Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

Unbounded operators generated by a given spectral measure

Let T be a closed densely-defined operator on a Banach space X and let E(·) be a spectral measure whose range $\text{E}$ is a complete Boolean algebra of projections in X. Then T is of the form ∝f(λ) dE(λ) if and only if T commutes with $\text{E}$ and leaves invariant every invariant subspace of $\text{E}$.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

The asymptotics of nonexpansive iterations

Let D be a closed subset of a Banach space X and T: D → D a nonexpansive mapping. Conditions are given (on the space X) for T to satisfy the following property of ergodic type: $\text{{T}{}^{\mathrm{n}}\text{x}\text{n}\text{}}$ converges (either weakly or strongly) to a vector v. Rather unexpectedly, D is not assumed to be convex, nor is I – T assumed to satisfy any range condition. In addition, it is shown that ?v is the unique point of least norm in the closure of R(I – T) if and only if I – T satisfies a certain range condition at infinity. Several interesting applications to accretive operator and nonlinear semigroup theory are also included.     

Absolute Lipschitz extendability

A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space Y?X, where the loss in the Lipschitz constant in the extension is independent of $\text{Y,}\text{Z}$, and f. We show that various classes of natural metric spaces are absolutely Lipschitz extendable. To cite this article: J.R. Lee, A. Naor, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Caractérisation spectrale des algèbres de Banach de dimension finie

Let f be an analytic function mapping a domain in $\text{C}$ into a complex Banach algebra. Using potential theory and a new result on almost continuity of the spectrum, which extends the theorem of Newburgh, we prove that either the set of λ such that the spectrum of f(λ) is finite is of outer capacity zero, or there exists an integer n such that the spectrum of f(λ) has at most n elements for every λ. From this we get extensions of a theorem given, in the complex case, by Kaplansky in 1954 and Hirschfeld and Johnson in 1972. More precisely we show that, if the spectrum is finite for every element of an open set of a real algebra or of the set of Hermitian elements of an algebra with an involution, then the quotient of this algebra by its radical is finite-dimensional.     

Zeros for accretive operators satisfying certain boundary conditions

Let X be a Banach space with the dual space $\text{X}{}^1$ to be uniformly convex, let D ? X be open, and let $\text{T:}\text{D}\text{?}\text{→ X}$ be strongly accretive (i.e., for some k < 1: (λ ? k)∥ u ? v∥ ? ∥(λ ? 1)(u ? v)+ T(u) ? T(v)∥ for all $\text{u, v ?}\text{D}\text{?}$ and λ > k). Suppose T is demicontinuous and strongly accretive and suppose there exists z?D satisfying: T(x) t(x ? z) for all x??D and t < 0. Then it is shown that T has a unique zero in $\text{D}\text{?}$. This result is then applied to the study of existence of zeros of accretive mappings under apparently different types of boundary conditions on T.     

A remark on commutativity of the image of A-Valued analytic function

Let BL(?) be a complex Banach algebra of all bounded linear operators acting on the Hilbert space ?. J. Globevnik and I. Vidav proved that when the range of an operator-valued analytic function with domain of definition D ? ? consists of the normal operators, then f(D) is a commutative set in the algebra BL(?). The paper strengthens this result for the topological algebras.     

Optimal solution of a Monge-Kantorovitch transportation problem

Suppose that P and Q are probabilities on a separable Banach space $\text{E}$. It is known that if (P, Q) satisfies certain regularity conditions and a random variable X has law P, then there exists a function $\text{f :}\text{E}\text{→}\text{E}$, such that the function f(X) has the law Q and the random pair (X, f(X)) is an optimal coupling for the Monge-Kantorovitch problem. In this paper we provide an approximation of the function f when the law Q is discrete. Thenwe extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations.     

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.     

The voltage and current transfer ratios of RLC operator networks

The subject of this work is the voltage and current transfer ratios of three-terminal networks having no mutual coupling and whose impedances are analytic functions taking their values in an abelian self-adjoint algebra of bounded linear operators on a Hilbert space. Each such value is also assumed to be invertible. It is shown that these ratios have the form [I + A(ζ)]^?1, where, for each ζ in a sufficiently small open cone in the right-half complex plane with apex at the origin and the real axis as its bisector, the numerical range of A(ζ) is contained in a compact subset of the open right-half plane. This implies that the ratios are strictly contractive for each ζ in the cone. The angle of the cone is $\text{π}\text{(2k + 2)}$, where k is the number of internal nodes of a certain “surrogate” network; this result is best possible. For two-terminal-pair networks the ratios are shown to be strictly contractive for each ζ in a similar cone with angle $\text{π}\text{(2k + 4)}$.     

Manifolds of linear involutions

The set of bounded linear involutions on a complex Banach space $\text{X}$ is equipped with a Banach manifold structure and an affine connection compatible with its embedding into $\text{B}$($\text{X}$). Geodesic lines are characterized. Moreover, if $\text{X}$ is a Hilbert space and the topology of the self-adjoint part of the manifold is strengthened to be compatible with the Hilbert-Schmidt metric, these geodesics are identified as minimal arcs between pairs of self-adjoint involutions whose straight line distance is less than 2.     

Compact holomorphic mappings on Banach spaces and the approximation property

Let $\text{H}$(E) be the space of complex valued holomorphic functions on a complex Banach space E. The approximation property for $\text{H}$(E), endowed with various natural locally convex topologies, is studied. For example, $\text{H}$(E) with the compact-open topology has the approximation property if and only if E has the approximation property. In order to characterize when $\text{H}$(E) has the approximation property for topologies other than the compact-open, the notion of a compact holomorphic map between Banach spaces in introduced and studied.     

Positive integrable elements relative to a left Hilbert algebra

Let $\text{U}$ be an achieved left Hilbert algebra. Let $\text{η∈D}{}^{\mathrm{?}}$ be an element such that π′(η) is a positive operator. Then, following M. A. Rieffel, η is called integrable if sup{(η|e)e∈U and ee^?e²} < + ∞. It is shown that η is integrable if and only if there is an element ζ∈D^flat; such π′(ζ) is positive and ζ is a square root of η in an appropriate sense. This is shown to be a generalization of Godement's well known theorem on the existence of a convolution square root for a continuous square-integrable positive-definite function on a locally compact group. An “integral” and an “L¹-norm” are then defined on the linear span of the positive integrable elements and the completion of this space, denoted by L¹($\text{U}$), is shown to be the predual of $\text{l}$($\text{U}$). “Godement's theorem” is then used to investigate square-integrable representations of $\text{U}$.     

Random fixed points of K-set- and pseudo-contractive random maps

Let $\text{(Ω,Σ)}$ be a measurable space, X and Y separable Banach spaces, and C a weakly compact subset of X. Let $\text{f}\text{:}\text{Ω×C→Y}$ and $\text{T}\text{:}\text{Ω×C→Y}$ be continuous random operators. Then the deterministic solvability of the equation$\text{f(ω,x)−T(ω,x)=0}\text{(ω∈Ω,x∈C)}$implies the stochastic solvability of it provided that (f−T)(ω,.) is demiclosed at zero and T(ω,C) is bounded for each $\text{ω∈Ω}$. As applications, random fixed points of various types of pseudo-contractive and k-set-contractive random operators are obtained.