Characterizations of strictly singular operators on Banach lattices

New characterizations of strictly singular operators between Banach lattices are given. It is proved that, for Banach lattices X and Y such that X has finite cotype and Y satisfies a lower 2-estimate, an operator T : X → Y is strictly singular if and only if it is disjointly strictly singular and 2-singular. Moreover, if T is regular then the same equivalence holds provided that Y is just order continuous. Furthermore, it is shown that these results fail if the conditions on the lattices are relaxed. Introduction Strictly singular operators were introduced by Kato [18] in connection with the perturbation theory of Fredholm operators. Recall that an operator T : X → Y between Banach spaces is strictly singular if it is not an isomorphism when restricted to any infinite-dimensional (closed) subspace of X. Strictly singular operators constitute a closed two-sided operator ideal that contains the ideal of compact operators. Moreover, an operator T : X → Y is strictly singular if and only if, for every infinite-dimensional subspace M of X, there exists an infinite-dimensional subspace N of M such that the restriction T |N is compact. In the context of Banach lattices, a weaker notion is the following: given a Banach lattice X, a Banach space Y , and an operator T : X → Y , we say that T is disjointly strictly singular if it is not an isomorphism when restricted to the closed linear span of any disjoint sequence in X. This notion is quite a useful tool in the study of strictly singular operators on Banach lattices, for example, in the context of domination problems for positive operators (cf. [10]), and for comparing structures of rearrangement invariant spaces (cf. [13, 14]). Several properties of disjointly strictly singular operators have been studied in [8, 9, 11]. In this paper, we are interested in giving characterizations of the strict singularity of operators acting between Banach lattices. Since strictly singular operators are disjointly strictly singular, we are mainly interested in converse statements. Our motivation stems from the following facts. First, it is well known that an endomorphism of Lp = Lp[0, 1], with 1 p < ∞, is strictly singular if and only if it is p-singular and 2-singular [22, 25]. In other words, an endomorphism T on Lp is strictly singular if and only if it is disjointly strictly singular and 2-singular. Recall that an operator between Banach spaces is called p-singular for some 1 p ∞ if it is not an isomorphism when restricted to any subspace isomorphic to p. For recent results on p-singular operators, we refer to [16]. Given an order continuous Banach lattice X, if an operator T : X → Y is disjointly strictly singular and p-singular for every 1 p 2, then T is strictly singular. This can be seen using the Kadeč–Pe lczyński disjointification method and Aldous’ theorem on subspaces of L1 (see [2]). Furthermore, in the special case of X (or Y ) being a Banach lattice with type 2, Received 8 July 2008; published online 24 March 2009. 2000 Mathematics Subject Classification 46B42, 47B07, 47B60. The first, second, and fourth authors were partially supported by Spanish grants MTM2005-00082, UCM910346, and PR34/07-15837. The third author was supported by NSF grant DMS-0555670. The fourth author was partially supported by MEC grant AP-2004-4841. CHARACTERIZATIONS OF STRICTLY SINGULAR OPERATORS 613 if T : X → Y is disjointly strictly singular and 2-singular, then T is strictly singular. A similar statement also holds for inclusion operators between rearrangement invariant spaces [12]. One of our main results in this direction is the following. Theorem A. Let X and Y be Banach lattices such that X has finite cotype and Y satisfies a lower 2-estimate. Then an operator T : X → Y is strictly singular if and only if it is both disjointly strictly singular and 2-singular. Then, we consider the class of regular operators, that is, those that are a difference of positive operators, proving that, for this class, the equivalence given above in Theorem A is also true under much weaker conditions on the lattices. Theorem B. Let X and Y be Banach lattices such that X has finite cotype and Y is order continuous. Then a regular operator T : X → Y is strictly singular if and only if it is both disjointly strictly singular and 2-singular. Both Theorems A and B are obtained by means of the following general result. If X is a Banach lattice with finite cotype, Y a Banach space, and T : X → Y is disjointly strictly singular and AM-compact, then T is strictly singular (see Theorem 2.4). Recall that, for a Banach lattice X, an operator T : X → Y is called AM-compact if the image of every order interval is a relatively compact set. The connection between AM-compact operators and 2-singular operators is studied in Section 2 (see Propositions 2.5 and 2.6). Let us remark that the motivation of this kind of result dates from Rosenthal [24], where it was proved that, for endomorphisms on L1spaces, being AM-compact and 2-singular are equivalent notions. As an application of these characterizations, a domination result for positive strictly singular operators is easily obtained, improving a result of [10]. Precisely, given two operators 0 R T : X → Y , with T strictly singular, then we have that R is also strictly singular provided that X has finite cotype and Y is order continuous (Corollary 2.8). In Section 3, we prove that the hypothesis in Theorem A on the range lattice Y cannot be weakened, in the sense that the result is no longer true for Banach lattices Y with a lower q-estimate for some q > 2. To this end, we consider the Banach lattice Lr( q) that consists of sequences x = (x1, x2, . . .) of elements in Lr such that ‖x‖Lr( q) = ∥∥∥∥∥ ( ∞ ∑ i=1 |xi| ∥∥∥∥∥ Lr < ∞. Theorem C. Consider the Banach lattices Lp and Lr( q), where 1 < r < p < 2 < q < ∞. For each p < s < 2, there exists an operator T : Lp → Lr( q) such that it is p-singular and 2-singular, but not s-singular. In particular, the operator T is disjointly strictly singular and 2-singular, but not strictly singular. The proof of this fact requires some preliminary results. First, we present some technical lemmas that make use of known estimates for independent and identically distributed (i.i.d.) s-stable random variables for 1 < s < 2, given in [15] (see Proposition 3.1). We consider the atomic lattice representation Hr of the space Lr, associated to the unconditional Haar basis (hi), in order to define a suitable operator R from Hr to Lr( q) that, restricted to Lp, satisfies 614 J. FLORES, F. L. HERNÁNDEZ, N. J. KALTON AND P. TRADACETE the required conditions. More precisely, let (wn) denote a block basis of the Haar basis (hi) of the form wn = qn ∑ i=qn−1+1 aihi, which is equivalent to a sequence of i.i.d. s-stable random variables in both Lp and in Lr (for 1 < r < p < s < 2). We can consider the operator R : Hr → Lr( q), defined by R ( (ci)i=1 ) = ⎛⎝ qn ∑ i=qn−1+1 ci hi ⎞⎠∞ n=1 . The operator Ts : Lp → Lr( q) is now defined as the composition RLJ , where J is the canonical inclusion Lp ↪→ Lr and L is the isomorphism between Hr and Lr mapping each sequence in Hr to the corresponding expansion as a series with respect to the Haar system. We also show that the characterization for regular operators given in Theorem B fails if the order continuity of the range lattice is missing. 1. Preliminaries Let us start with some notation and definitions. We refer the reader to the monographs [4, 20, 21, 28] for unexplained terminology from Banach lattices and positive operator theory. Throughout, we will write SS and DSS for strictly singular and disjointly strictly singular operators, respectively. Let us recall that a Banach space X has cotype q for some 2 q < ∞ if there exists a constant C < ∞ so that, for every finite set of vectors (xj)j=1 in X, we have ⎛⎝ n ∑