ON THE MODULUS OF C. J. READ’S OPERATOR

VLADIMIR G. TROITSKY

Abstract. Let T: ℓ_{1} → ℓ_{1} be the quasinilpotent operator without an invariant subspace constructed by C. J. Read in [R3]. We prove that the modulus of this operator has an invariant subspace (and even an eigenvector). This answers a question posed by Y. Abramovich, C. Aliprantis and O. Burkinshaw in [AAB1, AAB3].

During the last several years there has been a noticeable increase of interest in the invariant subspace problem for positive operators on Banach lattices. A rather complete and comprehensive survey on this topic is presented in [AAB3], to which we refer the reader for details and for an extensive bibliography. In particular, the following theorem was proved in [AAB1].

Theorem 1 ([AAB1, AAB3]). If the modulus of a continuous operator T: ℓ_{p} → ℓ_{p} (1 ⩽ p < ∞) exists and is quasinilpotent, then T has a non-trivial closed invariant subspace which is an ideal.

It follows that each positive quasinilpotent operator on ℓ_{p} (1 ⩽ p < +∞) has a nontrivial closed invariant subspace. In the same papers the authors posed the following problem.

Problem. Does every positive operator on ℓ_{1} have an invariant subspace?

Keeping in mind that each operator on ℓ_{1} has a modulus and that C. J. Read in [R1, R2, R3] has constructed several operators on ℓ_{1} without invariant subspaces, it was suggested in [AAB1, AAB3] that the modulus of some of these operators might be a natural candidate for a counterexample to the above problem. Following this suggestion, we will be dealing in this paper with the modulus of the quasinilpotent operator T constructed in [R3]. It turns out, quite surprisingly, that not only does |T | have an invariant subspace but it even has a positive eigenvector. This result increases the chances for an affirmative answer to the above problem.

The paper is organized as follows. After introducing some necessary notation and terminology we prove a general theorem on the existence of an invariant subspace for the modulus

Positivity, 2 (1998), no. 3, 257–264

1 2 V. G. TROITSKY

1991 Mathematics Subject Classification. 47A15, 47B60, 47B65.

Key words and phrases. Banach lattice, invariant subspaces, positive operator.

Supported in part by NSF Grant DMS 96-22454.

of a quasinilpotent operator. The rest of the paper will be devoted to the verification that C. J. Read’s operator, constructed in [R3], satisfies all the hypotheses of this theorem and so its modulus does have an invariant subspace.

For terminology and notation regarding operators and Banach lattices we refer to [AB, Sch]. All operators considered in this work are linear and continuous. The space of all operators on a Banach space X is denoted by B(X), while K(X) stands for the subspace of all compact operators. A linear operator on a Banach lattice is said to be positive if it maps positive vectors to positive vectors. By an invariant subspace (invariant ideal) of an operator we mean a closed nontrivial subspace (resp. closed nontrivial ideal) which is invariant under the operator.

Together with the usual operator norm ∥S∥ of an operator S: X → X on a Banach space, we will also consider the (essential) seminorm ∥S∥_{e} given by

∥S∥_{e} = inf{ ∥S − K∥ : K ∈ K(X) }.

The essential spectral radius of S is computed via the formula r_{e}(S) = lim_{n} \sqrt[n]{∥S^{n}∥_{e}}. This, of course, is an analogue of the familiar formula for the usual spectral radius r(S) = lim_{n} \sqrt[n]{∥S^{n}∥}.

It is obvious that ∥S∥_{e} ⩽ ∥S∥ and r_{e}(S) ⩽ r(S). It is known that if r_{e}(S) = 0, then every nonzero point of σ(S) is an eigenvalue. Further details on essential spectral radius can be found in [N1, CPY]. We will use the following important version of the Krein-Rutman theorem established by R. Nussbaum.

Theorem 2. [N2, Corollary 2.2] Let S be a positive operator on a Banach lattice such that r_{e}(S) < r(S), then r(S) is an eigenvalue of S corresponding to a positive eigenvector.^{1}

We use this fact in the proof of the following simple but rather unexpected result.

Theorem 3. Suppose that a quasinilpotent operator S on ℓ_{p} has no invariant ideals and S^{−} is compact. Then r(|S|) is a positive eigenvalue of |S| corresponding to a positive eigenvector. In particular, |S| has an invariant subspace.

Proof. First observe that the operator |S| cannot be quasinilpotent. Indeed, if it were, then by Theorem 1 the operator S itself would have an invariant closed ideal contrary to our hypothesis. Thus, r(|S|) > 0.

Next we claim that r_{e}(|S|) = 0. To prove this, notice that |S| = S + 2S^{−}, and so

|S|^{n} = (S + 2S^{−})^{n} = S^{n} + RS^{−},

1In [N2] a more general form of this theorem is given which is valid for ordered Banach spaces.

ON THE MODULUS OF C. J. READ’S OPERATOR 3

where R is some polynomial in S and S^{−}. Hence RS^{−} is compact, whence

∥|S|^{n}∥_{e} = ∥S^{n} + RS^{−}∥_{e} ⩽ ∥S^{n}∥,

and consequently

r_{e}(|S|) ⩽ lim_{n→∞} \sqrt[n]{∥S^{n}∥} = r(S) = 0.

An application of Theorem 2 finishes the proof. □

Corollary 4. Under the hypotheses of the above theorem the operator S^{+} also has a nontrivial closed invariant subspace.

Proof. There are two possibilities: either S^{+} is quasinilpotent or it is not. If S^{+} is quasinilpotent, then applying Theorem 1 again, we see that S^{+} has an invariant ideal.

Assume that S^{+} is not quasinilpotent. Since S^{+} = S + S^{−}, the same argument as in the proof of Theorem 3 shows that r_{e}(S^{+}) = 0 and we can again apply Theorem 2. □

Theorem 3 is strong enough to enable us to prove Corollary 7 about the modulus of C. J. Read’s operator. But first we would like to mention a nice generalization of Theorem 3. Recall that a positive operator on a Banach lattice is called compact-friendly if it commutes with another positive operator which dominates some non-zero operator which in turn is dominated by a positive compact operator. The class of compact-friendly operators was introduced and studied by Abramovich, Aliprantis, and Burkinshaw in [AAB2]. This class includes positive operators that dominate or are dominated by a non-zero compact operator, or commute with a non-zero compact operator. Also, every positive operator on any discrete Banach lattice and every positive kernel operator is compact-friendly. It follows from [AAB3, Theorem 11.2] that if a positive compact-friendly operator is quasinilpotent, then it has an invariant ideal. Mimicking the proof of Theorem 3 we can obtain the following theorem.

Theorem 5. Let B be a compact-friendly operator with r_{e}(B) = 0. Then B has a nontrivial invariant subspace.

To illustrate this theorem we mention the following result: If S is a quasinilpotent kernel operator and S^{−} (resp. S^{+}) is compact, then |S| and S^{+} (resp. S^{−}) have invariant subspaces.

Recall that an operator S: X → Y between two Banach spaces is called nuclear if it can be written in the form S = ∑_{i=0}^{∞} x_{i}^{∗} ⊗ y_{i} with x_{i}^{∗} in X^{∗}, y_{i} in Y and ∑_{i=0}^{∞}∥x_{i}^{∗}∥∥y_{i}∥ < ∞. Here, as usual, the elementary tensor x^{∗} ⊗ y : X → Y is given by (x^{∗} ⊗ y)(x) = x^{∗}(x)y. The nuclear norm ν(S) is defined by ν(S) = inf ∑_{i=0}^{∞}∥x_{i}^{∗}∥∥y_{i}∥, where the infimum is taken over all nuclear representations of S. For a nuclear operator S we have ∥S∥ ⩽ ν(S). This implies

4 V. G. TROITSKY

in particular that every nuclear operator can be approximated by finite-rank operators and, therefore, is compact.

Following [R1, R2, R3] we denote the standard unit vectors of ℓ_{1} by (f_{i})_{i=0}^{∞}. It is well known that we can consider each S ∈ B(ℓ_{1}) as an infinite matrix S = (s_{ij})_{i,j=0}^{∞}. Let S_{(i)} denote the i-th row of this matrix. If x ∈ ℓ_{1}, then (Sx)_{i} = ⟨S_{(i)}, x⟩, so that Sx = ∑_{i=0}^{∞}⟨S_{(i)}, x⟩f_{i}. This gives a nuclear representation S = ∑_{i=0}^{∞} S_{(i)} ⊗ f_{i}, where the rows S_{(i)} of S are considered as linear functionals on ℓ_{1}. It follows that

ν(S) ⩽ ∑_{i=0}^{∞}∥S_{(i)}∥_{∞}∥f_{i}∥_{1} = ∑_{i=0}^{∞}∥S_{(i)}∥_{∞},

so that S is nuclear if the last sum is finite.

In spite of the fact that the construction of a quasinilpotent operator on ℓ_{1} without an invariant subspace in [R3] is far from being simple, the presentation in [R3] is very clearly structured. In Sections 2 and 3 of [R3] C. J. Read presents the construction of operator T, and in the rest of the paper he proves that T is bounded (Lemma 5.1), quasinilpotent (Theorem 6.5), and has no invariant subspaces (Theorem 7). In what follows we will restate (practically verbatim and retaining the notation) the definition of T. In the proof of [R3, Lemma 5.1] C. J. Read reveals a lot of information about the structure of the infinite matrix of T. Since we also are interested in this structure, we incorporate some fragments of the proof of [R3, Lemma 5.1] in our proof of Lemma 6.

The symbol F denotes the linear subspace of ℓ_{1}, spanned by f_{i}’s, and thus, F is dense and consists of all eventually vanishing sequences. Let d = (a_{1}, b_{1}, a_{2}, b_{2}, …) be a strictly increasing sequence of positive integers. Also let a_{0} = 1, v_{0} = 0, and v_{n} = n(a_{n} + b_{n}) for n ⩾ 1. Then there is a unique sequence (e_{i})_{i=0}^{∞} ⊂ F with the following properties:

0) f_{0} = e_{0};

A) if integers r, n, and i satisfy 0 < r ⩽ n, i ∈ [0, v_{n−r}] + ra_{n}, then f_{i} = (n^{ra_{n}}e_{i} −

e_{i−ra_{n}})(n − r)^{i−ra_{n}}a_{n−r};

B) if integers r, n, and i satisfy 0 < r < n, i ∈ (ra_{n} + v_{n−r}, (r + 1)a_{n}), (respectively, 1 ⩽ n, i ∈ (v_{n−1}, a_{n})), then f_{i} = n^{i}2^{(h−i)/\sqrt{a_{n}}}e_{i}, where h = (r + \frac{1}{2})a_{n} (respectively,

h = \frac{1}{2}a_{n});

C) if integers r, n, and i satisfy 0 < r ⩽ n, i ∈ [r(a_{n} + b_{n}), na_{n} + rb_{n}], then f_{i} =

n^{i}e_{i} − b_{n}n^{i−b_{n}}e_{i−b_{n}};

D) if integers r, n, and i satisfy 0 ⩽ r < n, i ∈ (na_{n} + rb_{n}, (r + 1)(a_{n} + b_{n})), then

f_{i} = n^{i}2^{(h−i)/\sqrt{b_{n}}}e_{i}, where h = (r + \frac{1}{2})b_{n}.

ON THE MODULUS OF C. J. READ’S OPERATOR 5

Indeed, since f_{i} = ∑_{j=0}^{i} λ_{ij}e_{j} for each i ⩾ 0 and λ_{ii} is always nonzero, this linear relation is invertible. Further,

lin{e_{i} : i = 1, …, n} = lin{f_{i} : i = 1, …, n} for every n ⩾ 0.

In particular all e_{i} are linearly independent and span F. Then C. J. Read defines T: F → F to be the unique linear map such that T e_{i} = e_{i+1}, and in Lemma 5.1 he proves that ∥T f_{i}∥ ⩽ 1 for every i ⩾ 0 provided d increases sufficiently rapidly, i. e., satisfies several conditions of the form

a_{n} ⩾ G(n, a_{0}, b_{0}, a_{1}, b_{1}, …, a_{n−1}, b_{n−1}), and
b_{n} ⩾ H(n, a_{0}, b_{0}, a_{1}, b_{1}, …, a_{n−1}, b_{n−1}, a_{n}),

where G and H are some positive integer-valued functions. It follows that T can be extended to a bounded operator on ℓ_{1}. Finally, in Theorems 6.5 and 7, C. J. Read proves that this extension, which is also denoted by T, is quasinilpotent and has no invariant subspaces, provided d increases sufficiently rapidly.

Our plan is as follows: we will prove that the negative part of T is nuclear, hence compact. Then Theorem 3 will imply that r(|T |) is a positive eigenvalue of |T |, corresponding to a positive eigenvector.

Lemma 6. The operator T^{−} is nuclear, provided d increases sufficiently rapidly.

Proof. Similarly to the proofs of [R3, Lemma 5.1] and [R2, Lemma 6.1] we study the matrices (t_{ij})_{i,j=0}^{∞} and (t_{ij}^{−})_{i,j=0}^{∞} of T and T^{−} respectively. Recall that t_{ki} = (T f_{i})_{k}, so that it suffices to look at the images of the standard unit vectors under T. We will see that the matrix of T is quite sparse and has the following structure: every entry on the diagonal right under the main diagonal is strictly positive, there is no nonnegative entries below this diagonal, and there are some entries above it. We consider consecutively all the cases mentioned above.

0) T f_{0} = e_{1} = 2^{(1−a_{1}/2)/\sqrt{a_{1}}}f_{1}, so that T^{−}f_{0} = 0.

A) If i < v_{n−r} + ra_{n}, i. e. i is not the right end point of the interval [ra_{n}, v_{n−r} + ra_{n}], then T f_{i} = (n − r)^{−1}f_{i+1}, so that T^{−}f_{i} = 0. The only nontrivial case here is when i is the right end of the interval, i. e. i = v_{n−r} + ra_{n}. Then we have

T f_{i} = a_{n−r}n^{ra_{n}}(n − r)^{v_{n−r}}e_{1+ra_{n}+v_{n−r}} − a_{n−r}(n − r)^{v_{n−r}}e_{1+v_{n−r}} = ε_{1}f_{1+v_{n−r}+ra_{n}} − ε_{2}f_{1+v_{n−r}},

where ε_{1} > 0 and ε_{2} is given by
ε_{2} = (n − r + 1)^{−1−v_{n−r}}2^{(1+v_{n−r}−a_{n−r+1}/2)/\sqrt{a_{n−r+1}}}a_{n−r}(n − r)^{v_{n−r}},

6 V. G. TROITSKY

so that

T^{−}f_{v_{n−r}+ra_{n}} = ε_{2}f_{1+v_{n−r}}.  (1)

B) Similarly, if ra_{n} + v_{n−r} < i < (r + 1)a_{n} − 1 or v_{n−1} < i < a_{n} − 1, then T f_{i} = n^{−1}2^{1/\sqrt{a_{n}}}f_{i+1}, so that T^{−}f_{i} = 0. If i = (r+1)a_{n}−1, then T f_{i} = n^{i}2^{(1−a_{n}/2)/\sqrt{a_{n}}}e_{(r+1)a_{n}}. To express this in terms of the f_{i}’s, we notice that f_{(r+1)a_{n}} = a_{n−r−1}(n^{(r+1)a_{n}}e_{(r+1)a_{n}} − e_{0}), which implies

e_{(r+1)a_{n}} = n^{−(r+1)a_{n}}(a_{n−r−1}^{−1}f_{(r+1)a_{n}} + f_{0}),  (2)

In this case T^{−}f_{i} = 0. Analogously, if i = a_{n} − 1, then T f_{i} = n^{i}2^{(1−a_{n}/2)/\sqrt{a_{n}}}e_{a_{n}}. It follows from f_{a_{n}} = a_{n−1}(n^{a_{n}}e_{a_{n}} − e_{0}) that

T f_{i} = n^{−1}2^{(1−a_{n}/2)/\sqrt{a_{n}}}(a_{n−1}^{−1}f_{a_{n}} + f_{0}),

and again T^{−}f_{i} = 0. Thus, case (B) produces no nontrivial entries in T^{−}. C) If i is not the right end of the interval, i.e. i < na_{n} + rb_{n}, then T f_{i} = n^{−1}f_{i+1}, so

that T^{−}f_{i} = 0. If i = na_{n} + rb_{n}, then

T f_{i} = n^{na_{n}+rb_{n}}e_{1+na_{n}+rb_{n}} − b_{n}n^{na_{n}+(r−1)b_{n}}e_{1+na_{n}+(r−1)b_{n}} = ε_{1}f_{1+na_{n}+rb_{n}} − ε_{2}f_{1+na_{n}+(r−1)b_{n}},

where ε_{1} > 0 and ε_{2} = b_{n}n^{−1}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}}. It follows that

T^{−}f_{na_{n}+rb_{n}} = b_{n}n^{−1}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}}f_{1+na_{n}+(r−1)b_{n}}.  (3)

D) If i < (r + 1)(a_{n} + b_{n}) − 1, then T f_{i} = n^{−1}2^{1/\sqrt{b_{n}}}f_{i+1}, so that T^{−}f_{i} = 0. If i = (r + 1)(a_{n} + b_{n}) − 1 then

T f_{i} = n^{i}2^{(−a_{n}/2−(r+1)a_{n}/2+1)/\sqrt{b_{n}}}e_{(r+1)(a_{n}+b_{n})}.

Using (C) inductively we obtain the following identity:

e_{(r+1)(a_{n}+b_{n})} = n^{−(r+1)(a_{n}+b_{n})}{f_{(r+1)(a_{n}+b_{n})} + b_{n}f_{(r+1)a_{n}+rb_{n}} + … + n_{n}^{r}f_{(r+1)a_{n}+b_{n}}} + b_{n}^{r+1}n^{−(r+1)b_{n}}e_{(r+1)a_{n}}.

Substitute e_{(r+1)a_{n}} from (2) and notice that all the the coefficients are positive and, therefore, T^{−}f_{i} = 0. Thus, case (D) does not produce any nontrivial entries in T^{−}.

Summarizing the calculations, the only nonzero entries of T^{−} are given by (1) and (3):

t_{1+v_{n−r},v_{n−r}+ra_{n}}^{−} = (n − r + 1)^{−1−v_{n−r}}2^{(1+v_{n−r}−a_{n−r+1}/2)/\sqrt{a_{n−r+1}}}a_{n−r}(n − r)^{v_{n−r}}

and

t_{na_{n}+(r−1)b_{n}+1,na_{n}+rb_{n}}^{−} = b_{n}n^{−1}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}}

ON THE MODULUS OF C. J. READ’S OPERATOR 7

for all 0 < r ⩽ n. To show that T^{−} is nuclear it suffices to show ∑_{k=0}^{∞}∥T_{(k)}^{−}∥_{∞} < +∞. Look at the rows of T^{−} containing non-zero entries. Notice that

t_{1+v_{n−r},v_{n−r}+ra_{n}}^{−} ⩽ a_{n−r}2^{(1+v_{n−r}−a_{n−r+1}/2)/\sqrt{a_{n−r+1}}} ⩽ 2^{−(1+v_{n−r})}

for all 0 < r ⩽ n provided d increases sufficiently rapidly. It follows that ∥T_{(1+v_{m})}^{−}∥_{∞} ⩽

2^{−(1+v_{m})} for every m ⩾ 0 and ∑_{m=0}^{∞}∥T_{(1+v_{m})}^{−}∥_{∞} ⩽ ∑_{m=0}^{∞} 2^{−1−v_{m}} < 1.

Further, the entries t_{na_{n}+(r−1)b_{n}+1,na_{n}+rb_{n}}^{−} do not depend on r, and their contribution to ∑_{k=0}^{∞}∥T_{(k)}^{−}∥_{∞} does not exceed the sum of all of them, which can be easily estimated:

∑_{n=1}^{∞} ∑_{r=1}^{n} b_{n}n^{−1}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}} ⩽ ∑_{n=1}^{∞} b_{n}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}} ⩽ ∑_{n=1}^{∞} 2^{−n} = 1,

because b_{n}2^{(1+na_{n}−b_{n}/2)/\sqrt{b_{n}}} ⩽ 2^{−n} for all n ⩾ 1 provided d increases sufficiently rapidly. Thus, ν(T^{−}) ⩽ ∑_{k=0}^{∞}∥T_{(k)}^{−}∥_{∞} < 2 provided d increases sufficiently rapidly. □

Corollary 7. C. J. Read’s operator T satisfies the following properties, provided d increases sufficiently rapidly:

(1) |T |, T^{+}, and T^{−} have positive eigenvectors;

(2) Neither |T | nor T^{+} has an invariant ideal.

Proof. It follows from Theorem 3 and Lemma 6 that |T | has a positive eigenvector. It was noticed in the proof of Lemma 6 that T^{−}f_{0} = 0, so that T^{−} also has a positive eigenvector.

To prove (2), assume that J is a closed ideal in ℓ_{1} invariant under |T | or T^{+}, and that 0 ≠ x ∈ J, then x_{k} ≠ 0 for some k ⩾ 0, so that f_{k} ∈ J. It follows from the proof of Lemma 6 that both |T |f_{i} and T^{+}f_{i} have nonzero (i + 1)-th component, implying f_{k+1} ∈ J. Proceeding inductively, we see that f_{i} ∈ J for all i ⩾ k. Further, the proof of Lemma 6 also shows that (|T |f_{i})_{0} ≠ 0 and (T^{+}f_{i})_{0} ≠ 0 for infinitely many i’s, so that f_{0} ∈ J. It follows that f_{i} ∈ J for every i ⩾ 0, so that J = ℓ_{1}. In fact, (2) is a manifestation of the fact that a positive operator S on ℓ_{p} (1 ⩽ p < ∞) has no invariant ideals if and only if there is a path between every two columns of S (c.f. [AAB3, TV2]).

It follows from (2) and Theorem 1 that T^{+} cannot be quasinilpotent. On the other hand, since T^{+} = T + T^{−} then, analogously to the proof of Theorem 3, we have r_{e}(T^{+}) = 0. Then by Theorem 2 we conclude that r(T^{+}) is a positive eigenvalue of T^{+}, corresponding to a positive eigenvector. □

The last statement of Corollary 7 emphasizes that the hypothesis of not having invariant ideals in Theorem 3 is weaker than not having invariant subspaces. We do not know if the analogues of the results of this paper hold for the operators produced in [R1, R2].

I would like to thank Prof. Y. A. Abramovich for suggesting this problem to me and for our many discussions. I am thankful to Professors C. D. Aliprantis, V. J. Lomonosov, and C. J. Read for their interest in this work.

We remark in conclusion that in [TV1] we use one of C. J. Read’s operators to solve one more problem related to invariant subspaces. Namely, we construct operators S_{1}, S_{2}, and K (not multiples of the identity) on ℓ_{1} such that T commutes with S_{1}, S_{1} commutes with S_{2}, S_{2} commutes with K, and K is compact. This shows that the celebrated Lomonosov theorem cannot be extended to chains of four operators.

References

[AAB1] Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspaces of operators on ℓ_{p}- spaces, J. Funct. Anal. 115 (1993), 418–424.

[AAB2] Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspaces of positive operators, J. Funct. Anal. 124 (1994), 95–111.

[AAB3] Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, The Invariant Subspace Problem: Some Recent Advances, Rend. Inst. Mat. Univ. Trieste, to appear.

[AB] C. D. Aliprantis and O. Burkinshaw, Positive operators, Academic Press, New York & London, 1985.

[CPY] S. R. Caradus, W. E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators of Banach Spaces, Marcel Dekker, New York, 1974.

[N1] R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473–478.

[N2] R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, in E. Fadell and G. Fourier, Fixed Point Theory, Springer Verlag yellow series, 886 (1981), 309–330.

[R1] C. J. Read, A solution to the invariant subspace problem on the space l_{1}, Bull. Lond. Math. Soc. 17 (1985) 305–317.

[R2] C. J. Read, A short proof concerning the invariant subspace problem, J. Lond. Math. Soc. (2) 34 (1986) 335–348.

[R3] C. J. Read, Quasinilpotent Operators and the Invariant Subspace Problem, Jour. Lond. Math. Soc., to appear.

[Sch] H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin & New York, 1974.

[TV1] V. G. Troitsky, Lomonosov theorem cannot be extended to chains of four operators, submitted.

[TV2] V. G. Troitsky, Paths and invariant ideals, unpublished.

Mathematics Department, University of Illinois at Urbana-Champaign, 1409 West Green St, Urbana, IL 61801, USA

  E-mail address: vladimir@math.uiuc.edu