Isomorphic classification of Lp,q-spaces, II
Introduction
This paper is devoted to the isomorphic classification of Lorentz spaces on general σ-finite measure spaces, where , , , which is a continuation of previous studies on the isomorphic classification of on resonant measure spaces contained in [36], [54] and is motivated by earlier work on subspaces of due to Carothers and Dilworth (see e.g. [12], [13], [14], [15], [19]). For the isomorphic classification of -spaces (i.e., , , , ), we refer to [60, Part III] and [7, Chapter XII]. For the isomorphic classification of weak -spaces, we refer to [37], [38], [39], [40].
The Lorentz spaces were introduced by G.G. Lorentz in [46], [47], and their importance is demonstrated in several areas of analysis such as harmonic analysis, interpolation theory, etc. (see e.g. [8], [19] and references therein). Recall that if is a measure space, then for and , the Lorentz space is the collection of all measurable functions f on Ω such that where denotes the decreasing rearrangement of (see the next section). It is well-known that is separable when Ω is σ-finite. In the special case when coincides with equipped with the counting measure, the space coincides with the familiar symmetric sequence space [19], [42]. It is well-known that if , then is a norm, and if , then it is a complete quasi-norm which is equivalent to a complete norm [8], [19]. It is clear that is the Lebesgue space , and that it is important to note that -spaces arise in the Lions–Peetre K-method of interpolation [12], [19].
It is known that and are not isomorphic to each other for any , , , and does not embed into as a complemented subspace [13], [19]. A stronger result showing that does not embed isomorphically into was proved in [36] and [54] using the so-called “subsequence splitting lemma” introduced and studied in [56] (see also [5], [21], [23], [24], [26], [55]). It is interesting to note that there are symmetric function spaces on which contain isomorphic -copies, , [51]. It is also known [36], [54] that does not embed isomorphically into the space . Hence, if , , , then is the full list of pairwise non-isomorphic -spaces over a resonant measure space.
In the present paper, we extend results in [36], [54] to the setting of general σ-finite measure spaces. Below, we briefly introduce the structure of the present paper. Unless stated otherwise, we will always assume that , and .
In Section 3, we prove some results on the finite representability of , which are important auxiliary tools. In particular, using a type/cotype argument and Kadec–Pełczyński theorem, we show that does not embed uniformly into when or or (see Corollary 3.3).
It is well-known that any σ-finite measure space is the direct sum of an atomless measure space and an atomic measure space. Recall that all separable infinite (or finite) atomless measure spaces are isomorphic to each other [11, Theorem 9.3.4], and is isomorphic to , , if Ω is an atomic resonant measure space. The main concern of this paper is the case when the measures of atoms are not necessarily the same in an atomic measure space. In Section 4, we classify infinite-dimensional -spaces on different atomic measure spaces and show that any -space on such a measure space is isomorphic to a direct sum of -spaces of the following types (see below for definitions and Section 4) In Table 1, we present a complete isomorphic classification of these spaces. Note that all Banach spaces X of type are isomorphic to .
If the atoms in Ω satisfy the conditions that and , then the -space on such a measure space is said to be of type . Johnson et al. [28, p. 31] introduced the subspace of an arbitrary symmetric function space Y on , which is spanned by the characteristic functions of . Such spaces are natural generalizations of the space introduced by Rosenthal [53]. For any fixed symmetric function space Y on , it is proved in [28, Theorem 8.7] (see also [43, Proposition 2.f.7]) that does not depend on the particular sequence used (up to an isomorphism). When , the space coincides with an -space of type and, for simplicity, we denote . By using techniques different those in [36], [54], we show that does not embed into (indeed, it does not embed into , see Theorem 5.7 below), which is a far-reaching generalization of [36, Theorem 11] and [54, Theorem 10]. Another interesting case is when the atoms satisfy the condition (or ). Any -space on a measure space of such type is said to be of type (resp. ). It is well known that for any disjointly supported sequence of unit vectors in , there exists a subsequence equivalent to the unit vector basis of [19, Theorem 5] (see also [13]). We establish a quantitative version of this result (see Proposition 4.10, Proposition 4.25), giving a criterion for the normalized characteristic functions of atoms (in the setting of and ) to be equivalent to the unit vector basis of . We also show that is isomorphic to a space of the type (or ) for some particular choice of . In particular, we obtain that is a subspace of .
In Section 5, we consider -spaces on arbitrary σ-finite measure spaces with non-trivial atomless part. Up to isomorphism, any such -space is one of the following We give a full characterization of the isomorphic embeddings between -spaces of all types listed above. It is clear that any space of the first four types is a complemented subspace of . We show that does not embed into any space of the first four types, which extends [36, Theorem 11] and [54, Theorem 10] significantly. We also establish an -space version of a Theorem for Orlicz sequence spaces due to Lindenstrauss and Tzafriri [43, Theorem 2.c.14] (see [7, Ch. XII, Theorem 9] for the case of -spaces), showing that there exists an isomorphic embedding if and only if . In the language of graph theory, the tree below is the Hasse diagram for the partially ordered set consisting of the equivalence classes of under Banach isomorphism with the order relation. For any spaces listed in the tree following, X is isomorphic to a subspace (indeed, a complemented subspace) of Y if and only if X can be joined to Y through a descending branch (see Table 2).
Our notations and terminology are standard and all unexplained terms may be found in [1], [28], [42], [43].
We would like to thank Professor J. Arazy for helpful discussions concerning results presented in his paper [3], which contains useful techniques in the study of isomorphic embedding of Banach spaces. We also thank Professor W.B. Johnson for his comments on finite representability of -spaces and his help in proving Proposition 3.2. We thank A. Kuryakov for many joint discussions, and T. Scheckter for his careful reading of this paper, and Professor E. Semenov for useful comments and sharing the paper [10], and we thank D. Zanin for helpful discussions. The second author was supported by the Australian Research Council (FL170100052). Authors thank the anonymous referee for reading the paper carefully and providing thoughtful comments, which improved the exposition of the paper.
Section snippets
Preliminaries
Let be a measure space with a σ-finite measure μ, defined on σ-algebra Σ, and let be the algebra of all classes of equivalent measurable real-valued functions on . For any function , its distribution is given by Denote by the subalgebra of consisting of all functions f such that for some . For every , its non-increasing rearrangement is defined by In the special case when the measure
Some results on the uniform embedding of
Recall that is said to embed uniformly in a Banach space X if for every , there exists an operator such that . The connection between type/cotype and the uniform embedding of into a Banach space has been widely studied (see [50] and [49, Theorem 3.5]).
It is well-known that [13], [19]. However, is finitely representable in , and therefore, in . In particular, Carothers and Flinn [15] obtained a quantitative result in the sense
Isomorphic classification of : the atomic case
This section contains main results of the present paper. Throughout this section, we always assume that and is an atomic measure space and such that Ω consists of atoms , , with . We denote by and the characteristic functions in generated by are denoted by .
Isomorphic classification of : general σ-finite case with a non-trivial atomless part
In this section, we consider -spaces on a general σ-finite measure space with a non-trivial atomless part.
We state below results of isomorphic classification of -spaces obtained in this section and supply the proof which is also based on a number of results given later in this subsection. Theorem 5.1 Let , , . Let be a σ-finite measure space which is not purely atomic. Then, is of one of the following types (up to an isomorphism)
References (60)
Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces
J. Funct. Anal.
(1981)- et al.
Lack of isomorphic embeddings of symmetric function spaces into operator ideals
J. Funct. Anal.
(2021) - et al.
Equidistributed random variables in
J. Funct. Anal.
(1989) Special Banach lattices and their applications
- et al.
Isomorphic classification of -spaces
J. Funct. Anal.
(2015) - et al.
The classification problem for nonatomic weak spaces
J. Funct. Anal.
(2010) - et al.
Banach spaces with a unique unconditional basis
J. Funct. Anal.
(1969) Subspace structure of Lorentz spaces and strictly singular operators
J. Math. Anal. Appl.
(2010)- et al.
Topics in Banach Space Theory
(2006) - et al.
On symmetric basic sequences in Lorentz sequence spaces
Isr. J. Math.
(1973)
The Banach–Saks p-property
Math. Ann.
Series of independent random variables in rearrangement invariant space
Isr. J. Math.
Théorie des opérations linéaires
Interpolation of Operators
Probability and Measure
Some estimates of the Banach–Mazur distance between finite-dimensional spaces
Measure Theory, vols. I, II
Geometry of Lorentz spaces via interpolation
Subspaces of
Proc. Am. Math. Soc.
Embedding in
Proc. Am. Math. Soc.
Projections on Banach spaces with symmetric bases
Stud. Math.
Weak convergence in non-commutative symmetric spaces
J. Oper. Theory
A scale of linear spaces related to the scale
Ill. J. Math.
Non-commutative Banach function spaces
Math. Z.
Banach–Saks properties in symmetric spaces of measurable operators
Stud. Math.
On p-convexity and q-concavity in non-commutative symmetric spaces
Integral Equ. Oper. Theory
Vilenkin systems and generalized triangular truncation operator
Integral Equ. Oper. Theory
The Banach–Saks properties in rearrangement invariant spaces
Stud. Math.
The dimension of almost spherical sections of convex bodies
Acta Math.
Cited by (1)
(Non-)Dunford–Pettis operators on noncommutative symmetric spaces
2022, Journal of Functional AnalysisCitation Excerpt :It is clear that Image 12 [31, Lemma 2.1 and Theorem 4.19].