix α_{ν} such that |⟨x_{α} − x, ν⟩| < \frac{ε}{3} whenever α ⩾ α_{ν}. If μ ∈ Y is such that ∥μ − ν∥ < \frac{ε}{3} then for α ⩾ α_{ν} we have

|⟨x_{α} − x, μ⟩| ⩽ \frac{ε}{3}∥x_{α} − x∥ + |⟨x_{α} − x, ν⟩| < \frac{2ε}{3} + \frac{ε}{3} = ε.

Pick ν_{1}, …, ν_{n} ∈ A such that A ⊆ ⋃_{k=1}^{n} B(ν_{k}, \frac{ε}{3}). Then for every α ⩾ max\{α_{ν_{1}}, …, α_{ν_{n}}\}

we must have ρ_{ξ}(K^{∗}x_{α} − K^{∗}x) < ε. □

An operator T ∈ L(X) will be said w^{∗}-locally quasinilpotent at a pair (x_{0}, ξ_{0}), where x_{0} ∈ X and ξ_{0} ∈ Y, if |T^{n}x_{0}(ξ_{0})|^{\frac{1}{n}} → 0. Clearly, if T is locally quasinilpotent at x_{0} then T is w^{∗}-locally quasinilpotent at (x_{0}, ξ_{0}) for every ξ_{0} ∈ Y.

Theorem 9. Suppose that X = Y^{∗} for some Banach lattice Y, and C is a convex collection of positive adjoint operators on X. Suppose that there exists a pair (x_{0}, ξ_{0}) ∈ X_{+} × Y_{+} such that x_{0}(ξ_{0}) ≠ 0 and every operator from C is w^{∗}-locally quasinilpotent at (x_{0}, ξ_{0}). Then there exist non-zero x ∈ X_{+} and f ∈ X_{+}^{∗} such that ⟨f, T x⟩ ⩽ θ(T_{∗}) for all T ∈ C.

Proof. The proof of the theorem is similar to that of Theorem 5. We may assume that ∥x_{0}∥ = 1, ∥ξ_{0}∥ = 1, and C is closed under taking positive multiples. Put ρ_{ξ_{0}}(x) = |x|(ξ_{0}). Evidently ρ_{ξ_{0}}(x) ⩽ ∥x∥ for all x ∈ X. It is also clear that |x| ⩽ |y| implies ρ_{ξ_{0}}(x) ⩽ ρ_{ξ_{0}}(y).

Fix 0 < ε < \frac{x_{0}(ξ_{0})}{8}. Define

C_{ε} = {T ∈ C: θ(T_{∗}) < ε} and
G_{ε}(x) = {z ∈ X: |z| ⩽ T x for some T ∈ C_{ε}}, x ∈ X_{+}.

Suppose that G_{ε}(x) is not dense in X for some x ∈ X_{+}. Analogously to the proof of Theorem 5, we find a positive functional h ∈ X_{+}^{∗} such that ⟨h, T x⟩ ⩽ 1 for all T ∈ C_{ε}. Considering separately the cases θ(T_{∗}) = 0 and θ(T_{∗}) ≠ 0, we get the conclusion of the theorem.

Thus, we may assume that \overline{G_{ε}(x)} = X for all x > 0. Define

U_{0} = \{x ∈ X_{+} : x ⩽ x_{0} and ρ_{ξ_{0}}(x − x_{0}) ⩽ \frac{x_{0}(ξ_{0})}{2}\}.

A VERSION OF LOMONOSOV’S THEOREM 7

Clearly, U_{0} is w^{∗}-compact.

Let x ∈ U_{0} be arbitrary. Since \overline{G_{ε}(x)} = X, we can find T_{x} ∈ C_{ε} such that ρ_{ξ_{0}}(x_{0} − x_{0} ∧ T_{x}x) ⩽ ∥x_{0} − x_{0} ∧ T_{x}x∥ < ε. Fix an operator K_{x} adjoint to an AM-compact operator such that ∥T_{x} − K_{x}∥ < ε. By Lemma 8, we can find a relative (to U_{0}) w^{∗}- open neighborhood V_{x} ⊆ U_{0} of x such that ρ_{ξ_{0}}(K_{x}x − K_{x}z) < ε for all z ∈ V_{x}. Then for an arbitrary z ∈ V_{x}, we have

ρ_{ξ_{0}}(x_{0} − x_{0} ∧ T_{x}z) ⩽ ρ_{ξ_{0}}(x_{0} − x_{0} ∧ T_{x}x) + ρ_{ξ_{0}}(x_{0} ∧ T_{x}x − x_{0} ∧ K_{x}x) + ρ_{ξ_{0}}(x_{0} ∧ K_{x}x − x_{0} ∧ K_{x}z) + ρ_{ξ_{0}}(x_{0} ∧ K_{x}z − x_{0} ∧ T_{x}z)

< ε + ∥T_{x}x − K_{x}x∥ + ρ_{ξ_{0}}(K_{x}x − K_{x}z) + ∥K_{x}z − T_{x}z∥

< ε + ∥T_{x} − K_{x}∥ · ∥x∥ + ε + ∥T_{x} − K_{x}∥ · ∥z∥ < 4ε < \frac{x_{0}(ξ_{0})}{2}.

Take x_{1}, …, x_{m} in U_{0} such that ⋃_{k=1}^{m} V_{x_{k}} = U_{0}. Then T = T_{x_{1}} + ⋯ + T_{x_{k}} ∈ C satisfies

ρ_{ξ_{0}}(x_{0} − x_{0} ∧ T z) ⩽ \frac{x_{0}(ξ_{0})}{2} for all z ∈ U_{0}. Since x_{0} ∧ T z ⩽ x_{0}, we have x_{0} ∧ T z ∈ U_{0} for all z ∈ U_{0}.

Put z_{0} = x_{0} and z_{n+1} = x_{0} ∧ T z_{n}. By the w^{∗}-local quasinilpotence of T at (x_{0}, ξ_{0}) we have ρ_{ξ_{0}}(z_{n}) ⩽ ρ_{ξ_{0}}(T^{n}x_{0}) = |T^{n}x_{0}(ξ_{0})| → 0 as n → ∞ which is impossible by the definition of U_{0}. □

The following result is derived from Theorem 9 in the same way that Theorem 7 was deduced from Theorem 5.

Theorem 10. Suppose that X = Y^{∗} for some Banach lattice Y, and C is a collection of positive adjoint operators on X. If C is finitely quasinilpotent at some x_{0} > 0 then there exist non-zero x ∈ X_{+} and f ∈ X_{+}^{∗} such that ⟨f, T x⟩ ⩽ θ(T_{∗}) for all T ∈ E_{C}.

As every operator on ℓ_{p} (1 ⩽ p < ∞) is AM-compact, this theorem can be used as an alternative proof of the following (certainly known) result.

Corollary 11. Every collection of positive operators on ℓ_{p}, 1 < p < ∞, which is finitely quasinilpotent at a non-zero positive vector, has a non-trivial closed common invariant ideal.

Of course, Corollary 11 follows easily from Theorem 3 when 1 ⩽ p < ∞.

Corollary 12. Every collection of positive adjoint operators on ℓ_{∞} which is finitely quasinilpotent at a non-zero positive vector has a non-trivial closed common invariant ideal.

The following example shows that the assumptions in Theorems 7 and 10 in general do not guarantee the existence of an invariant subspace.

Example 13. There is a collection C of operators which satisfies all the conditions of Theorem 10 and has no common non-trivial invariant subspaces. Namely, in [4], the authors constructed a multiplicative semigroup S_{p} of positive square-zero operators acting on L_{p}[0, 1], 1 ⩽ p < ∞, having no common non-trivial invariant subspaces. It is not difficult to show that S_{p} is in fact finitely quasinilpotent at every positive vector. Hence for 1 < p < ∞, C = S_{p} satisfies the conditions of Theorem 10.

Remark 14. Even though Theorem 1 is not a special case of Theorem 2, in the case of an algebra of adjoint operators the former can be easily deduced from the latter, see [8, Corollary 1]. Similarly, we will show that in case of adjoint operators, Theorem 3 can be deduced from Theorem 10. Indeed, suppose that X = Y^{∗} for some Banach lattice Y, and C is a collection of positive adjoint operators which is finitely quasinilpotent at some x_{0} > 0 and some operator in it dominates a non-zero AM-compact positive^{1} adjoint operator K. We will show that there is a non-trivial closed ideal which is invariant under C and under all adjoint operators in [C⟩.

Clearly, K ∈ E_{C}. Let x and f be as in Theorem 10.

J_{1} = {z ∈ X : |z| ⩽ T_{1}KT_{2}x for some T_{1}, T_{2} ∈ E_{C}},
J_{2} = {z ∈ X : T |z| = 0 for all T ∈ E_{C}}, and
J_{3} = {z ∈ X : |z| ⩽ T x for some T ∈ E_{C}}.

It is easy to see that J_{1}, J_{2}, and J_{3} are ideals in X, invariant under C and under all adjoint operators in [C⟩. It is left to show that at least one of the three must be non-trivial. Clearly, J_{2} is closed and J_{2} ≠ X. Suppose that J_{2} = {0}. In particular, x ∉ J_{2}. It follows that J_{3} ≠ {0}. Suppose that J_{3} is dense in X. It follows from Theorem 10 that J_{1} ⊆ ker f; hence \overline{J_{1}} is proper. Assume that J_{1} = \{0\}. Hence, T_{1}KT_{2}x = 0 for all T_{1}, T_{2} ∈ E_{C}. Since J_{2} = {0}, it follows that K vanishes on E_{C}x and, therefore, on J_{3}. Since J_{3} is dense in X it follows that K = 0; a contradiction.

Acknowledgment. We would like to thank Victor Lomonosov for helpful discussions.

1Unlike in Theorem 3, we also require that K ⩾ 0 here.

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Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1. Canada.

E-mail address: apopov@math.ualberta.ca
E-mail address: vtroitsky@math.ualberta.ca