NORM CLOSED OPERATOR IDEALS IN LORENTZ SEQUENCE SPACES

ANNA KAMINSKA, ALEXEY I. POPOV, EUGENIU SPINU, ADI TCACIUC, AND VLADIMIR G. TROITSKY

Abstract. In this paper, we study the structure of closed algebraic ideals in the algebra of operators acting on a Lorentz sequence space.

1. Introduction

1.1. Ideals. This paper is concerned with the study of the structure of closed algebraic ideals in the algebra L(X) of all bounded linear operatrors on a Banach space X.

Throughout the paper, by a subspace of a Banach space we mean a closed subspace; a vector subspace of X which is not necessarily closed will be referred to as linear subspace. A (two-sided) ideal in L(X) is a linear subspace J of L(X) such that AT B ∈ J whenever T ∈ J and A, B ∈ L(X). The ideal J is called proper if J ≠ L(X). The ideal J is non-trivial if J is proper and J ≠ {0}.

The spaces for which the structure of closed ideals in L(X) is well-understood are very few. It was shown in [7] that the only non-trivial closed ideal in the algebra L(ℓ_{2}) is the ideal of compact operators. This result was generalized in [13] to the spaces ℓ_{p} (1 ⩽ p < ∞) and c_{0}. A space constructed recently in [5] is another space with this property. In [15] and [16], it was shown that the algebras L((⊕_{k=1}^{∞}ℓ_{2}^{k})_{c_{0}}) and L((⊕_{k=1}^{∞}ℓ_{2}^{k})_{ℓ_{1}}) have exactly two non-trivial closed ideals. There are no other separable spaces for which the structure of closed ideals in L(X) is completely known.

Partial results about the structure of closed ideals in L(X) were obtained in [20, 5.3.9] for X = L_{p}[0, 1] (1 < p < ∞, p ≠ 2) and in [22] and [23] for L(ℓ_{p} ⊕ ℓ_{q}) (1 ⩽ p, q < ∞). The purpose of this paper is to investigate the structure of ideals in L(d_{w,p}) where d_{w,p} is a Lorentz sequence space (see the definition in Subsection 1.3).

For two closed ideals J_{1} and J_{2} in L(X), we will denote by J_{1} ∧ J_{2} the largest closed ideal J in L(X) such that J ⊆ J_{1} and J ⊆ J_{2} (that is, J_{1} ∧ J_{2} = J_{1} ∩ J_{2}), and we will

Date: August 12, 2011.

2010 Mathematics Subject Classification. Primary: 47L20. Secondary: 47B10, 47B37.

Key words and phrases. Lorentz space, operator ideal, strictly singular operator.

The forth and fifth authors were supported by NSERC.

2 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

denote by J_{1} ∨ J_{2} the smallest closed ideal J in L(X) such that J_{1} ⊆ J and J_{2} ⊆ J. We say that J_{2} is a successor of J_{1} if J_{1} ⊊ J_{2}. If, in addition, no closed ideal J in L(X) satisfies J_{1} ⊊ J ⊊ J_{2}, then we call J_{2} an immediate successor of J_{1}.

It is well-known that if X is a Banach space then every non-zero ideal in the algebra L(X) must contain the ideal F(X) of all finite-rank operators on X. It follows that, at least in the presence of the approximation property (in particular, if X has a Schauder basis), every non-zero closed ideal in L(X) contains the closed ideal K(X) of all compact operators.

Two ideals closely related to K(X) are the closed ideal SS(X) of strictly singular operators and the closed ideal FSS(X) of finitely strictly singular operators on X. Recall that an operator T ∈ L(X) is called strictly singular if no restriction T |_{Z} of T to an infinite-dimensional subspace Z of X is an isomorphism. An operator T is finitely strictly singular if for any ε > 0 there is N ∈ ℕ such that any subspace Z of X with dim Z ⩾ N contains a vector z ∈ Z satisfying ∥T z∥ < ε∥z∥. It is not hard to show that K(X) ⊆ FSS(X) ⊆ SS(X) (see [17, 19, 22, 4] for more information about these classes of operators).

If X is a Banach space and T ∈ L(X) then the ideal in L(X) generated by T is denoted by J_{T}. It is easy to see that J_{T} = {∑_{i=1}^{n} A_{i}T B_{i} : A_{i}, B_{i} ∈ L(X)}. It follows that if S ∈ L(X) factors through T, i.e., S = AT B for some A, B ∈ L(X) then J_{S} ⊆ J_{T}.

1.2. Basic sequences. The main tool in this paper is the notion of a basic sequence. In this subsection, we will fix some terminology and remind some classical facts about basic sequences. For a thorough introduction to this topic, we refer the reader to [9] or [12].

If (x_{n}) is a sequence in a Banach space X then its closed span will be denoted by [x_{n}]. We say that a basic sequence (x_{n}) dominates a basic sequence (y_{n}) and write (x_{n}) ⪰ (y_{n}) if the convergence of a series ∑_{n=1}^{∞} a_{n}x_{n} implies the convergence of the series ∑_{n=1}^{∞} a_{n}y_{n}. We say that (x_{n}) is equivalent to (y_{n}) and write (x_{n}) ∼ (y_{n}) if (x_{n}) ⪰ (y_{n}) and (y_{n}) ⪰ (x_{n}).

Remark 1.1. It follows from the Closed Graph Theorem that (x_{n}) ⪰ (y_{n}) if and only if the linear map from span{x_{n}} to span{y_{n}} defined by the formula T: x_{n} ↦ y_{n} is bounded.

If (x_{n}) is a basis in a Banach space X, z = ∑_{i=1}^{∞} z_{i}x_{i} ∈ X, and A ⊆ ℕ then the vector ∑_{i∈A} z_{i}x_{i} will be denoted by z|_{A} (provided the series converges; this is always the case when the basis is unconditional). We will refer to z|_{A} as the restriction of

OPERATOR IDEALS IN LORENTZ SPACES 3

z to A. The restrictions z|_{[n,∞)∩ℕ} and z|_{(n,∞)∩ℕ}, where n ∈ ℕ, will be abbreviated as z|_{[n,∞)} and z|_{(n,∞)}, respectively. We say that a vector v is a restriction of z if there exists A ⊆ ℕ such that v = z|_{A}. The vector z = ∑_{i=1}^{∞} z_{i}x_{i} will also be denoted by z = (z_{i}). If z = ∑_{i=1}^{∞} z_{i}x_{i} then the support of z is the set supp z = {i ∈ ℕ: z_{i} ≠ 0}.

Every 1-unconditional basis (x_{n}) in a Banach space X defines a Banach lattice order on X by ∑_{i=1}^{∞} a_{i}x_{i} ⩾ 0 if and only if a_{i} ⩾ 0 for all i ∈ ℕ (see, e.g., [18, page 2]). For x ∈ X, we have |x| = x ∨ (−x). A Banach lattice is said to have order continuous norm if the condition x_{α} ↓ 0 implies ∥x_{α}∥ → 0. For an introduction to Banach lattices and standard terminology, we refer the reader to [1, §1.2].

If (x_{n}) is a basic sequence in a Banach space X, then a sequence (y_{n}) in span{x_{n}} is a block sequence of (x_{n}) if there is a strictly increasing sequence (p_{n}) in ℕ and a sequence of scalars (a_{i}) such that y_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}x_{i} for all n ∈ ℕ.

The following two facts are classical and will sometimes be used without any references. The first fact is known as the Principle of Small Perturbations (see, e.g., [12, Theorem 4.23]).

Theorem 1.2. Let X be a Banach space, (x_{n}) a basic sequence in X, and (x_{n}^{∗}) the correspondent biorthogonal functionals defined on [x_{n}]. If (y_{n}) is a sequence such that ∑_{n=1}^{∞}∥x_{n}^{∗}∥ · ∥x_{n} − y_{n}∥ < 1 then (y_{n}) is a basic sequence equivalent to (x_{n}). Moreover, if [x_{n}] is complemented in X then so is [y_{n}]. If [x_{n}] = X then [y_{n}] = X.

The next fact, which is often called the Bessaga-Pełczyński selection principle, is a result of combining the “gliding hump” argument (see, e.g., [9, Lemma 5.1]) with the Principle of Small Perturbations.

Theorem 1.3. Let X be a Banach space with a seminormalized basis (x_{n}) and let (x_{n}^{∗}) be the correspondent biorthogonal functionals. Let (y_{n}) be a seminormalized sequence in X such that x_{n}^{∗}(y_{k}) ⟶^{k→∞} 0 for all n ∈ ℕ. Then (y_{n}) has a subsequence (y_{n_{k}}) which is basic and equivalent to a block sequence (u_{k}) of (x_{n}). Moreover, y_{n_{k}} − u_{k} → 0, and u_{k} is a restriction of y_{n_{k}}.

1.3. Lorentz sequence spaces. Let 1 ⩽ p < ∞ and w = (w_{n}) be a sequence in ℝ such that w_{1} = 1, w_{n} ↓ 0, and ∑_{i=1}^{∞} w_{i} = ∞. The Lorentz sequence space d_{w,p} is a Banach space of all vectors x ∈ c_{0} such that ∥x∥_{d_{w,p}} < ∞, where

∥(x_{n})∥_{d_{w,p}} = ( ∑_{n=1}^{∞} w_{n}x_{n}^{∗p})^{1/p}

4 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

is the norm in d_{w,p}. Here (x_{n}^{∗}) is the non-increasing rearrangement of the sequence (|x_{n}|). An overview of properties of Lorentz sequence spaces can be found in [17, Section 4.e].

The vectors (e_{n}) in d_{w,p} defined by e_{n}(i) = δ_{ni} (n, i ∈ ℕ) form a 1-symmetric basis in d_{w,p}. In particular, (e_{n}) is 1-unconditional, hence d_{w,p} is a Banach lattice. We call (e_{n}) the unit vector basis of d_{w,p}. The unit vector basis of ℓ_{p} will be denoted by (f_{n}) throughout the paper.

Remark 1.4. It is proved in [3, Lemma 1] and [10, Lemma 15] that if (u_{n}) is a seminormalized block sequence of (e_{n}) in d_{w,p}, u_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}e_{i}, such that a_{i} → 0, then there is a subsequence (u_{n_{k}}) such that (u_{n_{k}}) ∼ (f_{n}) and [u_{n_{k}}] is complemented in d_{w,p}. Further, it was shown in [3, Corollary 3] that if (y_{n}) is a seminormalized block sequence of (e_{n}) then there is a seminormalized block sequence (u_{n}) of (y_{n}) such that u_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}e_{i}, with a_{i} → 0. Therefore, every infinite dimensional subspace of d_{w,p} contains a further subspace which is complemented in d_{w,p} and isomorphic to ℓ_{p} ([10, Corollary 17]).

Remark 1.5. Remark 1.4 yields, in particular, that d_{w,p} does not contain copies of c_{0}. Since the basis (e_{n}) of d_{w,p} is unconditional, the space d_{w,p} is weakly sequentially complete by [2, Theorem 4.60] (see also [17, Theorem 1.c.10]). Also, [2, Theorem 4.56] guarantees that d_{w,p} has order continuous norm. In particular, if x ∈ d_{w,p} then ∥x|_{[n,∞)}∥ → 0 as n → ∞.

Remark 1.6. It was shown in [14] that if p > 1 then d_{w,p} is reflexive. This can also be easily obtained from Remark 1.4 (cf. [17, Theorem 1.c.12]).

Remark 1.7. The unit vector basis (e_{n}) of d_{w,p} is weakly null. Indeed, by Rosenthal’s ℓ_{1}-theorem (see [21]; also [17, Theorem 2.e.5]), (e_{n}) is weakly Cauchy. Since it is symmetric, (e_{n}) ∼ (e_{2n} − e_{2n−1}).

The next proposition will be used often in this paper.

Proposition 1.8 ([3, Proposition 5 and Corollary 2]). If (u_{n}) is a seminormalized block sequence of (e_{n}) then (f_{n}) ⪰ (u_{n}). If (u_{n}) does not contain subsequences equivalent to (f_{n}) then also (u_{n}) ⪰ (e_{n}).

The following lemma is standard.

Lemma 1.9. Let (x_{n}) be a block sequence of (e_{n}), x_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}e_{i}. If (y_{n}) is a basic sequence such that y_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} b_{i}e_{i}, where |b_{i}| ⩽ |a_{i}| for all i ∈ ℕ, then (x_{n}) is basic and (x_{n}) ⪰ (y_{n}).

OPERATOR IDEALS IN LORENTZ SPACES 5

Proof. Let

γ_{i} = \{ \begin{matrix}\frac{b_{i}}{a_{i}} , & if a_{i} ≠ 0, \\ 0, & if a_{i} = 0. \end{matrix}

Define an operator T ∈ L(d_{w,p}) by T (∑_{i=1}^{∞} c_{i}e_{i}) = ∑_{i=1}^{∞} c_{i}γ_{i}e_{i}. Then T is, clearly, linear and, since the basis (e_{n}) is 1-unconditional, T is bounded with ∥T ∥ ⩽ 1. In particular, T |_{[x_{n}]} is bounded. Also, T (x_{n}) = y_{n} for all n ∈ ℕ, hence (x_{n}) ⪰ (y_{n}). □

1.4. Outline of the results. The purpose of the paper is to uncover the structure of ideals in L(d_{w,p}). We show that (some of) these ideals can be arranged into the following diagram.

[figure: SS; \{0\}; K ⊊ \overline{J^{j}}; \overline{J^{ℓ_{p}}} ∧ SS; \overline{J^{ℓ_{p}}} ∨ SS; _{SS_{d_{w,p}}}; L(d_{w,p}); \overline{J^{ℓ_{p}}}]

(the notations will be defined throughout the paper). On this diagram, a single arrow between ideals, J_{1} ⟶ J_{2}, means that J_{1} ⊆ J_{2}. A double arrow between ideals, J_{1} ⟹ J_{2}, means that J_{2} is the only immediate successor of J_{1} (in particular, J_{1} ≠ J_{2}), whereas a dotted double arrow between ideals, \xymatrix{J_{1} \ar@{=>}[r] & J_{2}} , only shows that J_{2} is an immediate successor for J_{1} (in particular, J_{1} may have other immediate successors).

While working with the diagram above, we obtain several important characterizations of some ideals in L(d_{w,p}). In particular, we show that FSS(d_{w,p}) = SS(d_{w,p}) (Theorem 3.5). We also characterize the ideal of weakly compact operators (Theorem 3.6) and Dunford-Pettis operators (Theorem 5.7) on d_{w,p}. We show in Theorem 4.7 that \overline{J^{j}} is the only immediate successor of K under some assumption on the weights w. In the last section of the paper, we show that all strictly singular operators from ℓ_{1} to d_{w,1} can be approximated by operators factoring through the formal identity operator j: ℓ_{1} → d_{w,1} (see Section 4 for the definition). We also obtain a result on factoring positive operators from SS(d_{w,p}) through the formal identity operator (Theorem 6.12).

2. Operators factorable through ℓ_{p}

Let X and Y be Banach spaces and T ∈ L(X). We say that T factors through Y if there are two operators A ∈ L(X, Y) and B ∈ L(Y, X) such that T = BA.

The following two lemmas are standard. We present their proofs for the sake of completeness.

6 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Lemma 2.1. Let X and Y be Banach spaces and T ∈ L(X, Y), S ∈ L(Y, X) be such that ST = id_{X}. Then T is an isomorphism and Range T is a complemented subspace of Y isomorphic to X.

Proof. For all x ∈ X, we have ∥x∥ = ∥ST x∥ ⩽ ∥S∥∥T x∥, so ∥T x∥ ⩾ \frac{1}{∥S∥}∥x∥. This shows that T is an isomorphism. In particular, Range T is a closed subspace of Y isomorphic to X.

Put P = T S ∈ L(Y). Then P^{2} = T ST S = T id_{X}S = T S = P, hence P is a projection. Clearly, Range P ⊆ Range T. Also, P T = T ST = T, so Range T ⊆ Range P. Therefore Range P = Range T, and Range T is complemented. □

Lemma 2.2. Let X and Y be Banach spaces such that Y is isomorphic to Y ⊕Y. Then the set J = {T ∈ L(X) : T factors through Y } is an ideal in L(X).

Proof. It is clear that J is closed under multiplication by operators in L(X). In particular, J is closed under scalar multiplication. Let A, B ∈ J. Write A = A_{1}A_{2} and B = B_{1}B_{2}, where A_{1}, B_{1} ∈ L(Y, X) and A_{2}, B_{2} ∈ L(X, Y). Then A + B = UV where V : x ∈ X ↦ (A_{2}x, B_{2}x) ∈ Y ⊕ Y and U: (x, y) ∈ Y ⊕ Y ↦ A_{1}x + B_{1}y ∈ Y. Clearly, UV factors through Y ⊕ Y ≃ Y. Hence A + B ∈ J. □

We will denote the set of all operators in L(d_{w,p}) which factor through a Banach space Y by J^{Y}.

Theorem 2.3. The sets J^{ℓ_{p}} and \overline{J^{ℓ_{p}}} are proper ideals in L(d_{w,p}).

Proof. Since ℓ_{p} is isomorphic to ℓ_{p} ⊕ ℓ_{p}, it follows from Lemma 2.2 that J^{ℓ_{p}} is an ideal in L(d_{w,p}). Let us show that J^{ℓ_{p}} ≠ L(d_{w,p}).

Assume that J^{ℓ_{p}} = L(d_{w,p}), then the identity operator I on d_{w,p} belongs to J. Write I = ST where T ∈ L(d_{w,p}, ℓ_{p}) and S ∈ L(ℓ_{p}, d_{w,p}). By Lemma 2.1, the range of T is complemented in ℓ_{p} and is isomorphic to d_{w,p}. This is a contradiction because all complemented infinite-dimensional subspaces of ℓ_{p} are isomorphic to ℓ_{p} (see, e.g., [17, Theorem 2.a.3]), while d_{w,p} is not isomorphic to ℓ_{p} (see [6] for the case p = 1 and [14] for the case 1 < p < ∞; see also [17, p. 176]).

Being the closure of a proper ideal, \overline{J^{ℓ_{p}}} is itself a proper ideal (see, e.g., [11, Corollary VII.2.4]). □

Proposition 2.4. There exists a projection P ∈ L(d_{w,p}) such that Range P is isomorphic to ℓ_{p}. For every such P we have J_{P} = J^{ℓ_{p}}.

OPERATOR IDEALS IN LORENTZ SPACES 7

Proof. Such projections exist by Remark 1.4. Let Y = Range P, U: Y → ℓ_{p} be an isomorphism onto, and i: Y → d_{w,p} be the inclusion map. It is easy to see that P = (iU^{−1})(UP), hence P ∈ J^{ℓ_{p}}, so that J_{P} ⊆ J^{ℓ_{p}}.

On the other hand, if T ∈ J^{ℓ_{p}} is arbitrary, T = AB with A ∈ L(ℓ_{p}, d_{w,p}), B ∈ L(d_{w,p}, ℓ_{p}), then one can write T = (AUP)P (iU^{−1}B), so that T ∈ J_{P}. Thus J^{ℓ_{p}} ⊆ J_{P}. □

Corollary 2.5. The ideal \overline{J^{ℓ_{p}}} properly contains the ideal of compact operators K(d_{w,p}).

Proof. It was already mentioned in the introductory section that compact operators form the smallest closed ideal in L(d_{w,p}). Since a projection onto a subspace isomorphic to ℓ_{p} is not compact, it follows that K(d_{w,p}) ≠ \overline{J^{ℓ_{p}}}. □

3. Strictly singular operators

In this section we will study properties of strictly singular operators in L(d_{w,p}). Since projections onto the subspaces of d_{w,p} isomorphic to ℓ_{p} are clearly not strictly singular, it follows from Proposition 2.4 that SS(d_{w,p}) ≠ J^{ℓ_{p}}. Moreover, SS ≠ \overline{J^{ℓ_{p}}} ∨ SS and \overline{J^{ℓ_{p}}} ∧ SS ≠ \overline{J^{ℓ_{p}}}. So, the ideals we discussed so far can be arranged as follows:

[figure: SS; ≠; \{0\}; K; \overline{J^{ℓ_{p}}} ∧ SS; ≠; \overline{J^{ℓ_{p}}} ∨ SS; L(d_{w,p}); \overline{J^{ℓ_{p}}}]

The following theorem shows that there can be no other closed ideals between SS and \overline{J^{ℓ_{p}}} ∨ SS on this diagram.

Theorem 3.1. Let T ∈ L(d_{w,p}). If T ∉ SS(d_{w,p}) then J^{ℓ_{p}} ⊆ J_{T}.

Proof. Let T ∉ SS(d_{w,p}). Then there exists an infinite-dimensional subspace Y of d_{w,p} such that T |_{Y} is an isomorphism. By Remark 1.4, passing to a subspace, we may assume that Y is complemented in d_{w,p} and isomorphic to ℓ_{p}. Let (x_{n}) be a basis of Y equivalent to the unit vector basis of ℓ_{p}. Define z_{n} = T x_{n}, then (z_{n}) is also equivalent to the unit vector basis of ℓ_{p}. By Remark 1.4, (z_{n}) has a subsequence (z_{n_{k}}) such that [z_{n_{k}}] is complemented in d_{w,p} and isomorphic to ℓ_{p}.

Denote W = [x_{n_{k}}]. Then W and T (W) are both complemented subspaces of d_{w,p} isomorphic to ℓ_{p}. Let P and Q be projections onto W and T (W), respectively. Put S = (T |_{W} )^{−1}, S ∈ L(T (W), d_{w,p}). Then it is easy to see that P = (SQ)T P. Since SQ and P are in L(d_{w,p}), we have J_{P} ⊆ J_{T}. By Proposition 2.4, J^{ℓ_{p}} ⊆ J_{T}. □

8 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Corollary 3.2. \overline{J^{ℓ_{p}}} ⋁ SS(d_{w,p}) is the only immediate successor of SS(d_{w,p}) and \overline{J^{ℓ_{p}}} is an immediate successor of \overline{J^{ℓ_{p}}} ∧ SS(d_{w,p}).

Now we will investigate the ideal of finitely strictly singular operators on d_{w,p}. To prove the main statement (Theorem 3.5), we will need the following lemma due to Milman [19] (see also a thorough discussion in [22]). This lemma will be used more than once in the paper.

Lemma 3.3 ([19]). If F is a k-dimensional subspace of c_{0} then there exists a vector x ∈ F such that x attains its sup-norm at at least k coordinates (that is, x^{∗} starts with a constant block of length k).

We will also use the following simple lemma.

Lemma 3.4. Let s_{n} = ∑_{i=1}^{n} w_{i} (n ∈ ℕ) where w = (w_{i}) is the sequence of weights for d_{w,p}. If x ∈ d_{w,p}, y = x^{∗}, and N ∈ ℕ then 0 ⩽ y_{N} ⩽ \frac{∥x∥}{s_{N}^{1/p}}.

Proof. ∥x∥^{p} = ∥y∥^{p} = ∑_{i=1}^{∞} y_{i}^{p}w_{i} ⩾ y_{N}^{p} ∑_{i=1}^{N} w_{i} = y_{N}^{p}s_{N}. □

Theorem 3.5. Let X and Y be subspaces of d_{w,p}. Then FSS(X, Y) = SS(X, Y). In particular, FSS(ℓ_{p}, d_{w,p}) = SS(ℓ_{p}, d_{w,p}) and FSS(d_{w,p}) = SS(d_{w,p}).

Proof. Let T ∈ L(X, Y). Suppose that T is not finitely strictly singular. We will show that it is not strictly singular. Since T is not finitely strictly singular, there exists a constant c > 0 and a sequence F_{n} of subspaces of X with dim F_{n} ⩾ n such that for each n and for all x ∈ F_{n} we have ∥T x∥ ⩾ c∥x∥.

Fix a sequence (ε_{k}) in ℝ such that 1 > ε_{k} ↓ 0. We will inductively construct a sequence (x_{k}) in X and two strictly increasing sequences (n_{k}), (m_{k}) in ℕ such that:

(i) (x_{k}) and (T x_{k}) are seminormalized; we will denote T x_{k} by u_{k}; (ii) for all k ∈ ℕ, supp x_{k} ⊆ [n_{k}, ∞) and supp u_{k} ⊆ [m_{k}, ∞); (iii) if k ⩾ 2 then ∥x_{k−1}|_{[n_{k},∞)}∥ < ε_{k}, ∥u_{k−1}|_{[m_{k},∞)}∥ < ε_{k}, and all the coordinates of

u_{k−1} where the sup-norm is attained are less than m_{k}; (iv) for each k ∈ ℕ, the vector u_{k}^{∗} begins with a constant block of length at least k.

That is, (x_{n}) and (u_{n}) are two almost disjoint sequences and u_{n}’s have long “flat” sections.

Take x_{1} to be any nonzero vector in F_{1} and put n_{1} = m_{1} = 1. Suppose we have already constructed x_{1}, …, x_{k−1}, n_{1}, …, n_{k−1}, and m_{1}, …, m_{k−1} such that the conditions (i)– (iv) are satisfied. Choose n_{k} ∈ ℕ and m_{k} ∈ ℕ such that n_{k} > n_{k−1}, m_{k} > m_{k−1} and the condition (iii) is satisfied.

OPERATOR IDEALS IN LORENTZ SPACES 9

Consider the space

V = {y = (y_{i}) ∈ F_{n_{k}+m_{k}+k} : y_{i} = 0 for i < n_{k}} ⊆ F_{n_{k}+m_{k}+k}.

It follows from dim F_{n_{k}+m_{k}+k} ⩾ n_{k} +m_{k} +k that dim V ⩾ m_{k} +k. Since V ⊆ F_{n_{k}+m_{k}+k}, ∥T y∥ ⩾ c∥y∥ for all y ∈ V. In particular, dim(T V) ⩾ m_{k} + k. Define

Z = {z = (z_{i}) ∈ T V : z_{i} = 0 for i < m_{k}}.

It follows that dim Z ⩾ k.

Clearly, supp y ⊆ [n_{k}, ∞) for all y ∈ V and supp z ⊆ [m_{k}, ∞) for all z ∈ Z. By Lemma 3.3, we can choose u_{k} ∈ Z such that u_{k} is normalized and u_{k}^{∗} starts with a constant block of length k. Put x_{k} = (T |_{V})^{−1}(u_{k}) ∈ Y. Since x_{k} ∈ V and ∥u_{k}∥ = 1, it follows that \frac{1}{∥T ∥} ⩽ ∥x_{k}∥ ⩽ \frac{1}{c}, so the conditions (i)–(iv) are satisfied for (x_{k}).

For each k ∈ ℕ, let x′_{k} = x_{k}|_{[n_{k},n_{k+1})} and u′_{k} = u_{k}|_{[m_{k},m_{k+1})}. Passing to tails of sequences, if necessary, we may assume that both (x′_{k}) and (u′_{k}) are seminormalized block sequences of (e_{n}).

Since the non-increasing rearrangement of each u′_{k} starts with a constant block of length k by (iii), the coefficients in u′_{k} converge to zero by Lemma 3.4. Therefore, passing to a subsequence, we may assume by Remark 1.4 that (u′_{k}) is equivalent to the unit vector basis (f_{n}) of ℓ_{p}. Using Theorem 1.2 and passing to a further subsequence, we may also assume that (x_{k}) ∼ (x′_{k}) and (u_{k}) ∼ (u′_{k}).

By Proposition 1.8, the sequence (x′_{k}) is dominated by (f_{n}). Notice that the condition u_{k} = T x_{k} implies (x_{k}) ⪰ (u_{k}). Therefore, we get the following chain of dominations and equivalences of basic sequences:

(f_{n}) ⪰ (x′_{k}) ∼ (x_{k}) ⪰ (u_{k}) ∼ (u′_{k}) ∼ (f_{n}).

It follows that all the dominations in this chain are, actually, equivalences. In particular, (x_{k}) ∼ (u_{k}). Thus, T is an isomorphism on the space [x_{k}], hence T is not strictly singular. □

Recall that an operator T on a Banach space X is weakly compact if the image of the unit ball of X under T is relatively weakly compact. Alternatively, T is weakly compact if and only if for every bounded sequence (x_{n}) in X there exists a subsequence (x_{n_{k}}) of (x_{n}) such that (T x_{n_{k}}) is weakly convergent.

If 1 < p < ∞ then d_{w,p} is reflexive, and, hence, every operator in L(d_{w,p}) is weakly compact. In case p = 1 we have the following.

Theorem 3.6. Let T ∈ L(d_{w,1}). Then T is weakly compact if and only if T is strictly singular.

10 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Proof. Suppose that T is strictly singular. We will show that T is weakly compact.

Let (x_{n}) be a bounded sequence in X. By Rosenthal’s ℓ_{1}-theorem, there is a subsequence (x_{n_{k}}) of (x_{n}) such that (x_{n_{k}}) is either equivalent to the unit vector basis (f_{n}) of ℓ_{1} or is weakly Cauchy. In the latter case, (T x_{n_{k}}) is also weakly Cauchy. If (x_{n_{k}}) ∼ (f_{n}) then, since T is strictly singular, (T x_{n_{k}}) cannot have subsequences equivalent to (f_{n}). Hence, using Rosenthal’s theorem one more time and passing to a further subsequence, we may assume that, again, (T x_{n_{k}}) is weakly Cauchy. Since d_{w,1} is weakly sequentially complete, the sequence (T x_{n_{k}}) is weakly convergent. It follows that T is weakly compact.

Conversely, let J be the closed ideal of weakly compact operators in L(d_{w,1}). By the first part of the proof, J is a successor of SS(d_{w,1}). Suppose that J ≠ SS(d_{w,1}). By Theorem 3.1, J^{ℓ_{1}} ⊆ J. This, however, is a contradiction since a projection onto a copy of ℓ_{1} (which belongs to J^{ℓ_{1}} by Proposition 2.4) is not weakly compact. □

4. Operators factorable through the formal identity

The operator j: ℓ_{p} → d_{w,p} defined by j(e_{n}) = f_{n} is called the formal identity operator from ℓ_{p} to d_{w,p}. It follows immediately from the definition of the norm in d_{w,p} that ∥j∥ = 1.

We will denote by the symbol J^{j} the set of all operators T ∈ L(d_{w,p}) which can be factored as T = AjB where A ∈ L(d_{w,p}) and B ∈ L(d_{w,p}, ℓ_{p}).

Proposition 4.1. J^{j} is an ideal in L(d_{w,p}).

Proof. It is clear from the definition that the set J^{j} is closed under both right and left multiplication by operators from L(d_{w,p}). We have to show that if T_{1} and T_{2} are in J^{j} then T_{1} + T_{2} is in J^{j}, as well.

Write T_{1} = A_{1}jB_{1}, T_{2} = A_{2}jB_{2} with A_{1}, A_{2} ∈ L(d_{w,p}) and B_{1}, B_{2} ∈ L(d_{w,p}, ℓ_{p}). Let A ∈ L(d_{w,p} ⊕ d_{w,p}, d_{w,p}) and B ∈ L(d_{w,p}, ℓ_{p} ⊕ ℓ_{p}) be defined by

A(x_{1}, x_{2}) = A_{1}x_{1} + A_{2}x_{2} and Bx = (B_{1}x, B_{2}x).

Define also U: ℓ_{p} → ℓ_{p} ⊕ ℓ_{p} and V : d_{w,p} → d_{w,p} ⊕ d_{w,p} by

U((x_{n})) = ((x_{2n−1}), (x_{2n})), and V ((x_{n})) = ((x_{2n−1}), (x_{2n})).

Since the bases of ℓ_{p} and d_{w,p} are both unconditional, U and V are bounded.

Now observe that for each x = (x_{n}) ∈ d_{w,p} we can write

AV jU^{−1}Bx = AV jU^{−1}(B_{1}x, B_{2}x) =

A(jB_{1}x, jB_{2}x) = A_{1}jB_{1}x +A_{2}jB_{2}x = T_{1}x + T_{2}x.

OPERATOR IDEALS IN LORENTZ SPACES 11

This shows that T_{1} +T_{2} = AV jU^{−1}B with AV ∈ L(d_{w,p}) and U^{−1}B ∈ L(d_{w,p}, ℓ_{p}), hence T_{1} + T_{2} ∈ J^{j}. □

As we already mentioned before, the space d_{w,p} contains many complemented copies of ℓ_{p}. Consider the operator jUP ∈ L(d_{w,p}) where P is a projection onto any subspace Y isomorphic to ℓ_{p} and U: Y → ℓ_{p} is an isomorphism onto. It turns out that the ideal generated by any such operator does not depend on the choice of Y and, in fact, coincides with J^{j}.

Proposition 4.2. Let Y be a complemented subspace of d_{w,p} isomorphic to ℓ_{p}, P ∈ L(d_{w,p}) be a projection with range Y, and U: Y → ℓ_{p} be an isomorphism onto. If T = jUP then J_{T} = J^{j}.

Proof. Clearly, J_{T} ⊆ J^{j}. Let S ∈ J^{j}. Then S = AjB where A ∈ L(d_{w,p}) and B ∈ L(d_{w,p}, ℓ_{p}). It follows that

S = AjB = Aj(UP U^{−1})B = AT (U^{−1}B) ∈ J_{T} .

□

The next goal is to show that the ideal \overline{J^{j}} “sits” between K(X) and SS(X) ∧ \overline{J^{ℓ_{p}}}.

Theorem 4.3. The formal identity operator j: ℓ_{p} → d_{w,p} is finitely strictly singular.

Proof. Let ε > 0 be arbitrary. Take n ∈ ℕ such that \frac{1}{n} ∑_{i=1}^{n} w_{i} < ε; such n exists by w_{n} → 0. Since (w_{n}) is also a decreasing sequence, it follows that w_{i} < ε for all i ⩾ n.

Let Y ⊆ ℓ_{p} be a subspace with dim Y ⩾ n. By Lemma 3.3, there exists a vector x ∈ Y such that ∥x∥_{ℓ_{p}} = 1 and x attains its sup-norm at at least n coordinates. Denote δ = ∥x∥_{sup} > 0. Then ∥x∥_{ℓ_{p}} ⩾ n^{1/p}δ, so δ ⩽ n^{−1/p}.

Observe that the non-increasing rearrangement x^{∗} of x satisfies the condition that x_{i}^{∗} = δ for all 1 ⩽ 1 ⩽ n. Therefore

∥jx∥_{d_{w,p}}^{p} = ∑_{i=1}^{∞} x_{i}^{∗p}w_{i} ⩽ δ^{p} ∑_{i=1}^{n} w_{i} + ε ∑_{i=n+1}^{∞} x_{i}^{∗p} ⩽ δ^{p}nε + ε∥x∥_{ℓ_{p}}^{p} ⩽ 2ε.

Hence ∥jx∥_{d_{w,p}} ⩽ (2ε)^{1/p}. □

Corollary 4.4. The following inclusions hold: K(d_{w,p}) ⊊ \overline{J^{j}} and J^{j} ⊆ SS(d_{w,p}) ∧ J^{ℓ_{p}}.

Proof. Let Y, P, and U be as in Proposition 4.2. Then jUP ∈ J^{j}. If x_{n} = U^{−1}f_{n} ∈ d_{w,p} then (x_{n}) is seminormalized and jUP x_{n} = e_{n}. Hence the sequence (jUP x_{n}) has no convergent subsequences, so that jUP is not compact.

The inclusion J^{j} ⊆ SS(d_{w,p}) ∧ J^{ℓ_{p}} is obvious since j is strictly singular. □

12 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Conjecture 4.5. The ideal \overline{J^{j}} is the only immediate successor of K(d_{w,p}).

In [3] and [10] (see also [17]), conditions on the weights w = (w_{n}) are given under which d_{w,p} has exactly two non-equivalent symmetric basic sequences. We will show that the conjecture holds true in this case.

Lemma 4.6. If T ∈ SS(d_{w,p})\K(d_{w,p}) then there exists a seminormalized basic sequence (x_{n}) in d_{w,p} such that (f_{n}) ⪰ (x_{n}) and (T x_{n}) is weakly null and seminormalized.

Proof. Let (z_{n}) be a bounded sequence in d_{w,p} such that (T z_{n}) has no convergent subsequences. Then (z_{n}) has no convergent subsequences either. Applying Rosenthal’s ℓ_{1}-theorem and passing to a subsequence, we may assume that (z_{n}) is either equivalent to the unit vector basis of ℓ_{1} or is weakly Cauchy.

Case: (z_{n}) is equivalent to the unit vector basis of ℓ_{1}. Since a reflexive space cannot contain a copy of ℓ_{1}, we conclude that p = 1, so (z_{n}) ∼ (f_{n}). Again, by Rosenthal’s theorem, (T z_{n}) has a subsequence which is either equivalent to (f_{n}) or is weakly Cauchy. If (T z_{n_{k}}) ∼ (f_{n}) then T is an isomorphism on the space [z_{n_{k}}], contrary to the assumption that T ∈ SS(d_{w,p}). Therefore, (T z_{n_{k}}) is weakly Cauchy. Put x_{k} = z_{n_{2k}} − z_{n_{2k−1}}. Then (x_{k}) is basic and (T x_{k}) is weakly null. Passing to a further subsequence of (z_{n_{k}}) we may assume that (T x_{k}) is seminormalized. Also, (x_{k}) is still equivalent to (f_{n}), hence is dominated by (f_{n}).

Case: (z_{n}) is weakly Cauchy. Clearly, (T z_{n}) is also weakly Cauchy. Consider the sequence (u_{n}) in d_{w,p} defined by u_{n} = z_{2n} − z_{2n−1}. Then both (u_{n}) and (T u_{n}) are weakly null. Passing to a subsequence of (z_{n}), we may assume that (T u_{n}) and, hence, (u_{n}) are seminormalized. Applying Theorem 1.3, we get a subsequence (u_{n_{k}}) of (u_{n}) which is basic and equivalent to a block sequence (v_{n}) of (e_{n}). Denote x_{k} = u_{n_{k}}. By Proposition 1.8, (f_{n}) dominates (v_{n}) and, hence, (x_{k}). □

Theorem 4.7. If d_{w,p} has exactly two non-equivalent symmetric basic sequences, then \overline{J^{j}} is the only immediate successor of K(d_{w,p}).

Proof. Let T be a non-compact operator on d_{w,p}. It suffices to show that J^{j} ⊆ J_{T}. We may assume that T is strictly singular because, otherwise, we have J^{j} ⊆ J^{ℓ_{p}} ⊆ J_{T} by Theorem 3.1.

Let (x_{n}) be a sequence as in Lemma 4.6. Passing to a subsequence and using Theorem 1.3, we may assume that (T x_{n}) is basic and equivalent to a block sequence (h_{n}) of (e_{n}) such that T x_{n} − h_{n} → 0. We claim that (h_{n}) has no subsequences equivalent to (f_{n}). Indeed, otherwise, for such a subsequence (h_{n_{k}}) of (h_{n}), we would have

OPERATOR IDEALS IN LORENTZ SPACES 13

(f_{n}) ∼ (f_{n_{k}}) ⪰ (x_{n_{k}}) ⪰ (T x_{n_{k}}) ∼ (h_{n_{k}}) ∼ (f_{n}), so (x_{n_{k}}) ∼ (T x_{n_{k}}), contrary to T ∈ SS(d_{w,p}). By [10, Theorem 19], (h_{n}) has a subsequence which spans a complemented subspace in d_{w,p} and is equivalent to (e_{n}). Therefore, by Theorem 1.2, we may assume (by passing to a further subsequence) that (T x_{n}) ∼ (e_{n}) and [T x_{n}] is complemented in d_{w,p}.

We have proved that there exists a sequence (x_{n}) in d_{w,p} such that [T x_{n}] is complemented in d_{w,p} and

(f_{n}) ⪰ (x_{n}) ⪰ (T x_{n}) ∼ (e_{n}).

Let A ∈ L(ℓ_{p}, d_{w,p}) and B ∈ L([T x_{n}], d_{w,p}) be defined by Af_{n} = x_{n} and B(T x_{n}) = e_{n}. Let Q ∈ L(d_{w,p}) be a projection onto [T x_{n}]. Then for all n ∈ ℕ, we obtain: BQT Af_{n} = BQT x_{n} = BT x_{n} = e_{n}. It follows that BQT A = j, so that J^{j} ⊆ J_{T}. □

In order to prove Conjecture 4.5 without additional conditions on w, it suffices to show that if T ∈ \overline{J^{j}} \ K(d_{w,p}) then J^{j} ⊆ \overline{J_{T}}. We will prove a weaker statement: if T ∈ J^{j} \ K(d_{w,p}) then J^{j} ⊆ J_{T}.

Recall (see [3, p.148]) that if x = (a_{n}) ∈ d_{w,p} then a block sequence (y_{n}) of (e_{n}) is called a block of type I generated by x if it is of the form y_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i−p_{n}}e_{i} for all n. A set A ⊆ d_{w,p} will be said to be almost lengthwise bounded if for each ε > 0 there exists N ∈ ℕ such that ∥x^{∗}|_{[N,∞)}∥ < ε for all x ∈ A. We will usually use it in the case when A = {x_{n}} for some sequence (x_{n}) in d_{w,p}. We need the following result, which is a slight extension of [3, Theorem 3]. We include the proof for completeness.

Theorem 4.8. Let (x_{n}) be a seminormalized block sequence of (e_{n}) in d_{w,p}.

(i) If (x_{n}) is not almost lengthwise bounded then there exists a subsequence (x_{n_{k}}) such that (x_{n_{k}}) ∼ (f_{n}).

(ii) If (x_{n}) is almost lengthwise bounded, then there exists a subsequence (x_{n_{k}}) equivalent to a block of type I generated by a vector u = ∑_{i=1}^{∞} b_{i}e_{i} ∈ d_{w,p} with b_{i} ↓ 0. Moreover, if the sequence (x_{n}) is bounded in ℓ_{p} then u is in ℓ_{p}.^{1}

Proof. (i) Without loss of generality, ∥x_{n}∥ ⩽ 1 for all n ∈ ℕ. By the assumption, there exists ε > 0 with the property that for each k ∈ ℕ, there is n_{k} ∈ ℕ such that ∥x_{n_{k}}^{∗}|_{(k,∞)}∥ ⩾ ε. Let u_{k} be a restriction of x_{n_{k}} such that u_{k}^{∗} = x_{n_{k}}^{∗}|_{[1,k]} and v_{k} = x_{n_{k}} −u_{k}.

Clearly, each nonzero entry of u_{k} is greater than or equal to the greatest entry of v_{k}. By Lemma 3.4, the k-th coordinate of u_{k}^{∗} is less than or equal to _{\frac{1}{s_{k}^{1/p}}} where s_{k} = ∑_{i=1}^{k} w_{i}.

1As a sequence space, ℓ_{p} is a subset of d_{w,p}. That is, we can identify ℓ_{p} with Range j. More precisely, we claim here that if (j^{−1}x_{n}) is bounded in ℓ_{p} then u is in Range j. Being a block sequence of (e_{n}), (x_{n}) is contained in Range j.

14 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

It follows that (v_{k}) is a block sequence of (e_{n}) such that ε ⩽ ∥v_{k}∥ ⩽ 1 and absolute values of the entries of v_{k} are all at most _{\frac{1}{s_{k}^{1/p}}}. Since lim_{k} s_{k} = +∞ by the definition of d_{w,p}, passing to a subsequence and using Remark 1.4 we may assume that (v_{k}) is equivalent to (f_{n}). By Proposition 1.8, (f_{n}) dominates (x_{n_{k}}). Using also Lemma 1.9, we obtain the following diagram:

(f_{n}) ⪰ (x_{n_{k}}) ⪰ (v_{k}) ∼ (f_{n}).

Hence (x_{n_{k}}) is equivalent to (f_{n}).

(ii) Suppose that x_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}e_{i}. Clearly, the sequence (a_{i}) is bounded. Without loss of generality, a_{p_{n}+1} ⩾ … ⩾ a_{p_{n+1}} ⩾ 0 for each n. Put y_{n} = x_{n}^{∗}. Using a standard diagonalization argument and passing to a subsequence, we may assume that (y_{n}) converges coordinate-wise; put b_{i} = lim_{n→∞} y_{n,i}. It is easy to see that b_{i} ⩾ b_{i+1} for all i. Put u = (b_{i}).

Case: the sequence (p_{n+1} −p_{n}) is bounded. Passing to a subsequence, we may assume that N := p_{n_{k}+1} −p_{n_{k}} is a constant. Note that supp u ⊆ [1, N] and supp y_{n_{k}} ⊆ [1, N] for all k. Put u_{k} = ∑_{i=p_{n_{k}} +1}^{p_{n_{k}+1}} b_{i−p_{n_{k}}} e_{i}, then u = u_{k}^{∗} and (u_{k}) as a block of type I generated by u. By compactness, ∥x_{n_{k}} − u_{k}∥ = ∥y_{n_{k}} − u∥ → 0. Therefore, passing to a further subsequence, we have (x_{n_{k}}) ∼ (u_{k}). Being a vector with finite support, u belongs to ℓ_{p}.

Case: the sequence (p_{n+1} − p_{n}) is unbounded. We will construct the required subsequence (x_{n_{k}}) and a sequence (N_{k}) inductively. Put n_{1} = N_{1} = 1 and let k > 1. Suppose that n_{1}, …, n_{k−1} and N_{1}, …, N_{k−1} have already been selected. Since (x_{n}) is almost lengthwise bounded, we can find N_{k} > N_{k−1} such that ∥y_{n}|_{(N_{k},∞)}∥ < \frac{1}{k} for all n. Put v_{k} = u|_{[1,N_{k}]}. Using coordinate-wise convergence, we can find n_{k} > n_{k−1} such that ∥y_{n_{k}}|_{[1,N_{k}]} −v_{k}∥_{ℓ_{p}} < \frac{1}{k} and p_{n_{k}} +N_{k} ⩽ p_{n_{k}+1}. Put u_{k} = ∑_{i=p_{n_{k}} +1}^{p_{n_{k}} +N_{k}} b_{i−p_{n_{k}}} e_{i}. Then u_{k}^{∗} = v_{k}, so that

(1)  ∥x_{n_{k}}|_{(p_{n_{k}} ,p_{n_{k}} +N_{k}]} − u_{k}∥_{ℓ_{p}} = ∥y_{n_{k}}|_{[1,N_{k}]} − v_{k}∥_{ℓ_{p}} < \frac{1}{k}

and

∥x_{n_{k}}|_{(p_{n_{k}} +N_{k},p_{n_{k}+1}]}∥ = ∥y_{n_{k}}|_{(N_{k},∞)}∥ < \frac{1}{k}.

It follows that ∥x_{n_{k}} − u_{k}∥ → 0. Passing to a subsequence, we get (x_{n_{k}}) ∼ (u_{k}). Next, we show that u ∈ d_{w,p}. Since ∥·∥ ⩽ ∥·∥_{ℓ_{p}}, it follows from (1) that

∥v_{k}∥ = ∥u_{k}∥ ⩽ ∥x_{n_{k}}|_{(p_{n_{k}} ,p_{n_{k}} +N_{k}]}∥ + \frac{1}{k} ⩽ ∥x_{n_{k}}∥ + \frac{1}{k}.

Since (x_{n}) is bounded, so is (v_{k}). Since supp v_{k} = N_{k} → ∞, we have u ∈ d_{w,p}. For the “moreover” part, we argue in a similar way. By (1), we have

∥v_{k}∥_{ℓ_{p}} ⩽ ∥u_{k}∥_{ℓ_{p}} ⩽ ∥x_{n_{k}}|_{(p_{n_{k}} ,p_{n_{k}} +N_{k}]}∥_{ℓ_{p}} + \frac{1}{k} ⩽ ∥x_{n_{k}}∥_{ℓ_{p}} + \frac{1}{k}.

Therefore, if (x_{n}) is bounded in ℓ_{p} then so is (v_{k}), hence u ∈ ℓ_{p}. □

Lemma 4.9. Suppose that (u_{n}) is a block of type I in d_{w,p} generated by some u = ∑_{i=1}^{∞} b_{i}e_{i}. If b_{i} ↓ 0 and u ∈ ℓ_{p} then (u_{n}) has a subsequence equivalent to (e_{n})

Proof. By Corollary 4 of [3], we may assume that the basic sequence (u_{n}) is symmetric. It suffices to show that [u_{n}] is isomorphic to d_{w,p} because all symmetric bases in d_{w,p} are equivalent; see e.g., Theorem 4 of [3]. Without loss of generality, ∥u∥ = 1. Lemma 4 of [3] asserts that [u_{n}] is isomorphic to d_{w,p} iff (s_{n}^{(u)}) ∼ (s_{n}), where s_{n} = ∑_{i=1}^{n} w_{i}, s_{n}^{(u)} = ∑_{i=1}^{∞} b_{i}^{p}(s_{ni} − s_{n(i−1)}), and (α_{n}) ∼ (β_{n}) means that there exist positive constants A and B such that Aα_{n} ⩽ β_{n} ⩽ Bα_{n} for all n. Let’s verify that this condition is, indeed, satisfied. On one hand, taking only the first term in the definition of s_{n}^{(u)}, we get s_{n}^{(u)} ⩾ b_{1}^{p}s_{n}. On the other hand, it follows from w_{i} ↓ that s_{ni} − s_{n(i−1)} ⩽ s_{n} for every i, hence s_{n}^{(u)} ⩽ ∑_{i=1}^{∞} b_{i}^{p}s_{n} = ∥u∥_{ℓ_{p}}^{p}s_{n}. □

Lemma 4.10. Let (x_{n}) be a block sequence of (f_{n}) in ℓ_{p} such that the sequences (x_{n}) and (jx_{n}) are seminormalized in ℓ_{p} and d_{w,p}, respectively. Then there exists a subsequence (x_{n_{k}}) such that (jx_{n_{k}}) ∼ (e_{n}).

Proof. Clearly, (x_{n}) ∼ (f_{n}). It follows that (jx_{n}) ≁ (f_{n}) because, otherwise, j would be an isomorphism on [x_{n}], which is impossible because j is strictly singular by Theorem 4.3. Applying Theorem 4.8 to (jx_{n}) and passing to a subsequence, we may assume that (jx_{n}) ∼ (u_{n}), where (u_{n}) is a block of type I generated by some u = ∑_{i=1}^{∞} b_{i}e_{i} such that b_{i} ↓ 0 and u ∈ ℓ_{p}. Applying Lemma 4.9 and passing to a subsequence, we get (u_{n}) ∼ (e_{n}). □

Theorem 4.11. If T ∈ J^{j} \ K(d_{w,p}) then J^{j} ⊆ J_{T}.

Proof. Write T = AjB where B: d_{w,p} → ℓ_{p} and A: d_{w,p} → d_{w,p}. Let (x_{n}) be as in Lemma 4.6. The sequence (Bx_{n}) is bounded, hence we may assume by passing to a subsequence that it converges coordinate-wise. Since (T x_{n}) is weakly null and seminormalized, it has no convergent subsequences. It follows that, after passing to a subsequence of (x_{n}), we may assume that (T z_{n}) is seminormalized, where z_{n} = x_{2n} − x_{2n−1}. In particular, (z_{n}), (Bz_{n}), and (jBz_{n}) are seminormalized. Also, (Bz_{n}) converges to zero coordinate-wise. Using Theorem 1.3 and passing to a further subsequence, we may assume that (Bz_{n}) is equivalent to a block sequence (u_{n}) of (f_{n}) and Bz_{n}−u_{n} → 0. It follows from (f_{n}) ⪰ (x_{n}) that (f_{n}) ⪰ (z_{n}) ⪰ (Bz_{n}) ∼ (u_{n}) ∼ (f_{n}). In particular, (z_{n}) ∼ (f_{n}).

16 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Since Bz_{n} −u_{n} → 0 and (jBz_{n}) is seminormalized, we may assume that the sequence (ju_{n}) is seminormalized. By Lemma 4.10, passing to a further subsequence, we may assume that (ju_{n}) and, hence, (jBz_{n}) are equivalent to (e_{n}).

Passing to a subsequence and using Theorem 1.3, we may assume that (T z_{n}) is equivalent to a block sequence (v_{n}) of (e_{n}) such that T z_{n} −v_{n} → 0. Since T ∈ SS(d_{w,p}), no subsequence of (T z_{n}) and, therefore, of (v_{n}) is equivalent to (f_{n}). By Proposition 1.8, (v_{n}) ⪰ (e_{n}). It follows from (jBz_{n}) ∼ (e_{n}) that (e_{n}) ⪰ (T z_{n}), hence (T z_{n}) ∼ (e_{n}) ∼ (v_{n}).

Write v_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{n}e_{n}. By Remark 1.4, a_{n} ↛ 0. Hence, passing to a subsequence and using [10, Remark 9], we may assume that [v_{n}] is complemented. By Theorem 1.3, we may assume that [T z_{n}] is complemented. Let P ∈ L(d_{w,p}) be a projection onto [T z_{n}] and U ∈ L(ℓ_{p}, d_{w,p}) and V ∈ L([T z_{n}], d_{w,p}) be defined by Uf_{n} = z_{n} and V T z_{n} = e_{n}. Then we can write j = V P T U. Therefore J^{j} ⊆ J_{T}. □

5. d_{w,p}-strictly singular operators

The ideals in L(d_{w,p}) we have obtained so far can be arranged into the following diagram.

[figure: SS; \{0\}; K ⊊ \overline{J^{j}}; \overline{J^{ℓ_{p}}} ∧ SS; \overline{J^{ℓ_{p}}} ∨ SS; L(d_{w,p}); \overline{J^{ℓ_{p}}}]

(see the Introduction for the notations). In this section, we will characterize the greatest ideal in the algebra L(d_{w,p}), that is, a proper ideal in L(d_{w,p}) that contains all other proper ideals in L(d_{w,p}).

If X and Y are two Banach spaces, then an operator T ∈ L(X) is called Y -strictly singular if for any subspace Z of X isomorphic to Y, the restriction T |_{Z} is not an isomorphism. The set of all Y -strictly singular operators in L(d_{w,p}) will be denoted by SS_{Y}.

According to this notation, the symbol SS_{d_{w,p}} stands for the set of all d_{w,p}-strictly singular operators in L(d_{w,p}) (not to be confused with SS(d_{w,p})).

Lemma 5.1. Suppose that T ∈ SS_{d_{w,p}} and (x_{n}) is a basic sequence in d_{w,p} equivalent to the unit vector basis (e_{n}). Then T x_{n} → 0.

Proof. Suppose, by way of contradiction, that T x_{n} ↛ 0. Then there is a subsequence (x_{n_{k}}) such that (T x_{n_{k}}) is seminormalized. Since (x_{n}) is weakly null (Remark 1.7), we

OPERATOR IDEALS IN LORENTZ SPACES 17

may assume by using Theorem 1.3 and passing to a further subsequence that (T x_{n_{k}}) is a basic sequence equivalent to a block sequence (z_{k}) of (e_{n}).

By Proposition 1.8, either (z_{k}) has a subsequence equivalent to (f_{n}) or (z_{k}) ⪰ (e_{n}). Since (T x_{n_{k}}) cannot have subsequences equivalent to (f_{n}) (this would contradict boundedness of T), the former is impossible. Therefore (z_{k}) ⪰ (e_{n}). We obtain the following diagram:

(e_{n}) ∼ (x_{n_{k}}) ⪰ (T x_{n_{k}}) ∼ (z_{k}) ⪰ (e_{n}).

Therefore T |_{[x_{n_{k}}]} is an isomorphism. This contradicts T being in SS_{d_{w,p}}. □

Corollary 5.2. Let T ∈ SS_{d_{w,p}}. If Y ⊆ d_{w,p} is a subspace isomorphic to d_{w,p} then there is a subspace Z ⊆ Y such that Z is isomorphic to d_{w,p} and T |_{Z} is compact.

Proof. Let (x_{n}) be a basis of Y equivalent to (e_{n}). By Lemma 5.1, T x_{n} → 0. There is a subsequence (x_{n_{k}}) of (x_{n}) such that ∑_{k=1}^{∞} \frac{∥T x_{n_{k}} ∥}{∥x_{n_{k}} ∥} is convergent. Let Z = [x_{n_{k}}]. It follows that Z is isomorphic to d_{w,p} and T |_{Z} is compact (see, e.g., [8, Lemma 5.4.10]). □

Theorem 5.3. The set SS_{d_{w,p}} of all d_{w,p}-strictly singular operators in L(d_{w,p}) is the greatest proper ideal in the algebra L(d_{w,p}). In particular, SS_{d_{w,p}} is closed.

Proof. First, let us show that SS_{d_{w,p}} is an ideal. Let T ∈ SS_{d_{w,p}}. If A ∈ L(d_{w,p}) then, trivially, AT ∈ SS_{d_{w,p}}. If T A ∉ SS_{d_{w,p}} then there exists a subspace Y of d_{w,p} such that Y and T A(Y) are both isomorphic to d_{w,p}. Then A|_{Y} is bounded below, hence A(Y) is isomorphic to d_{w,p}. It follows that T is an isomorphism on a copy of d_{w,p}, contrary to T ∈ SS_{d_{w,p}}. So, SS_{d_{w,p}} is closed under two-sided multiplication by bounded operators.

Let T, S ∈ SS_{d_{w,p}}. We will show that T + S ∈ SS_{d_{w,p}}. Let Y be a subspace of d_{w,p} isomorphic to d_{w,p}. By Corolary 5.2, there exists a subspace Z of Y such that Z is isomorphic to d_{w,p} and T |_{Z} is compact. Applying Corolary 5.2 again, we can find a subspace V of Z such that V is isomorphic to d_{w,p} and S|_{V} is compact. Therefore (T + S)|_{V} is compact, so that (T + S)|_{Y} is not an isomorphism. So, SS_{d_{w,p}} is an ideal.

Clearly, the identity operator I does not belong to SS_{d_{w,p}}, so SS_{d_{w,p}} is proper. Let us show that SS_{d_{w,p}} is the greatest ideal in L(d_{w,p}).

Let T ∉ SS_{d_{w,p}}. Then there exists a subspace Y of d_{w,p} such that Y and T (Y) are isomorphic to d_{w,p}. By [10, Corollary 12], there exists a complemented (in d_{w,p}) subspace Z of T (Y) such that Z is isomorphic to d_{w,p}. Let P ∈ L(d_{w,p}) be a projection onto Z. Put H = T^{−1}(Z). It follows that H is isomorphic to d_{w,p}. Let U: d_{w,p} → H and V : Z → d_{w,p} be surjective isomorphisms. Then S ∈ L(d_{w,p}) defined by S = (V P)T U is an invertible operator. Clearly S ∈ J_{T}, hence J_{T} = L(X).

18 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

The fact that SS_{d_{w,p}} is closed follows from [11, Corollary VII.2.4]. □

The next theorem provides a convenient characterization of d_{w,p}-strictly singular operators.

Lemma 5.4. Let T ∈ L(d_{w,p}) be such that T e_{n} → 0. Suppose that (x_{n}) is a bounded block sequence of (e_{n}) in d_{w,p} such that (x_{n}) is almost lengthwise bounded. Then T x_{n} → 0.

Proof. Write x_{n} = ∑_{i=p_{n}+1}^{p_{n+1}} a_{i}e_{i}. Since (x_{n}) is bounded, there is C > 0 such that |a_{i}| ⩽ C for all i and n ∈ ℕ. Let ε > 0. Find N ∈ ℕ such that ∥x_{n}^{∗}|_{[N,∞)}∥ < ε for all n ∈ ℕ. Let u_{n} be a restriction of x_{n} such that u_{n}^{∗} = x_{n}^{∗}|_{[1,N)} and v_{n} = x_{n} − u_{n}. It is clear that ∥v_{n}∥ = ∥x_{n}^{∗}|_{[N,∞)}∥ < ε. Also, ∥T u_{n}∥ ⩽ NC · max_{p_{n}+1⩽i⩽p_{n+1}}∥T e_{i}∥.

Pick M ∈ ℕ such that ∥T e_{k}∥ < \frac{ε}{N} for all k ⩾ M. Then

∥T x_{n}∥ ⩽ ∥T u_{n}∥ + ∥T v_{n}∥ ⩽ NC \frac{ε}{N} + ε∥T ∥ = ε(C + ∥T ∥)

for all n such that p_{n} > M. It follows that T x_{n} → 0. □

Theorem 5.5. An operator T ∈ L(d_{w,p}) is d_{w,p}-strictly singular if and only if T e_{n} → 0.

Proof. Suppose that T e_{n} → 0 but T ∉ SS_{d_{w,p}}. Then there exists a subspace Y of d_{w,p} such that Y is isomorphic to d_{w,p} and T |_{Y} is an isomorphism. Let (x_{n}) be a basis of Y equivalent to (e_{n}). By Remark 1.7, x_{n} ⟶^{w} 0. Using Theorem 1.3 and passing to a subsequence, we may assume that (x_{n}) is equivalent to a block sequence (z_{n}) of (e_{n}) such that x_{n} − z_{n} → 0. Since (z_{n}) is equivalent to (e_{n}), it is almost lengthwise bounded by Theorem 4.8. By Lemma 5.4, T z_{n} → 0. Since x_{n} − z_{n} → 0, we obtain T x_{n} → 0. This is a contradiction since (x_{n}) is seminormalized and T |_{[x_{n}]} is an isomorphism.

The converse implication follows from Lemma 5.1. □

Remark 5.6. In Theorem 5.3 we showed, in particular, that SS_{d_{w,p}} is closed under addition. Alternatively, we could have deduced this from Theorem 5.5.

Recall that an operator T on a Banach space X is called Dunford-Pettis if for any sequence (x_{n}) in X, x_{n} ⟶^{w} 0 implies T x_{n} → 0. If 1 < p < ∞ then the class of Dunford-Pettis operators on d_{w,p} coincides with K(d_{w,p}) because d_{w,p} is reflexive. For the case p = 1 we have the following result.

Theorem 5.7. Let T ∈ L(d_{w,1}). Then T is d_{w,1}-strictly singular if and only if T is Dunford-Pettis.

OPERATOR IDEALS IN LORENTZ SPACES 19

Proof. If T is Dunford-Pettis then then T is d_{w,1}-strictly singular by Theorem 5.5 because (e_{n}) is weakly null.

Conversely, suppose that T is d_{w,1}-strictly singular. Let (x_{n}) be a weakly null sequence. Suppose that (T x_{n}) does not converge to zero. Then, passing to a subsequence, we may assume that (x_{n}) is a seminormalized weakly null basic sequence equivalent to a block sequence (u_{n}) of (e_{n}) such that x_{n} − u_{n} → 0. Clearly, (u_{n}) is weakly null. In particular, (u_{n}) has no subsequences equivalent to (f_{n}). By Theorem 4.8, (u_{n}) is almost lengthwise bounded. Hence, by Lemma 5.4, T u_{n} → 0. It follows that T x_{n} → 0, contrary to the choice of (x_{n}). □

6. Strictly singular operators between ℓ_{p} and d_{w,p}.

We do not know whether the ideals \overline{J^{j}}, SS ∧\overline{J^{ℓ_{p}}}, and SS are distinct. In this section, we discuss some connections between these ideals.

Conjecture 6.1. \overline{J^{j}} = SS ∧ \overline{J_{ℓ_{p}}}. In particular, every strictly singular operator in L(d_{w,p}) which factors through ℓ_{p} can be approximated by operators that factor through j.

The following statement is a refinement of Lemma 1.9. Recall that d_{w,p} is a Banach lattice with respect to the coordinate-wise order.

Lemma 6.2. Suppose that (x_{n}) and (y_{n}) are seminormalized sequences in d_{w,p} such that |x_{n}| ⩾ |y_{n}| for all n ∈ ℕ and x_{n} → 0 coordinate-wise. Then there exists an increasing sequence (n_{k}) in ℕ such that (x_{n_{k}}) and (y_{n_{k}}) are basic and (x_{n_{k}}) ⪰ (y_{n_{k}}).

Proof. Clearly, y_{n} → 0 coordinate-wise. By Theorem 1.3, we can find a sequence (n_{k}) and two block sequences (u_{k}) and (v_{k}) of (e_{n}) such that (x_{n_{k}}) and (y_{n_{k}}) are basic, (x_{n_{k}}) ∼ (u_{k}), (y_{n_{k}}) ∼ (v_{k}), x_{n_{k}} − u_{k} → 0, y_{n_{k}} − v_{k} → 0, and for each k ∈ ℕ, the vector u_{k} (v_{k}, respectively) is a restriction of (x_{n_{k}}) (of (y_{n_{k}}), respectively).

For each k ∈ ℕ, define h_{k} ∈ d_{w,p} by putting its i-th coordinate to be equal to h_{k}(i) = sign (v_{k}(i)) · (|u_{k}(i)| ∧ |v_{k}(i)|). Then (h_{k}) is a block sequence of (e_{n}) such that |h_{k}| ⩽ |u_{k}|. A straightforward verification shows that |h_{k} − v_{k}| ⩽ |u_{k} − x_{n_{k}}|. It follows that h_{k} − v_{k} → 0. By Theorem 1.2, passing to a subsequence, we may assume that (h_{k}) is basic and (h_{k}) ∼ (v_{k}). By Lemma 1.9, (u_{k}) ⪰ (h_{k}). Hence (x_{n_{k}}) ⪰ (y_{n_{k}}). □

The next lemma is a version of Theorem 4.8 for the case (x_{n}) is an arbitrary bounded sequence.

20 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

Lemma 6.3. If the bounded sequence (x_{n}) in d_{w,p} is not almost lengthwise bounded, then there is a subsequence (x_{n_{k}}) such that (x_{n_{2k}} −x_{n_{2k−1}}) is equivalent to the unit vector basis (f_{n}) of ℓ_{p}.

Proof. We can assume without loss of generality that no subsequence of (x_{n}) is equivalent to the unit vector basis of ℓ_{1}. Indeed, if (x_{n_{k}}) is equivalent to the unit vector basis of ℓ_{1} then p = 1. It follows that (x_{n_{k}}) is equivalent to (f_{n}) and hence (x_{n_{2k}} − x_{n_{2k−1}}) is equivalent to (f_{n}), as well.

Without loss of generality, sup_{n}∥x_{n}∥ = 1. Since (x_{n}) is not almost lengthwise bounded, there exists c > 0 such that

(2)  ∀N ∈ ℕ ∃n ∈ ℕ ∥x_{n}^{∗}|_{[N,∞)}∥ > c.

Let \frac{c}{4} > ε_{k} ↓ 0. We will inductively construct increasing sequences (n_{k}) and (N_{k}) in ℕ and a sequence (y_{k}) in d_{w,p} such that the following conditions are satisfied for each k:

(i) ∥x_{n_{k}}|_{[N_{k+1},∞)}∥ < ε_{k}; (ii) y_{k} is supported on [N_{k}, N_{k+1}); (iii) y_{k} is a restriction of x_{n_{k}}; (iv) ∥y_{k}∥ > \frac{c}{2};

(v) ∥y_{k}∥_{∞} ⩽ s_{N_{k}}^{−1/p} where s_{N} is as in Lemma 3.4. For k = 1, we put N_{1} = 1, and define n_{1} to be the first number n such that ∥x_{n}∥ > c; such an n exists by (2). Pick N_{2} ∈ ℕ such that ∥x_{n_{1}}|_{[N_{2},∞)}∥ < ε_{1}. Put y_{1} = x_{n_{1}}|_{[N_{1},N_{2})}. It follows that 1 ⩾ ∥y_{1}∥ > c − ε_{1} > \frac{c}{2}, and the coordinates of y_{1} are all at most 1 (= s_{1}^{−1/p}), hence all the conditions (i)–(v) are satisfied for k = 1.

Suppose that appropriate sequences (n_{i})_{i=1}^{k}, (N_{i})_{i=1}^{k+1}, and (y_{i})_{i=1}^{k} have been constructed. Use (2) to find n_{k+1} such that ∥x_{n_{k+1}}^{∗}|_{[2N_{k+1},∞)}∥ > c. Let z be the vector obtained from x_{n_{k+1}} by replacing its N_{k+1} largest (in absolute value) entries with zeros. Then ∥z|_{[N_{k+1},∞)}∥ ⩾ ∥z^{∗}|_{[N_{k+1},∞)}∥ = ∥x_{n_{k+1}}^{∗}|_{[2N_{k+1},∞)}∥ > c. By Lemma 3.4, ∥z∥_{∞} ⩽ s_{N_{k+1}}^{−1/p}. Choose N_{k+2} such that ∥x_{n_{k+1}}|_{[N_{k+2},∞)}∥ < ε_{k+1}. It follows that ∥z|_{[N_{k+2},∞)}∥ < ε_{k+1}. Put y_{k+1} = z|_{[N_{k+1},N_{k+2})}. Then ∥y_{k+1}∥ ⩾ c − ε_{k+1} > \frac{c}{2}, and the inductive construction is complete.

The sequence (y_{k}) constructed above is a seminormalized block sequence of (e_{n}) such that the coordinates of (y_{k}) converge to zero by condition (v). Using Remark 1.4 and passing to a subsequence, we may assume that (y_{k}) is equivalent to the unit vector basis (f_{n}) of ℓ_{p}.

Since (x_{n}) contains no subsequences equivalent to the unit vector basis of ℓ_{1}, using the Rosenthal’s ℓ_{1}-theorem and passing to a further subsequence, we may assume that

OPERATOR IDEALS IN LORENTZ SPACES 21

(x_{n_{k}}) is weakly Cauchy. For all m > k ∈ ℕ, we have: ∥x_{n_{k}}|_{[N_{m},∞)}∥ ⩽ ∥x_{n_{k}}|_{[N_{k+1},∞)}∥ ⩽ ε_{k}. Therefore ∥x_{n_{m}} − x_{n_{k}}∥ ⩾ ∥(x_{n_{m}} − x_{n_{k}})|_{[N_{m},∞)}∥ ⩾ ∥x_{n_{m}}|_{[N_{m},∞)}∥ − ε_{k} ⩾ ∥y_{m}∥ − ε_{k} ⩾ \frac{c}{2} − ε_{k} > \frac{c}{4}. It follows that the sequence (u_{k}) defined by u_{k} = x_{n_{2k}} − x_{n_{2k−1}} is seminormalized and weakly null. Passing to a subsequence of (x_{n_{k}}), we may assume that (u_{k}) is equivalent to a block sequence of (e_{n}). By Proposition 1.8, (f_{n}) ⪰ (u_{k}).

Let v_{k} = x_{n_{2k}} − (x_{n_{2k−1}}|_{[1,N_{2k})} ). Then ∥u_{k} − v_{k}∥ = ∥x_{n_{2k−1}}|_{[N_{2k},∞)}∥ < ε_{2k−1} → 0. By Theorem 1.2, passing to a subsequence of (x_{n_{k}}), we may assume that (v_{k}) is basic and (v_{k}) ∼ (u_{k}). Also, (v_{k}) is weakly null. Note that |y_{2k}| ⩽ |v_{k}| for all k ∈ ℕ, since supp y_{2k} ⊆ [N_{2k}, N_{2k+1}), so that y_{2k} is a restriction of v_{k}. By Lemma 6.2, passing to a subsequence, we may assume that (v_{k}) ⪰ (y_{2k}). Therefore we obtain the following diagram:

(f_{k}) ⪰ (u_{k}) ∼ (v_{k}) ⪰ (y_{2k}) ∼ (f_{2k}) ∼ (f_{n}).

It follows that (u_{k}) is equivalent to (f_{k}). □

Corollary 6.4. If T ∈ SS(ℓ_{p}, d_{w,p}) then the sequence (T f_{n}) is almost lengthwise bounded.

Proof. Suppose that (T f_{n}) is not almost lengthwise bounded. By Lemma 6.3, there is a subsequence (f_{n_{k}}) such that (T f_{n_{2k}} − T f_{n_{2k−1}}) is equivalent to (f_{n}). It follows that T |_{[f_{n_{2k}} −f_{n_{2k−1}}]} is an isomorphism. □

Remark 6.5. If we view T as an infinite matrix, the vectors (T f_{n}) represent its columns.

Theorem 6.6. If T ∈ L(ℓ_{1}, d_{w,1}) is such that the sequence (T f_{n}) is almost lengthwise bounded, then for any ε > 0 there exists S ∈ L(ℓ_{1}) such that ∥T − jS∥ < ε, where j ∈ L(ℓ_{1}, d_{w,1}) is the formal identity operator.

Proof. Let ε > 0 be fixed. Find N ∈ ℕ such that ∥(T f_{n})^{∗}|_{[N,∞)}∥ < ε for all n. Let z_{n} ∈ d_{w,1} be the vector obtained from T f_{n} by keeping its largest N coordinates and replacing the rest of the coordinates with zeros.

Define S: ℓ_{1} → d_{w,1} by Sf_{n} = z_{n}. Note that ∥T − S∥ = sup_{n}∥(T − S)f_{n}∥ = sup_{n}∥T f_{n} − z_{n}∥ ⩽ ε; in particular, S is bounded. Let F = span{e_{1}, …, e_{N}}. Since dim F < ∞, there exists C > 0 such that

\frac{1}{C} ∥x∥_{ℓ_{1}} ⩽ ∥x∥_{d_{w,1}} ⩽ C∥x∥_{ℓ_{1}}

for all x ∈ F. Observe that for each n ∈ ℕ, the non-increasing rearrangement (Sf_{n})^{∗} is in F. Therefore, for all n ∈ ℕ, we have

∥Sf_{n}∥_{ℓ_{1}} = ∥(Sf_{n})^{∗}∥_{ℓ_{1}} ⩽ C∥(Sf_{n})^{∗}∥_{d_{w,1}} = C∥Sf_{n}∥_{d_{w,1}} ⩽ C∥S∥.

It follows that the operator S̃: ℓ_{1} → ℓ_{1} defined by S̃f_{n} = Sf_{n} belongs to L(ℓ_{1}). Obviously, S = jS̃. So, ∥T − jS̃∥ < ε. □

The next corollary follows immediately from Theorem 6.6 and Corollary 6.4. This corollary can be considered as a support for Conjecture 6.1.

Corollary 6.7. SS(ℓ_{1}, d_{w,1}) is contained in the closure of {jS : S ∈ L(ℓ_{1}, d_{w,1})}.

Question. Does Corollary 6.7 remain valid for p > 1?

The following fact is standard, we include its proof for convenience of the reader.

Proposition 6.8. If X is a Banach space then SS(X, ℓ_{1}) = K(X, ℓ_{1}).

Proof. Let T ∉ K(X, ℓ_{1}). Pick a bounded sequence (x_{n}) in X such that (T x_{n}) has no convergent subsequences. By Schur’s theorem, (T x_{n}) and, therefore, (x_{n}) have no weakly Cauchy subsequences. Applying Rosenthal’s ℓ_{1}-theorem twice, we find a subsequence (x_{n_{k}}) such that (x_{n_{k}}) and (T x_{n_{k}}) are both equivalent to the unit vector basis of ℓ_{1}. It follows that T is not strictly singular. □

Proposition 6.9. For all p ∈ [1, ∞), SS(d_{w,p}, ℓ_{p}) = K(d_{w,p}, ℓ_{p}).

Proof. By Proposition 6.8, we only have to consider the case p > 1. Let T ∉ K(X, ℓ_{p}). Pick a bounded sequence (x_{n}) in X such that (T x_{n}) has no convergent subsequences. Since d_{w,p} contains no copies of ℓ_{1}, by Rosenthal’s ℓ_{1}-theorem we may assume that (x_{n}) is weakly Cauchy. Passing to a further subsequence, we may assume that the sequence (T y_{n}), where y_{n} = x_{2n} − x_{2n−1}, is seminormalized. It follows that (y_{n}) is also seminormalized. Also, (y_{n}) and, therefore, (T y_{n}) are weakly null. Passing to a subsequence of (x_{n}), we may assume that (y_{n}) and (T y_{n}) are both basic, equivalent to block sequences of (e_{n}) and (f_{n}), respectively. By [3, Proposition 5] and [17, Proposition 2.a.1], (f_{n}) ⪰ (y_{n}) and (f_{n}) ∼ (T y_{n}). So, we obtain the diagram

(f_{n}) ⪰ (y_{n}) ⪰ (T y_{n}) ∼ (f_{n}).

Hence [y_{n}] is isomorphic to [T y_{n}], so that T is not strictly singular. □

The following lemma is standard.

Lemma 6.10. Let X be a Banach space. Every seminormalized basic sequence in X is dominated by the unit vector basis of ℓ_{1}.

OPERATOR IDEALS IN LORENTZ SPACES 23

Lemma 6.11. Let (x_{n}) and (y_{n}) be two sequences in a Banach space X such that (x_{n}) is equivalent to the unit vector basis of ℓ_{1} and (y_{n}) is convergent. Then the sequence (z_{n}) defined by z_{n} = x_{n} + y_{n} has a subsequence equivalent to the unit vector basis of ℓ_{1}.

Proof. Observe that (z_{n}) cannot have weakly Cauchy subsequences since (x_{n}) does not have such subsequences. Since (z_{n}) is bounded, the result follows from Rosenthal’s ℓ_{1}-theorem. □

Recall that an operator A between two Banach lattices X and Y is called positive if x ⩾ 0 entails T x ⩾ 0.

Conjecture 6.1 asserts, in particular, that if T ∈ SS(d_{w,p}) and T = AB for some A: d_{w,p} → ℓ_{p} and B: ℓ_{p} → d_{w,p} then T ∈ \overline{J^{j}}. In the next theorem, we prove this under the additional assumptions that p = 1 and both A and B are positive.

Theorem 6.12. Let T ∈ SS(d_{w,1}) be such that T = AB, where A ∈ L(ℓ_{1}, d_{w,1}), B ∈ L(d_{w,1}, ℓ_{1}), and both A and B are positive. Then T ∈ \overline{J^{j}}.

Proof. Define a sequence (A_{N}) of operators in L(ℓ_{1}, d_{w,1}) by the following procedure. For each n ∈ ℕ, let A_{N}f_{n} be obtained from Af_{n} by keeping the largest N coordinates and replacing the rest of the coordinates with zeros. Since Af_{n} ⩾ 0 for all n ∈ ℕ, this defines a positive operator ℓ_{1} → d_{w,1}. Also, ∥A_{N}f_{n}∥ ⩽ ∥Af_{n}∥ ⩽ ∥A∥ for all n ∈ ℕ, hence ∥A_{N}∥ ⩽ ∥A∥.

Define A′_{N} = A − A_{N}. It is clear that 0 ⩽ A′_{N}f_{n} ⩽ Af_{n} for all n ∈ ℕ, hence A′_{N} ⩾ 0 and ∥A′_{N}∥ ⩽ ∥A∥. We claim that A′_{N} → 0 in the strong operator topology (SOT). Indeed, since A′_{N}f_{n} is obtained from Af_{n} by removing the largest N coordinates, the elements of the matrix of A′_{N} are all smaller than \frac{∥A∥}{s_{N}} by Lemma 3.4. In particular, if 0 ⩽ x ∈ ℓ_{1}, then A′_{N}x ↓ 0; it follows that ∥A′_{N}x∥ → 0 because d_{w,1} has order continuous norm (see Remark 1.5). If x ∈ ℓ_{1} is arbitrary then ∥A′_{N}x∥ ⩽ ∥A′_{N}|x| ∥ → 0.

We will show that ∥A′_{N}B∥ → 0 as N → ∞, so that ∥AB − A_{N}B∥ → 0 as N → ∞. Since (A_{N}f_{n})_{n=1}^{∞} is almost lengthwise bounded (in fact, the vectors in the sequence (A_{N}f_{n})_{n=1}^{∞} all have at most N nonzero entries), the theorem will follow from Theorem 6.6.

Assume, by way of contradiction, that there are c > 0 and a sequence (N_{k}) in ℕ such that ∥A′_{N_{k}}B∥ > c. Then there exists a normalized positive sequence (x_{k}) in d_{w,p} such that ∥A′_{N_{k}}Bx_{k}∥ > c. By Rosenthal’s ℓ_{1}-theorem, we may assume that (x_{k}) is either weakly Cauchy or equivalent to (f_{n}).

Assume that (x_{k}) is weakly Cauchy. Then (Bx_{k}) is weakly Cauchy. Since (Bx_{k}) is a sequence in ℓ_{1}, it must converge to some z ∈ ℓ_{1} by the Schur property. Then

24 A. KAMINSKA, A. I. POPOV, E. SPINU, A. TCACIUC, AND V. G. TROITSKY

∥A′_{N_{k}}Bx_{k} − A′_{N_{k}}z∥ ⩽ ∥A′_{N_{k}}∥ · ∥Bx_{k} − z∥ ⩽ ∥A∥ · ∥Bx_{k} − z∥ → 0. Since A′_{N_{k}} → 0 in SOT, it follows that A′_{N_{k}}Bx_{k} → 0, contrary to the assumption. Therefore (x_{k}) must be equivalent to (f_{n}).

Since the entries of the matrix of A′_{N} are all less than \frac{∥A∥}{s_{N}}, the coordinates of the vector A′_{N_{k}}Bx_{k} are all less than \frac{∥A∥}{s_{N_{k}}}∥B∥ → 0. Hence, passing to a subsequence, we may assume that (A′_{N_{k}}Bx_{k}) is equivalent to a block sequence (u_{k}) of (e_{n}) such that each u_{k} is a restriction of A′_{N_{k}}Bx_{k}. In particular, the coordinates of (u_{k}) converge to zero. Passing to a further subsequence, we may assume by Remark 1.4 that (A′_{N_{k}}Bx_{k}) ∼ (f_{n}).

The sequence (T x_{k}) cannot have subsequences equivalent to (f_{n}) since T is strictly singular. Therefore, by Rosenthal’s ℓ_{1}-theorem, we may assume that (T x_{k}) is weakly Cauchy. Since d_{w,1} is weakly sequentially complete (Remark 1.5), the sequence (T x_{k}) weakly converges to a vector y ∈ d_{w,1}. Since the positive cone in a Banach lattice is weakly closed, y ⩾ 0.

Note that T x_{k} ⩾ A′_{N_{k}}Bx_{k} ⩾ u_{k} ⩾ 0 for every k. Since (u_{k}) is a seminormalized block sequence of (e_{n}), it follows that (T x_{k}) is not norm convergent. Write T x_{k} = y+h_{k}; then (h_{k}) converges to zero weakly but not in norm. Therefore, passing to a subsequence, we may assume that (h_{k}) is seminormalized and basic (but not, necessarily, positive).

Let r_{k} = A′_{N_{k}}Bx_{k} − (A′_{N_{k}}Bx_{k} ∧ y) ⩾ 0, k ∈ ℕ. Observe that A′_{N_{k}}Bx_{k} ∧ y ∈ [0, y] for all k. Since d_{w,1} has order continuous norm and the order in d_{w,1} is defined by a 1-unconditional basis, order intervals in d_{w,1} are compact (see, e.g., [24, Theorem 6.1]). Therefore, passing to a subsequence of (x_{n_{k}}), we may assume that (A′_{N_{k}}Bx_{k} ∧ y) is convergent, hence, passing to a further subsequence, (r_{k}) is equivalent to (f_{n}) by Lemma 6.11 and Theorem 1.2.

It follows from y+h_{k} ⩾ A′_{N_{k}}Bx_{k} ⩾ 0 that |h_{k}| ⩾ r_{k} for all k. Passing to a subsequence, we may assume by Lemma 6.2 that (h_{k}) ⪰ (r_{k}) ∼ (f_{n}). By Lemma 6.10, in fact (h_{k}) ∼ (f_{n}), and, hence, by Lemma 6.11, (ABx_{k}) ∼ (f_{n}). Since also (x_{k}) ∼ (f_{n}), this contradicts to T = AB ∈ SS(d_{w,1}). □

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(A. Kaminska) Department of Mathematical Sciences The University of Memphis, Memphis, TN 38152-3240. USA

E-mail address: kaminska@memphis.edu

(A. I. Popov, E. Spinu, and V. G. Troitsky) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1. Canada

E-mail address: apopov@math.ualberta.ca, spinu@ualberta.ca
E-mail address: troitsky@ualberta.ca

(A. Tcaciuc) Mathematics and Statistics Department, Grant MacEwan University, Edmonton, AB, T5J P2P, Canada

E-mail address: atcaciuc@ualberta.ca