On Narrow Operators from L p into Operator Ideals

. It is well known that every l 2 -strictly singular operator from L p , 1 < p < ∞ to any Banach space X with an unconditional basis is narrow. In this article, we extend this result to the setting of Banach spaces without an unconditional basis. We show that if 1 ≤ p, r < ∞ , then every (cid:2) 2 -strictly singular operator T from L p into the Schatten–von Neumann r -class C r is narrow. This is a noncommutative complement to results in Mykhaylyuk et al. (in Israel J Math 203:81–108, 2014).


Introduction
Let F (0, 1) be a separable symmetric function space of (classes of equivalent) Lebesgue measurable functions on (0, 1). Suppose that A is a measurable subset of (0, 1). By a sign on A we mean an element x ∈ L ∞ (0, 1) with supp x = A which takes values in {−1, 0, 1}. We say that x ∈ F (0, 1) is a mean zero sign on A if x is a sign on A and 1 0 xdμ = 0. Let X be a Banach space and L(F (0, 1), X) be the space of all bounded linear operators from F (0, 1) into X. We say that a linear operator T ∈ L(F (0, 1), X) is narrow if for every ε > 0 and every measurable set A ⊂ (0, 1), there exists a mean zero sign x on A, such that T x X < ε. The notion of narrow operators was formally introduced by Plichko and Popov [18] for operators acting on symmetric function spaces (in fact, the study of operators of this type goes back to Bourgain, Ghoussoub and Rosenthal, see [18,19,22] and references therein).
In 1990, Plichko and Popov [18] asked whether every 2 -strictly singular operator T : L p → X is necessarily narrow (see also [19] and [20,Problem F. Sukochev's research is supported by the Australian Research Council (FL170100052). 0123456789().: V,-vol 1.6]). Here, an operator T is said to be 2 -strictly singular if it is not an isomorphism when restricted to any isomorphic copy of 2 in L p .
When 1 ≤ p < 2 and p < r, it is proved in [13,Theorem 5] that every operator T : L p → r is narrow. It is also known that when 1 ≤ r < 2 and 1 ≤ p < ∞, every operator T : L p → r is narrow [18,Proposition 2]. The remaining cases were settled in [16]. Precisely, [16,Theorem A] shows that every T : L p → r is narrow when 1 ≤ p < ∞ and 1 ≤ r = 2 < ∞ or 1 ≤ p < 2 and r = 2. Moreover, when p > 2 and r = 2, there exists a non-narrow operator T : L p → 2 [16, Example 1.1]. Finally, [16,Theorem B] asserts that every 2 -strictly singular operator T : L p → X is necessarily narrow provided that X has an unconditional basis.
The main objective of this paper is to establish a noncommutative generalization of [16,Theorems A and B] for the situation where we deal with 2 -strictly singular operators T : L p → X if the space X does not possess an unconditional basis. For all unexplained notions and notations we refer the reader to [1,11,12].
Let E be a separable symmetric sequence space [2,14], that is, a Banach space of sequences such that the standard unit vectors e n 's, n = 1, 2, 3, . . ., (defined by e n (j) = δ n,j ) form a normalized, 1-symmetric basis of E. Let C E be the ideal in B( 2 ) corresponding to E (see [2,14]), i.e., the Banach space of all compact operators x on 2 for which s(x) ∈ E, normed by Here, s(x) = {s n (x)} ∞ n=1 is the sequence of s-numbers of x, i.e., the eigenvalues of |x| = (x * x) 1 2 arranged in a non-increasing ordering, counting multiplicity. In the case when E = p , the ideal C p is denoted simply by C p .
Let {e ij } i,j≥1 be the matrix unit of C E . Let T E be the upper triangular part of C E , i.e., x ∈ T E if and only if x : Note that all preceding results in this area are established only for spaces X either with unconditional bases or for Banach lattices [19], whereas the spaces C E do not even possess the local unconditional structure [7]. In particular, it follows from [10] that the ideal C E has an unconditional basis if and only if it coincides with the Hilbert-Schmidt ideal.
Our approach to the study of narrow operators T : L p → X when X = T E or X = C E is based on a fundamental fact that the spaces T E admit the finite-dimensional unconditional (Schauder) decomposition (UFDD) given by elements {span{e ij } 1≤i≤j } j≥1 (see for details [2,Proposition 4.9]). Recall also that T E is isomorphic to C E when E has non-trivial Boyd indices [2,Theorem 4.7] (see also [3,Proposition 2] and [15]). This setting has been already explored in [8], where assuming the socalled 2-co-lacunary property (see [25]), the authors of the present article obtained a noncommutative version of [18,Proposition 2]. Precisely, since C r , r ≤ 2, is 2-co-lacunary, it follows from [8,Theorem 4.3 and Remark 4.4] that every 2 -strictly singular operator T : L p → C r is narrow when 1 ≤ p < ∞ and 1 ≤ r ≤ 2. However, the case of r > 2 remains unresolved and this case cannot be treated by methods from [8].
The following theorem is a noncommutative analogue of [16,Theorem A], which answers Plichko and Popov's question for Banach spaces C r , r > 2 and resolves the unanswered cases in [8]. Our proof is motivated by an extended version of reproducibility hatched within noncommutative analysis in [4]. Based on a careful analysis of approach used in [16], we obtain a slight extension of [16, Proposition 3.1] concerning on the reproducibility (with respect to non-narrow operators) of the Haar basis in L p (see Proposition 2.2 below). Our approach is applicable to a much wider class of operator ideals than the class of (Schatten-von Neumann ideals) C r , 2 < r < ∞. The main result of the present paper, Theorem 1.1 below, is stated for ideals C E , for which the symmetric space E is satisfying an upper r-estimate (see, e.g. [12]), which extends and complements results in [6,8,13,16,18]. Theorem 1.1. Let 2 < r < ∞ and let F (0, 1) be a separable symmetric function space having the Khintchine property. If E is a separable symmetric sequence space satisfying an upper r-estimate, then every 2 -strictly singular operator T : Recall that T p := T Cp is isomorphic to C p when p > 1 [2, Theorem 4.7] (see also [3,Proposition 2] and [15]) and L p has the Khintchine property when p ∈ [1, ∞) (see, e.g. [12] or [21] The authors thank Professor Popov for discussion concerning results presented in [16] and [19]. We also thank the anonymous reviewer for his/her careful reading and helpful comments.

Proof of Theorem 1.1
Let S(0, 1) be the space of all Lebesgue measurable functions on (0, 1) equipped with Lebesgue measure m i.e. functions which coincide almost everywhere are considered identical.
We recall some basic terminology concerning Schauder decomposition [11, Chapter 1, Section g]. Let X be a Banach space. A sequence {X n } ∞ n=1 of closed subspaces of X is called a Schauder decomposition of X if every x ∈ X has a unique representation of the form , then the sequence {X n } ∞ n=1 is said to form an unconditional Schauder decomposition of X. Moreover, the operator M x := ∞ i=1 n x n is bounded and sup M < ∞ (see e.g. [4] or [1, Recall that a Schauder basis When K = 1, the basis is said to be precisely reproducible. It is well-known that the Haar system in an arbitrary separable symmetric function space on (0, 1) is precisely reproducible [12, Theorem 2.c.8] (see also [17]).
In noncommutative analysis, there is a more useful notion of reproducibility. A Schauder basis {x n } ∞ n=1 of a Banach space X is said to be precisely finite-dimensional decomposition (FDD)-reproducible, if for every isometric embedding of X into a space Y with a finite-dimensional decomposition {Y n } n≥1 and every ε > 0, there exists an increasing sequence {q n } ∞ n=1 of positive integers and a basic sequence of elements z n = qn≤k≤qn+1−1 λ k y k (y k ∈ Y k , λ k ∈ R), which is (1 + ε)-equivalent to {x n }. It was observed in [4, Theorem 5.2] that [12, Theorem 2.c.8] can be improved, that is, the Haar system in an arbitrary separable symmetric function space is precisely FDDreproducible.
In [16, Proposition 3.1], Mykhaylyuk et al. considered the reproducibility of the (L p -normalized) Haar system {h n } n≥1 with respect to non-narrow operators. Note that [16, Proposition 3.1] can be easily generalized from the case of L p -space, 1 ≤ p < ∞, to the case where F (0, 1) is an arbitrary separable symmetric space on (0, 1). The following proposition is a slight generalization of [16, Proposition 3.1] with respect to the FDD-reproducibility.
is an arbitrary increasing sequence of positive integers.
Then, for each ε > 0, there exist an operator S ∈ L(F (0, 1), X), an increasing sequence {p n } n≥1 of positive integers, a normalized basis (u n ) such that u n = pn≤k≤pn+1−1 λ k x k (x k ∈ X k , λ k ∈ R), and real numbers (a n ) such that (1) Sh n = a n u n for each n ∈ N with a 1 = 0; (2) Sx X ≥ δ for each mean zero sign x ∈ F (0, 1) on (0, 1); Proof. The proof is almost a verbatim repetition of that in [16, Proposition 3.1]. The only difference is that we need to consider a subsequence of the basis projections in the Banach space X while the proof in [16, Proposition 3.1] simply proceeded with the set of all basis projections.
More precisely, let (P n ) ∞ n=1 be the basis projections in X with respect to the basis {e k } k≥1 and P 0 = 0. Recall that X n = span{e k } qn+1−1 k=qn for every n ≥ 1. Let In particular, Q n is the projection onto clm{X k } n k=1 . Now, the claim of Proposition 2.2 can be obtained from the proof of [16, Proposition 3.1] (see also [19,Proposition 9.10]) by simply replacing P n with Q n throughout. Since our other arguments repeat [16, Proposition 3.1] we omit further details.
The following proposition is well known to experts [16,19]. We include a proof below for completeness.

Proposition 2.3. Let T and S be as in Proposition 2.2. If
T is 2 -strictly singular, then S is also 2 -strictly singular.
Proof. Assume by contradiction that S is not 2 -strictly singular. That is, there exists a sequence (x n ) ∞ n=1 in F (0, 1) which is equivalent to the natural basis of 2 , and a constant c > 0 such that and a constant C > 0 such that Let k ∈ N be so large that 1 k (1 + 1 k ) 2 ≤ 1 2 c −1 C −1 . By (4) of Proposition 2.2, there is a finite codimensional subspace X k of F (0, 1) such that for every x ∈ X k with x = 1.
Since every subspace with finite codimension in a Banach space is complemented [23,Lemma 4.21], it follows that there exists a bounded projection P from F (0, 1) onto X k . Since (x n ) is weakly null, it follows that (1 − P )x n is weakly null. Since X k is finite-codimensional, it follows that (1 − P )x n → 0 in · F (0,1) . Therefore, passing to a subsequence if necessary, we may assume that the sequence Passing to a subsequence if necessary, we may assume that (2.5) For any ∞ n=1 λ n y n ∈ F (0, 1) such that ∞ n=1 λ n y n F (0,1) = 1, we have (2.6) and, therefore, Hence, T is an isomorphism on span{V y n } · F (0,1) . However, by (2.4) and the fact that V is an isometry, span{V y n } · F (0,1) is isomorphic to 2 , which contracts the assumption that T is 2 -strictly singular.
Throughout this section by (h n ) ∞ n=1 is denoted the L ∞ -normalized Haar system. Let We say that a symmetric function space E(0, 1) has the Khintchine property if the Rademacher system in E(0, 1) is equivalent to the natural basis of 2 . In particular, L p (0, 1), p ≥ 1, has the Khintchine property (see e.g. [12,21]).

Proof of Theorem 1.1
Before proceeding to the proof of Theorem 1.1, we need one more auxiliary result. F (0, 1)

Proof. Let
(2.7) By Proposition 2.2, (Sr n ) is a sequence in T E , which are disjointly supported from the right. Indeed, every non-zero element in X n has its right support equal to e nn . By Proposition 2.2, r(Sh n ) ≤ pn+1−1 k=pn e ii . for some strictly increasing sequence {p n } n≥1 Therefore, Sr n 's are disjointly supported from the right.
Let n 1 = 1 and let y 1 = Sr 1 . By Proposition 2.2, there exists a positive integer N 1 large enough such that P N1 Sr 1 P N1 = Sr 1 .
We claim that P N1 Sr n → 0 in T E as n → ∞. Otherwise, if lim inf n→∞ P N1 Sr n TE > δ > 0, then, passing to a subsequence of (r n ) if necessary, we obtain the existence of an integer m with m ≤ N 1 such that Indeed, if lim inf n→∞ (P m − P m−1 )Sr n TE < δ N1 for all m ≤ N 1 , then by the triangle inequality, we have lim inf n→∞ P m Sr n TE < δ, which is a contradiction. Passing to a subsequence, we may assume that (P m − P m−1 )Sr n TE ≥ δ N1 for all n. Denoting e mm (Sr n )(e mm (Sr n )) * = e mm (Sr n )(Sr n ) * e mm = ae mm for some a ≥ 0, we have Hence, e mm (Sr n )(Sr n ) * e mm = ae mm ≥ ( δ N1 ) 2 e mm . Moreover, since Sr n 's are disjointly supported from the right, it follows that for any (α n ) ∈ 2 , we have Hence, for any (α n ) ∈ 2 , we have where c(p) is a positive constant depending on p. This contradicts the fact that S is 2 -strictly singular. Hence, in T E as n → ∞. There exists a positive integer n 2 such that By Proposition 2.2, there exists a positive integer N 2 sufficiently large such that We have Observe that (P N2 − P N1 )Sr n2 and y 1 are disjointly supported from the left and the right. We define y 2 := (P N2 − P N1 )Sr n2 .
Continuing the procedure, we construct a sequence of integers (n m ) and a sequence (y m ) of elements in C E which are disjointly supported from the left and the right such that Sr nm − y m TE ≤ δ 2 m . This completes the proof. Now, we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. Let r > 2. Let {e ij } j≥i be the natural Schauder basis of T E (e.g. in the induced rectangular ordering [15]). Suppose that T ∈ L (F (0, 1), T E ) is not narrow. Without loss of generality, we may assume that there exists a positive number δ such that T x TE ≥ 2δ for any mean zero sign x on (0, 1). Applying Proposition 2.2 by taking ε < T , X = T E and X n = span{e i,n } 1≤i≤n , we can choose an operator S ∈ L (F (0, 1), T E ) such that S ≤ 2 T , (2.9) and for which conditions (1)-(4) in Proposition 2.2 hold. By Proposition 2.3, S is 2 -strictly singular. By Proposition 2.4, we may assume that there exists a subsequence (n(m)) m≥1 of N such that n(m)'s are odd numbers and (Sr n(m) ) is equivalent to a basic sequence of elements in T E which are disjointly supported from the left and the right. We consider Let C > 0 and N ∈ N. We denote We define Since Sr n(m) is equivalent to a sequence of elements in T r which are disjointly supported from the left and the right and E satisfies an upper r-estimate with r > 2, we have Since r > 0, it follows that Our goal is to select a subset J of {2 n(m) + k} m≥1,0≤i≤2 n(m) −1 such that g = C √ N n∈J h n is close enough to a sign. Set Observe that if τ (ω) = 2 n(m) + k, then ω ∈ I k n(m) (see, e.g. the argument in [16, p.94] and [19]). Further, if there exists ω ∈ I k n(m) with τ (ω) ≥ 2 n(m) + k, then we have τ (ξ) ≥ 2 n(m) + k (2.10) for every ξ ∈ I k n(m) . Indeed, since ω ∈ I k n(m) , for every z < 2 n(m) + k and every ξ ∈ I k n(m) , we have h z (ω) = h z (ξ). Thus, Hence, τ (ξ) ≥ 2 n(m) + k. Now, we define a set Observe that for every ω ∈ [0, 1], we have for any ω / ∈ I k n(m) , it follows that for any ω ∈ [0, 1], we have where c E > 0 depends on E only. Assume that N is an odd number. By the definition of τ (ω) and g(ω), for every ω ∈ A, we have (2.14) and for every ω ∈ [0, 1] \ A, Indeed, since N is odd (that is, N −1 is even), it follows that N −1 j=1 r nj = ±1 everywhere, and therefore, for ω ∈ I k n(N ) , we have Now, we define a function g by setting g(ω) = sgn(g(ω)).
By the Central Limit Theorem, for a sufficiently large odd number N , we have Thus, .
Since r > 2, it follows that for every δ > 0, there exists a sufficiently large positive number C and a sufficiently large odd number N such that It remains to observe thatg is a mean zero sign on [0, 1]. Indeed, since N is an odd number, it follows from (2.14) and (2.15) that the support ofg is equal to [0, 1]. Observe that for every ω ∈ [0, 1] and every 2 n(m) +k ≤ 2 n(N )+1 , k ≤ 2 n(m) , we have Thus,g is a mean zero sign on [0, 1], which contradicts (2) of Proposition 2.2.

Remarks
It is shown in [6, Theorem 1.1] that if T is a regular operator from L p into an order continuous Banach lattice F , then "T is 2 -strictly singularity" ⇒ "T is narrow". Note that [16,Theorem B] is stated for 2 -strictly singular operator T : L p → X when 1 < p < ∞ and X has an unconditional basis. However, a careful analysis of its proof shows that it still holds for any p ∈ [1, ∞) (see also [19, p.110] The class of Banach spaces having unconditional finite-dimensional decompositions is very wide. For example, the operator ideal C E has an UFDD when E has non-trivial Boyd indices [2,Corollary 4.6]. Let E(0, 1) be a symmetric function space on the unit interval (0, 1) and let E(R) be the noncommutative operator space [14] affiliated with the hyperfinite II 1 -factor R.
Then, E(R) has an UFDD [4,24]. This opens an avenue for further extensions of Theorem 1.1. E(M, τ) be a noncommutative symmetric space and Y be an F -space. We call T : E(M, τ) → Y a narrow operator if for each projection p ∈ M and ε > 0, there exists a self-adjoint element x ∈ E(M, τ) such that x 2 = p, τ (x) = 0 and T (x) Y < ε.

Definition 3.2. Let
Assume that E(0, 1) is a symmetric function and X is a Banach space. It is clear (see e.g. the proof of [8,Corollary 4.5]) that if all elements of L(E(0, 1), X) are narrow, then every element of L(E(M), X) is narrow for an arbitrary atomless finite von Neumann algebra M equipped with a faithful normal tracial state τ . The following result is a direct consequence of Corollary 1.2. Recall that a Banach space X is said to have infratype q > 1 [19, p.216] if there exists a constant C > 0 such that for each n ∈ N and x 1 , . . . , x n ∈ X, we have It is clear that if a Banach space has type q then it has infratype q. Note that if q < 2, then the notions of type and infratype coincide [26].
Assume that p ≥ 2. Note that there exists a non-narrow operator T : L p → 2 [16, Example 1.1]. Since 2 is a complemented subspace of C r , it follows that there exists a non-narrow operator from L p into C r .
When p = r = 1, it is shown in [9] that all operators from L 1 into C 1 are Dunford-Pettis, and therefore, narrow.
The case for 1 < p < 2 and 1 ≤ r ≤ p seems to be open, i.e., we do not know whether there exists a non-narrow operator from L p into C r when 1 < p < 2 and 1 ≤ r ≤ p.
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