Banach Space Theory Research Articles

Infinite Representability of Schrödinger Operators with Ergodic Potential

Analogous to the notion of finite representability in the theory of Banach spaces, the notions of the representability and the infinite representability of self-adjoint operators are introduced. It is proved that the infinite representability of the operator A in B yields that the essential spectrum of B contains the spectrum of A . This result applied to ergodic Schrödinger operators yields a new proof for the nonrandomness of the spectrum and for the connection between the spectrum and the density of states. A formula for the spectrum of the Hamiltonian of a substitutional alloy is presented, which clarifies the bowing effect. Similar results were found independcntly by Kirsch and Martinelli.

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T: X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc. 22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L 1 semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l 1. As a consequence of the proof, it is shown that if X is a subspace of L 1, either L 1 embeds in X or l 1 embeds in L 1 X . The simpler result that L 1 does not semi-embed in c 0 is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.

Espaces 𝑙^{𝑝} dans les sous-espaces de 𝐿¹

It is shown that every subspace E E of L 1 {L^1} contains a subspace isomorphic to l p ( E ) {l^{p(E)}} , where p ( E ) p(E) is the upper bound of the set of real p p ’s such that E E is of type p p -Rademacher. As p ( E ) p(E) is also the upper bound of the set of real p p ’s such that E E embeds into L p {L^p} , this result answers a question of H. P. Rosenthal. The proof uses the theory of stable Banach spaces developed by J. L. Krivine and B. Maurey.

Bases in Banach Spaces

The publication of Banach's book [2] in 1932 might be regarded as marking the beginning of the systematic study of Banach spaces. Research activity in this area has expanded dramatically during the past two decades. Interesting new directions have developed and interplays between Banach space theory and other mathematics have proved to be very valuable. Most well-known classical problems have been solved, but some remain and important new problems have arisen. Our purpose is very limited. It is to discuss some of the most important and fundamental facts about bases in Banach spaces, particularly those that may be interesting and useful for mathematicians in other fields. Often, only sketches of proofs will be given, while others are given in more detail because they are relatively easy and may contribute to developing intuitive feeling for the concepts involved. It will be seen that many very important and beautiful theorems have very easy and natural proofs. A basis (or Schauder basis) for a Banach space X is a sequence (en: n > 1) of members of X which has the property that, for each x in X, there is exactly one sequence of scalars (xi} for which x = xiei in the sense that lim.nX lix - E ixieijj = 0. Some important Banach spaces have very natural bases. If (en} is a complete orthonormal sequence in Hilbert space H and x is any member of H, then there is exactly one sequence (xn} of scalars such that lim x yn- lxieill = 0. For this sequence of scalars, each xi is (x, ei) and n 00 ~~~~~1/2

Martingale proofs of some geometrical results in Banach space theory

Martingale proofs of some geometrical results in Banach space theory

On nonseparable Banach spaces

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of L 1 { − 1 , 1 } α {L^1}{\{ - 1,\,1\} ^\alpha } into X ∗ {X^{\ast }} implies, for any infinite cardinal α \alpha , the isomorphic embedding of l α 1 l_\alpha ^1 into X X , gets the following form: if α = ω \alpha = \omega , has been proved from Pełczynski; if α > ω + \alpha > {\omega ^ + } , the proof is given in this paper; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + C .H . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + {\text {C}}{\text {.H}}{\text {.}} , an example discovered by Haydon gives a negative answer; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + ⌝ C .H . + M .A . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + \urcorner {\text {C}}{\text {.H}}{\text {.}} + {\text {M}}{\text {.A}}{\text {.}} , is also proved in this paper.

An Ordinal L p -Index for Banach Spaces, with Application to Complemented Subspaces of L p

One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five simple examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an EP-space, that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns large countable ordinals to any

Ultraproducts in Banach space theory.

Ultraproducts in Banach space theory.

The Hanf Number of the First Order Theory of Banach Spaces

The Hanf Number of the First Order Theory of Banach Spaces

The Hanf number of the first order theory of Banach spaces

In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets).

Ultrapowers and local properties of Banach spaces

The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the L p {\mathcal {L}_p} -spaces). Many examples are given; also, as an application, it is proved that any L p {\mathcal {L}_p} -space, 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty , has an ultrapower which is isomorphic to an L p {L_p} -space.

Some recent discoveries in the isomorphic theory of Banach spaces

Some recent discoveries in the isomorphic theory of Banach spaces

Intersection Properties of Balls and Subspaces in Banach Spaces

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with L1 (u) dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with L1 (a) dual spaces whose unit ball contains extreme points. Introduction. A Banach space A is said to be a Cl-space if for every Banach space X containing A, there is a linear projection P from X onto A with IIPII A, there exists a linear extension t: Y -* A of T with II TII = I|I T1. An example of a space with this property is 1. (r) for some set r o 0. This is shown by application of the Hahn-Banach theorem coordinatewise. In 1950 Nachbin [39] and Goodner [18] characterized the real 91-spaces whose unit balls have extreme points: They are (up to isometry) the C(K)spaces for which the compact Hausdorff space K is extremally disconnected (i.e. the closure of every open set is open). In 1952 Kelley [30] showed that the assumption on the unit ball was superfluous. This was shown to hold also in the complex case by Hasumi [21] in 1958. In 1955 Grothendieck [19] showed that a real Banach space A is isometric to an LI (,a)-space for some measure t, if and only if its dual A* is a 0'1-space. This result was later proved for the complex case by Sakai [42]. Received by the editors August 20, 1975. AMS (MOS) subject classifications (1970). Primary 46B99; Secondary 46A40, 46B05, 46E15, 46E30. (1) This paper is a revised version of the author's doctoral dissertation at the University of Oslo, 1974. 0) American Mathematical Society 1977

Intersection properties of balls and subspaces in Banach spaces

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces whose unit ball contains extreme points.

Some applications of model theory in Banach space theory

Some applications of model theory in Banach space theory

Some applications of p-summing operators to Banach space theory

Some applications of p-summing operators to Banach space theory

The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

$L^p$-convolution operators and tensor products of Banach spaces

intimately connected with duality theory (the notation is that of [1]). In both cases the middle algebra is the closure of L(G) in the dual of the first algebra and also the predual of the third algebra (at least when G is amenable in the second case). Furthermore, the third algebra is closely connected with the multiplier algebra of the first algebra. For abelian groups, compact or discrete, Varopoulos [11], [12] showed to great effect how the second triple could be obtained and studied by starting with the tensor product ^0(G) (8)y^0(^)» 7 the greatest crossnorm. An analogous construction starting this time with^0(G ) ®x te0(G)9 A the least cross-norm, would produce the first triple. On the other hand, at least for amenable groups, the triples in (1) can be considered as the extreme case/?=1, 2, respectively, of a family {A(G), cv{G), B(G)}, 1 ^ / ? ^ 2 , associated with /^-convolution operator theory, and obtained by starting with the tensor product L'(G) <g)y L (G),pj±\, or<g0(G) ®y L\G), p=\. Indeed, Herz has shown that A{G) is a pointwise Banach algebra [6] while B(G), l < p ^ 2 , is both the multiplier algebra of A{G) and the Banach dual space of cv(G), G amenable. In these notes we outline a new approach to convolution operator theory, by starting with ^^G) ®a ^0(G)9 a a tensorial norm [5], rather than with L'{G) ®y L (G). A triple { f (G), &'(G), #(<7)} analogous to (1) is obtained. For //-convolution operator theory, a family of tensorial norms CLVQ is used. The two basic ideas are to exploit classical Banach space theory concerning L(iu)spaces, for example, forgetting about group structure, and then, when a group structure is imposed, to exploit standard ^ 0 ( ^ ) a n d L(G)-techniques because all the 'L^-theory' has been thrown into the norm aM,

A characterization of M-Summands

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection ifand an L-projection if, insteadA closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

Banach ideals of operators with applications

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the L p -spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L p (μ)-spaces.