Unbounded generalizations of left Hilbert algebras,II

The first purpose of this paper is to study the classification of unbounded left Hilbert algebras. The second purpose is to investigate the unbounded left Hilbert algebras generated by positive linear functional on a ¹-algebra. The final purpose is to study the classification of positive linear functionals.     

The singularity of positive linear functionals

Our purpose is to show that the various concepts of singularity of representable positive functionals on ^?-algebras coincide, moreover to present a new characterization of singularity by means of Choquet theory of the state space. In the context of singularity, the paper includes an equivalent condition for a representable positive functional to be pure.     

Topologies on BP-algebras

The class of locally convex ^*-algebra topologies on a BP^*-algebra which possess the same bounded hermitian idempotent subsets is considered and is shown to have a finest element. The cone of positive elements of a symmetric BP^*-algebra is studied and is seen to be closed in this finest topology. Order-bounded BP^*-algebras are considered, and it is seen that the positive linear functionals span the dual of such an algebra. The class of equivalent topologies on an order-bounded commutative BP^*-algebra for which every positive linear functional is continuous is considered, and there are found to be such topologies which are neither barreled nor Q-topologies, so the results of [6.], 15–28) are extended.     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998     

Ein globales Ergebnis für Bifurkationsaufgaben mit schwach differenzierbaren Operatoren

In this paper we are dealing with positive linear functionals on W-algebras. We introduce the notion of a positive linear functional with ∑-property (see Definition 1.1). It is shown that each positive linear functional on a W-algebra possesses the ∑-property. This fact allows to give a short proof of UHLMANN's conjecture on unitary mixing in the state space of a W-algebra. In proving our main theorem (see Theorem 1.2.) we obtain some results on positive linear functionals and orthoprojections which are useful in other context, too.     

Dividierte Differenzen und Monotonie von Quadraturformeln

Summary As a generalisation of divided differences we consider linear functionals vanishing for polynomials of given degree and with discrete support. It is shown that functionals of that type may be uniquely represented by a linear combination of divided differences. On the basis of this representation theorem we introduce the concept of positivity and definiteness of functions and linear functionals. Next we show that in many cases positivity follows from the number of sign changes of the coefficients of the given linear functional. These results may be applied to the problems of nonexistence of Newton-Côtes and Gegenbauer quadrature formulas with positive weights and to the monotony problem of Gauss and Newton Côtes quadrature.

     

Characterization of the D w -Laguerre-Hahn Functionals

We give some characterization theorems for the D w -Laguerre-Hahn linear functionals and we extend the concept of the class of the usual Laguerre-Hahn functionals to the D w -Laguerre-Hahn functionals, recovering the classic results when w tends to zero. Moreover, we show that some transformations carried out on the D w -Laguerre-Hahn linear functionals lead to new D w -Laguerre-Hahn linear functionals. Finally, we analyze the class of the resulting functionals and we give some applications relative to the first associated Charlier, Meixner, Krawtchouk and Hahn orthogonal polynomials.     

Transform formulae for linear functionals of affine processes and their bridges on positive semidefinite matrices

In this paper, we derive transform formulae for linear functionals of affine processes and their bridges whose state space is the set of positive semidefinite d×d$d\times d$ matrices. Particularly, we investigate the relationship between such transforms and certain integral equations. Our findings extend and unify the well known results of Cuchiero et al. (2011) [5] and Pitman and Yor (1982) [19], who analysed affine processes on positive semidefinite matrices and transforms of linear functionals of squared Bessel processes, respectively. We are, then, able to derive analytic expressions for Laplace transforms of some functionals of Wishart bridges.     

Generalized gradients of Lipschitz functionals

The purpose of this article is threefold: (i) to present in a unified fashion the theory of generalized gradients, whose elements are at present scattered in various sources; (ii) to give an account of the ways in which the theory has been applied; (iii) to prove new results concerning generalized gradients of summation functionals, pointwise maxima, and integral functionals on subspaces of L^∞. These last-mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control. Their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by Rockafellar, Ioffe and Levin, Valadier, and others.     

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.     

Generalized Brownian functionals and stochastic integrals

The purpose of this paper is two-fold; i) a new class of generalized Brownian functionals, in fact generalized linear functionals, is introduced and ii) generalized stochastic integrals based on creation operators are discussed. These topics are in line with the causal calculus of Brownian functionals.Communicated by H. H. Kuo     

Characterization of radon integrals as linear functionals

The problem of characterization of integrals as linear functionals is considered in this paper. It has its origin in the well-known result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and is directly connected with the famous theorem of J. Radon (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact in ? ⁿ . After the works of J. Radon, M. Fréchet, and F. Hausdorff, the problem of characterization of integrals as linear functionals has been concretized as the problem of extension of Radon??s theorem from ? ⁿ to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and abundant history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such eminent mathematicians as S. Banach (1937?C38), S. Saks (1937?C38), S. Kakutani (1941), P. Halmos (1950), E. Hewitt (1952), R. E. Edwards (1953), Yu. V. Prokhorov (1956), N. Bourbaki (1969), H. K¨onig (1995), V. K. Zakharov and A. V. Mikhalev (1997), et al. Essential ideas and technical tools were worked out by A. D. Alexandrov (1940?C43), M. N. Stone (1948?C49), D. H. Fremlin (1974), et al. The article is devoted to the modern stage of solving this problem connected with the works of the authors (1997?C2009). The solution of the problem is presented in the form of the parametric theorems on characterization of integrals. These theorems immediately imply characterization theorems of the above-mentioned authors.     

A superreflexive banach spaceX withL(X) admitting a homomorphism onto the Banach algebraC(βN)

A separable superreflexive Banach spaceX is constructed such that the Banach algebraL(X) of all continuous endomorphisms ofX admits a continuous homomorphism onto the Banach algebraC(βN) of all scalar valued functions on the Stone-Čech compacification of the positive integers with supremum norm. In particular: (i) the cardinality of the set of all linear multiplicative functionals onL(X) is equal to 2^c and (ii)X is not isomorphic to any finite Cartesian power of any Banach space.     

Oscillation theorems for self-adjoint matrix Hamiltonian systems involving general means

By use of monotone functionals and positive linear functionals, a generalized Riccati transformation and the general means technique, some new oscillation criteria for the following self-adjoint Hamiltonian matrix system

(E)     

Solution of a Ball–Murat problem

A necessary and sufficient condition for the W ^{1, p} -quasi-convexity of integrands to imply the lower semicontinuity of the corresponding integral functionals with respect to the weak convergence of sequences in W ^{1, p} is obtained. It is shown that the absence of the Lavrent’ev phenomenon in minimization problems with linear boundary data is sufficient, under a minor technical assumption, for the lower semicontinuity of integral functionals with quasi-convex integrands.     

Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone K⊂RΩ admits a finitely additive probability m such that supk∈Km(k)?0. Based on these, we characterise those linear functionals that are representable as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.     

A short proof on the cardinality of maximal positive bases

A positive basis is a minimal set of vectors whose nonnegative linear combinations span the entire space \mathbb Rⁿ{\mathbb R^{n}}. Interest in positive bases was revived in the late nineties by the introduction and analysis of some classes of direct search optimization algorithms. It is easily shown that the cardinality of every positive basis is bounded below by n + 1. There are proofs in the literature that 2n is a valid upper bound for the cardinality, but these proofs are quite technical and require several pages. The purpose of this note is to provide a simple demonstration that relies on a fundamental property of basic feasible solutions in linear programming theory.     

Integral type linear functionals on ordered cones

We introduce linear functionals on an ordered cone that are minimal with respect to a given subcone. Using concepts developed for Choquet theory we observe that the properties of these functionals resemble those of positive Radon measures on locally compact spaces. Other applications include monotone functionals on cones of convex sets, H-integrals on H-cones in abstract potential theory, and classical Choquet theory itself.

     

The Arrow-Barankin-Blackwell theorem in a dual space setting

In 1953 Arrow, Barankin, and Blackwell proved that, ifR ⁿ is equipped with its natural ordering and ifF is a closed convex subset ofR ⁿ , then the set of points inF that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points ofF. Many generalizations of this density result to infinite-dimensional settings have been given. In this note, we consider the particular setting where the setF is contained in the topological dualY ^* of a partially ordered, nonreflexive normed spaceY, and the support functionals are restricted to be either nonnegative or strictly positive elements in the canonical embedding ofY inY ^*. Three alternative density results are obtained, two of which generalize a space-specific result due to Majumdar for the dual system (Y,Y ^*)=(L ¹,L ).This research was supported in part by funds provided by the Provident Chair of Excellence in Applied Mathematics at the University of Tennessee, Chattanooga, Tennessee.