Operator Ranges in Banach Spaces,I

Let X be a Banach space. A subspace L of X is called an operator range if there exists a continuous linear operator T defined on some Banach space Y and such that TY = L. If Y = X then L is called an endomorphism range. The paper investigates operator ranges under the following perspectives: (1) Existence (Section 3), (2) Inclusion (Section 4), and (3) Decomposition (Section 5). Section 3 considers the existence in X of operator ranges satisfying certain conditions. The main result is the following: if X and Fare separable Banach spaces and T : Y → X is a continuous operator with nonclosed range, then there exists a nuclear operator R:Y→X such that T + R is injective and has nonclosed dense range (Theorem 3.2). Section 4 seeks to determine conditions under which every nonclosed operator range in a Banach space is contained in the range of some injective endomorphism with nonclosed dense range. Theorem 4.3 contains a sufficient condition for this. Examples of spaces satisfying this condition are c₀, l_p (1 < p < ∞), L_q (1 < q < 2) and their quotients. In particular, this answers a question posed by W. E. Longstaff and P. Rosenthal (Integral Equations and Operator Theory 9 , (1986), 820-830. Section 5 discusses the possibility of representing a given dense nonclosed operator range as the sum of a pair L_1, L₂ of operator ranges with zero intersection in the cases where (a) L₁ and L₂ are dense, (b) L₁ and L₂ are closed. The results generalize corresponding results, for endomorphisms in Hilbert space, of J. Dixmier (Bull. Soc. Math. France 77 (1949), 11-101 and P. A. Fillmore and J. P. Williams (Adv. Math. 7 (1971), 254-281. A final section is devoted to open problems.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Invariant subspaces of X** under the action of biconjugates

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X** under the action of the algebra of biconjugates of bounded operators on . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c ₀, and show in particular that any space which does not contain ℓ₁ and has property (u) of Pelczynski is simple.     

Extrapolation results for <Emphasis Type="Italic">q</Emphasis> < 1

Given a sublinear operator T such that is bounded, it can be shown that is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : L ^q (μ) → X is bounded, with constant C/(1−q), for every and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped. This work has been partially supported by MTM2004-02299 and by 2005SGR00556.     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Operator spaces with few completely bounded maps

We construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable 1-Hilbertian operator space X₀ such that, whenever X is an infinite dimensional quotient of X₀, X is a subspace of X, and is a completely bounded map, then T=I_X+S, where S is compact Hilbert-Schmidt and ||S||₂/16||S||_cb||S||₂. Moreover, every infinite dimensional quotient of a subspace of X₀ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property. Mathematics Subject Classification (2000):The first author was supported in part by the NSF grants DMS-9970369, 0296094, and 0200714.     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Borel properties of linear operators

Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T⁻¹ is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.     

Automatic continuity,bases, and radicals in metrizable algegbras

The automatic continuity of a linear multiplicative operator T: XY, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (x_i) (orthogonal in the sense that x_iX_j=0 if i j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1129–1132, August, 1992.     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

DISJOINTNESS OF OPERATOR RANGES IN BANACH SPACES

Abstract

Let X be a Banach space. A linear subspace of X is called an operator range if it coincides with the range of a bounded linear operator defined on some Banach space. The paper studies disjointness and inclusion properties of various types of operator ranges in a separable infinite dimensional Banach space X. One of the main results is the following: Let E be a non-closed operator range in X. Then X contains a non-closed dense operator range R with the properties E∩= {0}, and R is decomposable, i.e. R = M + N where M,N are closed and infinite dimensional and M ∩ N = {0} (Theorem 6.2).     

Semi-Fredholm operators and sequence conditions

A typical result of this paper is the following: A closed linear operator T between Banach spaces X and Y is lower semi-Fredholm iff it is a rk-(=relatively compact)surjection, i.e. for every bounded sequence (y_n) in Y there is a bounded sequence (x_n) in D(T) such that (y_n?Tx_n) is relatively compact in Y. The concept of c_o-(rk- and wk-)injections and surjections are introduced to characterize semi-Fredholm properties (in the usual or generalized sense). This is done also by new operator moduli.     

Universal measurability of the identity mapping of a Banach space in certain topologies

If X is a Banach space and X is its conjugate, then a subset Y of X is called madmissible for X if a) the topology (X, Y) is Hausdorff, b) the identity embedding of (X, (X, Y)) into X is universally measurable (Ref. Zh. Mat., 1975, 8B 75 8K). If X is separable, then the existence of an m-admissible set is well known. In this note it is shown that there exist nonseparable X having separable m-admissible sets. The properties of spaces with separable m-admissible sets are considered. It is proved, in particular, that a separable normalizing subset Y of X is m-admissible for X if and only if every (X, Y)-compact set is separable in X.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 305–314, February, 1978.     

Geometry of spaces of compact operators

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if for all reflexive Banach spaces Y. We show that X ^* has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.     

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).     

Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums ∑_nμ(A_n)F(t_n). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.     

Additions to the Periodic Decomposition Theorem

A pair of linear bounded commuting operators T₁, T₂ in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T₁)(I-T₂) = Ker (I-T₁) + Ker (I-T₂). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Révész that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included. This revised version was published online in August 2006 with corrections to the Cover Date.     

Pointwise compact sets of measurable functions

If X is a compact Radon measure space, and A is a pointwise compact set of real-valued measurable functions on X, then A is compact for the topology of convergence in measure (Corollary 2H). Consequently, if X_o,..., X_n are Radon measure spaces, then a separately continuous real-valued function on X_o×X₁×...×X_n is jointly measurable (Theorem 3E). If we seek to generalize this work, we encounter some undecidable problems (§4).