Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

More on convexity and smoothness of operators

Let X and Y be Banach spaces and T:Y→X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus δT of convexity and the T-modulus ρT of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel-Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

Perturbing the boundary conditions of the generator of a cosine family

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)???X and ??:D(A)???X (where ?X is another Banach space called boundary space). Putting conditions on L and ??, we show that the second order abstract Cauchy problem for the operator A _?? with A _?? u=Au and domain D(A _??):={u??D(A):Lu=??u} is well-posed and thus it generates a cosine operator function on the Banach space X.     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Integral mappings between Banach spaces

We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.     

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l₂^p(C).     

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

There exists a real hereditarily indecomposable Banach space X=X(C) (respectively X=X(H)) such that the algebra L(X)/S(X) is isomorphic to C (respectively to the quaternionic division algebra H).Up to isomorphism, X(C) has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek [J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (2) (1986) 221-226; S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1) (1986) 339-353; S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444], and proves that a theorem of G. Godefroy and N.J. Kalton [G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121-141] about isometric embeddings of separable real Banach spaces does not extend to the complex case.The quaternionic example X(H), on the other hand, has unique complex structure up to isomorphism; other examples with a unique complex structure are produced, including a space with an unconditional basis and non-isomorphic to l₂. This answers a question of S. Szarek in [S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444].     

Enlargements of positive sets

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set A we associate a family of enlargements E(A) and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of E(A) and the family of so-called representative functions of A. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.     

Sobolev embeddings into spaces of Campanato, Morrey, and Hölder type

Necessary and sufficient conditions on a rearrangement-invariant Banach function space X(Q) on a cube Q in , n?2, are given for the corresponding Sobolev space W¹X(Q) to be continuously embedded into (generalized) Campanato, Morrey, or Hölder spaces. The optimal such r.i. spaces X(Q) are found. As a by-product, sharp inclusion relations are proved among Campanato, Morrey, and Hölder type spaces.     

On multiplicative perturbation of integral resolvent families

In this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a Banach space X. Assuming that a(t) is a creep function which satisfies a(⁰⁺)>0, we prove that if (A,a) generates an integral resolvent, then (A(I+B),a) also generates an integral resolvent for all B∈B(X,Z), where Z belongs to a class of admissible Banach spaces. In special instances of a(t) the space Z is proved to be characterized by an extended class of Favard spaces.     

Computability of finite-dimensional linear subspaces and best approximation

We discuss computability properties of the set P_G(x) of elements of best approximation of some point x∈X by elements of G⊆X in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about P_G(x) as a closed set. In the case that G is finite-dimensional, one can compute negative information on P_G(x) as a compact set. This implies that one can compute the point in P_G(x) whenever it is uniquely determined. This is also possible for a wider class of subsets G, given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in P_G(x). We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace.     

Factorization Theorems for Multiplication Operators on Banach Function Spaces

Let X Y and Z be Banach function spaces over a measure space \({(\Omega, \Sigma, \mu)}\) . Consider the spaces of multiplication operators \({X^{Y'}}\) from X into the Köthe dual Y′ of Y, and the spaces X ^Z and \({Z^{Y'}}\) defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space \({X^Z \cdot Z^{Y'} \subseteq X^{Y'}}\) that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class \({d_{p,Z}^*}\) of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.     

Maurey-Rosenthal factorization of positive operators and convexity

We show that a Banach lattice X is r-convex, 1<r<∞, if and only if all positive operators T on X with values in some r-concave Köthe function spaces F(ν) (over measure spaces (Ω,ν)) factorize strongly through L_r(ν) (i.e., T=M_gR, where R is an operator from X to L_r(ν) and M_g a multiplication operator on L_r(ν) with values in F). This characterization of r-convexity motivates a Maurey-Rosenthal type factorization theory for positive operators acting between vector valued Köthe function spaces.     

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.      

Espaces de Banach superstables,distances stables et homeomorphismes uniformes

We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL ^p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol ^p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc ₀ does not uniformly imbed into any stable Banach space.     

On K 0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K ₀(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X ² is not a sufficient condition of K ₀(B(X)) = 0.