A uniformly convex hereditarily indecomposable banach space

Abstract

We construct a uniformly convex hereditarily indecomposable Banach space, using similar methods as Gowers and Maurey in [GM] and the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher and Weiss ( [5a]). n i=1 x i , then we say that X satisfies a lower f -estimate.Let q > 1 in R, q ′ such that 1/q + 1/q ′ = 1. Let θ ∈ ]0, 1[ , and p be the number defined by 1/p = 1 − θ + θ/q.Let S be the strip {z ∈ C/Re(z) ∈ [0, 1]}, δS its boundary, S 0 the line {z/Re(z) = 0}, S 1 the line {z/Re(z) = 1}. Let µ be the Poisson probability measure associated to the point θ for the strip S. We have µ(S 0 ) = 1 − θ. Let µ 0 be the probability measure on R defined by µ 0 (A) = µ(iA)/(1 − θ), µ 1 be the probability measure on R defined by µ 1 (A) = µ(1 + iA)/θ. Let A S be the set of analytic functions F on S, with values in c 00 , which are L 1 on δS for dµ and which satisfy the Poisson integral representation F (z 0 ) = δS F (z)dP z0 (z) on S ( this is well defined since such functions have finite ranges). If F is analytic and bounded on S, then F ∈ A S .We recall the definition of the interpolation space of a family of N -dimensional spaces from [5a]. Let . z for z in S be a family of norms on C N , equivalent with log-integrable constants, and such that z → x z is measurable for all x in C N . The interpolation space in θ is defined by the norm, where A N S denotes the image of the canonical projection from A S into the space of functions from S to C N .We generalize to the infinite-dimensional case as follows. Let {X z , z ∈ δS} be a family of Banach spaces in X , equipped with norms . z , such that for all x in c 00 , the function z → x z is measurable, and such that over vectors of finite range N , the norms . z are equivalent with log-integrable constants. Let X N z be E N X z , X N be the θ-interpolation space of the family X N z ; the interpolation space of the family in θ is completion(∪ N ∈N X N ). Now let {X t , t ∈ R} be a family of spaces in X , equipped with norms . t , such that for all t in R, X t satisfies a f -lower estimate and for all x in c 00 , the function t → x t is measurable. For vectors of range at most E N , we have f (N ) −1 x 1 ≤ x t ≤ x 1 , so that the norms . t are equivalent to . 1 with log-integrable constants. We are then allowed to define the θ-interpolation space of the family defined on δS as X t if z = it, l q if z = 1 + it. Let X θ be the class of spaces X obtained in that way.We shall sometimes use for z ∈ δS the notation . z , to mean . t if z = it, and . q if z = 1 + it. There will be no ambiguity from the context. We shall similarly use the notation . * z . The notation X N t stands for E N X t , and X N * t for E N X * t . Also, if not specified, the measure of a subset of R will be its measure for µ 0 .

Some facts from descriptive set theory concerning essential spectra and applications

“…space) if it does not contain any decomposable subspace. The class of hereditarily indecomposable Banach spaces was first introduced and investigated by Gowers and Maurey in [7] (see also [4]). One of the main facts relating to this class is the following result due to Gowers and Maurey [7].…”

“…[7,4]), strongly continuous semigroups exhibit a nice behavior. In particular, we establish that the infinitesimal generator of any strongly continuous group is necessarily bounded (which seems to be a feature of this class of spaces).…”

On Hereditarily Indecomposable Banach Spaces

“…Now, on one hand, Gowers and other analysts are using the "arbitrarily distortable" technique of HI spaces developed by Gowers, Maurey and other mathematicians to other mathematical topics (see, for example, [28,29,30,[31][32][33][34][35][36], [37] and [38]). On the other hand, many mathematicians have been investigating properties of HI spaces (see, [39,40,[41][42][43]), [44,45,46,47] and [48][49][50]). It has been already found that a space of HI type or the Gowers-Maurey type can be so "bad" that it has no subspace of infinite dimensions with a separable dual [14], and can be so "nice" that it has an equivalent uniformly convex norm [42].…”

“…On the other hand, many mathematicians have been investigating properties of HI spaces (see, [39,40,[41][42][43]), [44,45,46,47] and [48][49][50]). It has been already found that a space of HI type or the Gowers-Maurey type can be so "bad" that it has no subspace of infinite dimensions with a separable dual [14], and can be so "nice" that it has an equivalent uniformly convex norm [42]. Yet a lot of long-standing open problems would be resolved by constructing various types of hereditarily indecomposable Banach spaces, especially, non-separable HI (see, for instance, [27,51]).…”

A note on the method of minimal vectors

“…If the answer is positive then Theorem 2.2 will imply that every quasi-nilpotent operator on a super-reflexive Banach space has a non-trivial invariant subspace. Then, every strictly singular operator on a super-reflexive Hereditarily Indecomposable complex Banach space has a non-trivial invariant subspace (see [7]), and hence the space constructed in [6] would provide a positive solution to the I.S.P..…”