Decomposition of two functions in the orthogonality equation

The aim of this paper is to solve the orthogonality equation with two unknown functions. This problem was posed by J. Chmieliński during two international conferences.     

Isometries between Banach spaces of Lipschitz functions

The Banach spaces Lip ^a (S, Δ), lip ^a (S, Δ), Lip ^a (S, Δ;s ₀) and lip ^a (S, Δ;s ₀) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces. This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Dr. Y. Benyamini.     

James orthogonality and orthogonal decompositions of Banach spaces

We establish decompositions of a uniformly convex and uniformly smooth Banach space B and dual space B^∗ in the form B=M?J^∗M^⊥ and B^∗=M^⊥?JM, where M is an arbitrary subspace in B, M^⊥ is its annihilator (subspace) in B^∗, J:B→B^∗ and J^∗:B^∗→B are normalized duality mappings. The sign ? denotes the James orthogonal summation (in fact, it is the direct sums of the corresponding subspaces and manifolds). In a Hilbert space H, these representations coincide with the classical decomposition in a shape of direct sum of the subspace M and its orthogonal complement M^⊥: H=M⊕M^⊥.     

Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality

A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by $X{～}_{BJ}Y$, if there exists a (possibly nonlinear) bijection between X and Y that preserves Birkhoff-James orthogonality in both directions. It is shown that $X?Y$ if either X or Y is finite dimensional and $X{～}_{BJ}Y$, and that ${?}_p{?}_{BJ}{?}_q$ if $1<p<q<\infty$. Moreover, if H is a Hilbert space with $\dim ?H\ge 3$ and $H{～}_{BJ}X$, then $H=X$. In the two-dimensional case, it turns out that ${?}_{p,q}^2{～}_{BJ}{?}_2^2$, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Bump functions and differentiability in Banach spaces

We show that if a Banach space admits a continuous symmetrically Fréchet subdifferentiable bump function, then is an Asplund space.

     

Constructive utility functions on Banach spaces

In this paper we prove the existence of continuous order-preserving functions on subsets of ordered Banach spaces using a constructive approach.     

The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space are computed, and are used to show that our results are sharp.

     

Subdifferentials of perturbed distance functions in Banach spaces

In general Banach space setting, we study the perturbed distance function d_S^J(·){d_S^J(cdot)} determined by a closed subset S and a lower semicontinuous function J (·). In particular, we show that the Fréchet subdifferential and the proximal subdifferential of a perturbed distance function are representable by virtue of corresponding normal cones of S and subdifferentials of J (·).     

Implicit functions from topological vector spaces to Banach spaces

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.     

On a singular evolution equation in Banach spaces

We study the differential equation tu′(t) + Au(t) = f(t), 0 < t < ∞, in Banach spaces. We obtain existence and uniqueness theorems for the solutions as well as regularity properties in suitable interpolation spaces.     

Smooth approximation of convex functions in Banach spaces

This paper shows that every extended-real-valued lower semi-continuous proper (respectively Lipschitzian) convex function defined on an Asplund space can be represented as the point-wise limit (respectively uniform limit on every bounded set) of a sequence of Lipschitzian convex functions which are locally affine (hence, C^∞) at all points of a dense open subset; and shows an analogous for w^∗-lower semi-continuous proper (respectively Lipschitzian) convex functions defined on dual spaces whose pre-duals have the Radon-Nikodym property.     

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.     

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.