Finitely strictly singular operators between James spaces

Abstract. Let X and Y be separable Banach spaces and denote by SS(X, Y ) the subset of L(X, Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set SS(X, Y ). Our main results are summarized as follows. Firstly, we define a new rank r S on SS(X, Y ). We show that r S is a co-analytic rank and that dominates the rank ̺ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math., 169 (2009), 221-250]. Secondly, for every 1 ≤ p < +∞ we construct a Banach space Yp with an unconditional basis such that SS(ℓp, Yp) is a co-analytic non-Borel subset of L(ℓp, Yp) yet every strictly singular operator T : ℓp → Yp satisfies ̺(T ) ≤ 2. This answers a question of Argyros.

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“…for every countable ordinal ξ. Combining inequalities (8), (11), (12) and (13) we conclude that ̺(T ) ≤ r S (T ) as desired.…”

Section: Proof Of Theoremmentioning

confidence: 54%

“…In particular, the map T → ̺(T ) is an ordinal rank 1 on the set SS(X, Y ) of all strictly singular operators from X to Y . It was further studied in [3,8,11,23].…”

Section: Introductionmentioning

confidence: 99%

On Strictly Singular Operators Between Separable Banach Spaces

Beanland

Dodos

2010

Mathematika

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“…In fact the following refinement of this observation is true: every -dimensional subspace E of 0 contains a vector so that these coordinates have alternating signs [17]. More precisely, a finite or infinite sequence of real numbers in [−1 1] will be called a zigzag of order if it has a subsequence of the form (−1 1 −1 1 ) of length .…”

Section: Lemma 13mentioning

confidence: 99%

“…There are two proofs of Theorem 14, presented in [17], one based on combinatorial properties of polytopes (see [62]) and the other one based on the geometry of the set of all zigzags and algebraic topology (see [31,34]). Such a result can be used in order to provide a counterexample for the invariant subspace problem in the theory of finitely strictly singular operators.…”

Section: Theorem 14mentioning

confidence: 99%

An overview of some recent developments on the Invariant Subspace Problem

Chalendar

Partington

2012

Concrete Operators

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This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read's Banach space counter-example involving a finitely strictly singular operator. History of the problemThere is an important problem in operator theory called the invariant subspace problem. This problem is open for more than half a century, and there are many significant contributions with a huge variety of techniques, making this challenging problem so interesting; however the solution seems to be nowhere in sight. In this paper we first review the history of the problem, and then present an account of some recent developments with which we have been involved. Other recent approaches are discussed in [21]. The invariant subspace problem is the following: let be a complex Banach space of dimension at least 2 and let T ∈ ( ), i.e., T : → , linear and bounded. Is there any closed subspace ⊂ such that T ( ) ⊂ (i.e., is invariant for T ) and = {0} = (i.e., is not trivial)? In the sequel a "nontrivial invariant subspace" may be abbreviated as "ntis".• Here is a list of preliminary remarks:Assume that is of finite dimension ≥ 2, so that is isomorphic to C . Then T ∈ ( ) is a × matrix with complex entries, and thus T has eigenvectors. Each eigenvector generates a (nontrivial) invariant subspace of dimension 1. The Jordan form of T is of great help in describing the lattice of the invariant subspaces of T . Ifis not separable, then T ∈ ( ) has a nontrivial invariant subspace since for all ∈ , = 0, the closure of { (T ) : ∈ C[ ]} is invariant for T and nontrivial since it is a separable subspace by construction, containing a nonzero vector.Note also that for real Banach spaces, there exist operators with no nontrivial invariant subspace: take = R 2 and T a rotation of angle θ ∈ (0 2π) \

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“…The quasinilpotent one was constructed on 1 . The latest, which turned out to be finitely strictly singular [3], was defined on…”

Section: Introductionmentioning

confidence: 99%