The range of a vector measure

Let X be a completely regular Hausdorff space and E be a locally convex Hausdorff space. Then C_b(X) ? E is dense in (C_b(X, E), β₀), (C_b(X), β) ?_?E = (C_b(X) ? E, β) and (C_b(X), β₁) ?_?E = (C_b(X) ? E, β₁). For a separable space E, (C_b(X, E), β₀) is separable if and only if X is separably submetrizable. As a corollary, for a locally compact paracompact space X, if (C_b(X, E), β₀) is separable, then X is metrizable.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

On Dynkin's Markov property of random fields associated with symmetric processes

Let p(t, x, y) be a symmetric transition density with respect to a σ-finite measure m on (E, $\text{E}$), g(x,y)=∫p(t,x,y)dt, and $\text{M}\text{={σ-finite measures μ?0:∫g(x,y)μ(}\text{d}\text{x)μ(}\text{d}\text{y)<∞}}$. There exists a Gaussian random field $\text{Φ={?}{}_{\mathrm{\mu }}\text{:μ?}\text{M}\text{}}$ with mean 0 and covariance $\text{E}\text{?}{}_{\mathrm{\mu }}\text{?}{}_{\mathrm{\nu }}\text{=∫g(x,y)μ(}\text{d}\text{x)ν(}\text{d}\text{y)}$. Letting $\text{F}\text{(B)=σ{?}{}_{\mathrm{\mu }}\text{:μ(B}{}^{\mathrm{c}}\text{)=0}}$ we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: $\text{F}$(B), $\text{F}$(C) c.i. given $\text{F}$(B ∩ C). Of crucial importance is the following, proved by Dynkin: $\text{E}\text{{?}{}_{\mathrm{\mu }}\text{∣}\text{F}\text{(B)}=?}{}_{\mathrm{\mu B}}$, where μ_B is the hitting distribution of the process corresponding to p, m with initial law μ. Another important fact is that ?_μ=?_ν iff μ, ν have the same potential. Putting these together with an additional transience assumption, we present a potential theoretic proof of the following necessary and sufficient condition for (MP) on sets B, C: For every x?E, T_B∩C=T_B+T_C∮ θ_TB=T_C+T_B∮θ_TC a.s. P^x where, for D ? $\text{E}$, T_D is the hitting time of D for the process associated with p, m. This implies a necessary condition proved by Dynkin in a recent preprint for the case where B∪C=E and B, C are finely closed.     

Weakly and Separately Holomorphic Functions on Compact Determining Polydiscs in (DFN)-Spaces

The aim of this paper is to find some necessary and sufficient conditions by means of the property (DN) of E and the property (LB _∞) of F under which H _w (X, F) = H(X, F) and H _δ (X, H _b (F*)) = H(X, H _b (F*)) where E, F are Fréchet spaces and X is a compact determining polydisc in E*. At the same time, we also study the problem of the local Dirichlet representation of separately holomorphic functions.     

Sub-Stonean spaces and corona sets

A self-contained account of the theory of sub-Stonean spaces, and their relations to Stonean spaces and Rickart spaces is given. Of particular interest are the corona sets (of the form β(X) for locally compact, σ-compact spaces, because these highly nontrivial sub-Stonean spaces lend themselves to C?ech-cohomological considerations. The theory of sub-Stonean spaces is essential for our solution of the diagonalization problem for C(X)? M_n, found in K. Grove and G. K. Pedersen, Diagonalizing matrices over C(X), submitted for publication.     

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.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

Remarks on Banach spaces of compact operators

Let E and F be Banach spaces. We generalize several known results concerning the nature of the compact operators K(E, F) as a subspace of the bounded linear operators L(E, F). The main results are: (1) If E is a c₀ or l_p (1 < p < ∞) direct sum of a family of finite dimensional Banach spaces, then each bounded linear functional on K(E) admits a unique norm preserving extension to L(E). (2) If F has the bounded approximation property there is an isomorphism of $\text{L(E, F)}\text{into}\text{K(E, F)}{}^7$ such that its restriction to K(E, F) is the canonical injection. (3) If E is infinite dimensional and if F contains a complemented copy of c₀, K(E, F) is not complemented in L(E, F).     

A characterization of the minimum cycle mean in a digraph

Let C = (V,E) be a digraph with n vertices. Let f be a function from E into the real numbers, associating with each edge e ∈ E a weight$\text{?(e)}$. Given any sequence of edges σ = e₁,e₂,…,e_p define w(σ), the weight of σ, as $\text{∑}\text{i =}\text{1}\text{p}\text{?(e}{}_{\mathrm{i}}\text{)}$, and define m(σ), the mean weight of σ, as w(σ)?p. Let $\text{λ}{}^1\text{=}\text{min}{}_{\mathrm{C}}\text{m(C)}$ where C ranges over all directed cycles in G; λ¹ is called the minimum cycle mean. We give a simple characterization of λ¹, as well as an algorithm for computing it efficiently.     

Characterization of the closed convex hull of the range of a vector-valued measure

For a set K in a locally convex topological vector space X there exists a set T, a σ-algebra $\text{S}$ of subsets of T and a σ-additive measure m: $\text{S}$ → X such that K is the closed convex hull of the range {m(E): E ∈ $\text{S}$} of the measure m if and only if there exists a conical measure u on X so that K K_u,K_u, the set of resultants of all conical measures v on X such that v < u.     

On some results of Strichartz and of Rallis and Schiffman

Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ? F. We are interested in the following problem: is ($\text{i ? j, H}\text{}\text{?}\text{bo}{}_2\text{K, E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha F}}$) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ? j is always a continuous one to one map from $\text{H}\text{}\text{?}\text{bo}{}_2\text{K}$ into $\text{E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha }}\text{F}$. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=L^p(X,$\text{X}$,λ) for some σ-finite measure λ ? 0 then $\text{(i?j, H}\text{?}{}_2\text{K,L}{}^{\mathrm{p}}$(X,$\text{X}$,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L¹(X,$\text{X}$,λ)).     

A note on a theorem of heath

We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family $\text{C}\text{=?}{}_{\mathrm{m?N}}\text{C}{}_{\mathrm{m}}$ of closed sets, interior-preserving over each member C of which is a countable family {$\text{D}$_n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈$\text{C}$ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈$\text{D}$_v(Γ))?U.     

Characterization of similarities between two n-frames of projectors

An n-frame on a Banach space $\text{X}$ is E=(E₁,?, E_n) where the E_j's are bounded linear operators on $\text{X}$ such that E_j≠0,

$\text{∑}\text{j=1}\text{n}\text{E}{}_{\mathrm{j}}$

, and E_jE_k=δ_jkE_k (j, k=1,?, n). It is known that if two n-frames E and F are sufficiently close to each other, then they are similar, that is, F_j=TE_jT^-1 with T a bounded linear operator. Among the operators which realize the similarity of the two frames, there is the balanced transformation U(F, E)=(Σⁿ_j=1F_jE_j)(Σⁿ_j=1E_jF_jE_j)^{$\text{-1}\text{2}$}. One of our main results is a local characterization of the balanced transformation. Another operator which implements the similarity between E and F is the direct rotation R(F, E). It comes up in connection with the study of the set of all n-frames as a Banach manifold with an affine connection. Finally, it is shown that for quite a large set of pairs of 2-frames, the direct rotation has a global characterization.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Bessel functions and representation theory. I

In this paper a general theory of operator-valued Bessel functions is presented. These functions arise naturally in representation theory in the context of metaplectic representations, discrete series, and limits of discrete series for certain semi-simple Lie groups. In general, Bessel functions J_λ are associated to the action by automorphisms of a compact group U on a locally compact abelian group X, and are indexed by the irreducible representations λ of U that appear in the primary decomposition of the regular representation of U on L²(X). Then on the λ-primary constituent of L²(X), the Fourier transform is described by the Hankel transform corresponding to J_λ. More detailed information is available in the case in which (U, X) is an orthogonal transformation group which possesses a system of polar coordinates. In particular, when X=$\text{F}$^k×n,$\text{F}$ a real finite-dimensional division algebra, with k ? 2n and $\text{O}$(k, $\text{F}$), the representations λ of U are induced in a certain sense from representations π of GL(n, $\text{F}$). This leads to a characterization of J_λ as a reduced Bessel function defined on the component of 1 in GL(n, $\text{F}$) and to the connection between metaplectic representations and holomorphic discrete series for the group of biholomorphic automorphisms of the Siegel upper half-plane in the complexification of $\text{F}$^{n × n}.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).