The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.     

The construction of all locally compact extensions

The paper presents one of the ways to construct all the locally compact extensions of a given Tychonoff space T. First, there proved the “local” variant of the Stone-C?ech theorem on “completely regular” Riesz spaces X(T) of continuous bounded functions on T with no unit function, in general, but with a collection of local units. In Theorem 1 it is proved that all the functions from X(T) can be “completely regularly” extended on the largest locally compact extension β_xT. Theorem 3 states, that β_xT are presenting, in fact, all the locally compact extensions of T.     

Remarks on pseudo-contractive mappings

Let X be a Banach space, B a closed ball centered at the origin in X, and T: B → X a pseudo-contractive mapping (i.e., (λ ? 1) ∥x ? y∥ ? ∥(λI ? T)(x) ? (λI ? T) (y)∥ for all x, y?B and λ > 1). It is shown here that the antipodal boundary condition: T(x) = ?T(?x) for all x?δB assures existence of a fixed point of T in B provided that the ball B has the fixed point property with respect to non-expansive self-mappings. Also included are some fixed point theorems which involve the Leray-Schauder condition.     

Score vectors of tournaments

A tournament T on any set X is a dyadic relation such that for any x, y ∈ X (a) (x, x) ? T and (b) if x ≠ y then (x, y) ∈ T iff (y, x) ? T. The score vector of T is the cardinal valued function defined by R(x) = |{y ∈ X : (x, y) ∈ T}|. We present theorems for infinite tournaments analogous to Landau's necessary and sufficient conditions that a vector be the score vector for some finite tournament. Included also is a new proof of Landau's theorem based on a simple application of the “marriage” theorem.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

A pointwise abelian Ergodic theorem for Lp semigroups, 1 ⩽ p < ∞

Let (X, ∑, μ) be a σ-finite measure space and L_p(μ) = L_p(X, ∑, μ), 1 ? p ? ∞, the usual Banach spaces of complex-valued functions. Let {T_t: t ? 0} be a strongly continuous semigroup of positive L_p(μ) operators for some 1 ? p < ∞. Denote by R_λ the resolvent of {T_t}. We show that f?L_p(μ) implies λR_λf(x) → f(x) a.e. as λ → ∞.     

The inverse balayage problem for Markov chains

Let X be a Markov chain, let A be a finite sunset of its countable state space. let ?A consist of states in A′ that can be reached in one step from A and let v be a prescribed probability measure on ?A. In this paper we study the following inverse exit problem: describe and analyze the set M(v) of probability measures μ on A such that P^μ {X(T)?·}=v(·) where T= inl{k: X(k)?A′} is the first exit time from A. Characterizations are provided for elements of M(v), extreme points of M(v) and those measures in M(v) that are maximal with respect to a partial ordering induced by excessive functions. Potential theoretic aspects of the problem and one-dimensional birth and death processes are treated in detail, and examples are given that illustrate implications and limitations of the theory.     

Constructions for overlarge sets of disjoint pure directed triple systems

Let X be a v-set, v≥3. A transitive triple (x,y,z) on X is a set of three ordered pairs (x,y),(y,z) and (x,z) of X. A directed triple system of order v, denoted by DTS(v), is a pair (X,?), where X is a v-set and ? is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of ?. A DTS(v) is called pure and denoted by PDTS(v) if (x,y,z)∈? implies (z,y,x)??. An overlarge set of disjoint PDTS(v) is denoted by OLPDTS(v). In this paper, we establish some recursive constructions for OLPDTS(v), so we obtain some results.     

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.     

Weak limits of zeros of orthogonal polynomials

Letμ be a positive unit Borel measure with infinite support on the interval [?1, 1]. LetP _n(x, μ) denote the monic orthogonal polynomial of degreen associated withμ, and letv _n(μ) denote the unit measure with mass 1/n at each zero ofP _n(x, μ). A carrier is a Borel subset of the support ofμ having unitμ-measure, and a measurev is carrier related toμ when it has the same carriers asμ. We demonstrate that for each carrierB of positive capacity there is a measurev, which is carrier related toμ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v _n(v)} _n ^=1/∞ .     

Polar factorization and monotone rearrangement of vector-valued functions

Given a probability space (X, μ) and a bounded domain Ω in ?^d equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ L^p (X, μ; ?^d) there is a unique “polar factorization” u = ?Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u^?1(E)) = 0 for each Lebesgue negligible subset E of ?^d. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.     

Browder-Weyl theorems, tensor products and multiplications

A Banach space operator T∈B(X) satisfies Browder's theorem if the complement of the Weyl spectrum σw(T) of T in σ(T) equals the set of Riesz points of T; T is polaroid if the isolated points of σ(T) are poles (no restriction on rank) of the resolvent of T. Let Φ(T) denote the set of Fredholm points of T. Browder's theorem transfers from A,B∈B(X) to S=LARB (resp., S=A⊗B) if and only if A and B^∗ (resp., A and B) have SVEP at points μ∈Φ(A) and ν∈Φ(B) for which λ=μν∉σw(S). If A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to LARB; again, restricting ourselves to the completion of X⊗Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to A⊗B.     

Spectral integrals of operator-valued functions. II. From the study of stationary processes

This paper is a continuation of the study made in [38]. Using Douglas' operator range theorem and Crimmins' corollary we obtain several new results on the “square-integrability of operator-valued functions with respect to a nonnegative hermitian measure”. Using these facts we are able to extend in an important way theorems on the “spectral integral of an operator-valued function” which were obtained in [38], to wit, we are able to drop assumptions that functions are closed operator-valued. We apply these results to Wiener-Masani type infinite-dimensional stationary processes, representing a purely non-deterministic process as a “moving average” and obtaining a “factorization” of its spectral density. Next, anticipating global applications of our tools, we investigate the adjoint and generalized inverse of spectral integrals. Our definition of measurability for closed-operator-valued functions plays a key role here. Finally, we partially prove a conjecture (J. Multivariate Anal. (1974), 166–209) on simpler necessary and sufficient conditions on “when is a closed densely defined operator T from $\text{H}$^q to $\text{H}$^p a spectral integral T = fΦdE?”: Let q be finite and E be of countable multiplicity for $\text{H}$. Then (i) Tx ∈ $\text{S}$_x^p each x ∈ $\text{D}$_T (T is E-subordinate), and (ii) E(B)T ? TE(B) each B ∈ $\text{B}$ (T is E-commutative) implies L_x^pT ? TL_x^q each x ∈ $\text{H}$^q (T commutes with all the cyclic projections), and thus T = fΦdE.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

On a problem of Nirenberg concerning expanding maps

Let X be a Banach space and T:X → X a continuous map, which is expanding (i.e., ∥Tu ? Tv∥ ? ∥u ? v∥ for all u, v?X) and such that T(X) has a nonempty interior. Does this guarantee that T is onto? We give a counterexample in the case of X=L¹(N).     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Calculation of the Shannon information

Let ($\text{Ω, β, μ}{}_{\mathrm{X}}$) and $\text{(?, F, μ}{}_{\mathrm{N}}\text{)}$ be probability spaces, with $\text{f: Ω × ? ? ?}\text{a}\text{β × F|F}$ measurable map. Define μ_XY on β × $\text{F}$ by μ_XY(A) = μ_X ? μ_N{(x, y): (x, f(x, y)) ?A}, and let μ_Y = (μ_X ? μ_N)^of^?1. An expression is determined for computing the Shannon information in the measure μ_XY. This expression is used to compute the information for the non-linear additive Gaussian channel, and can be used to solve the channel capacity problem.