Coordinate-wise Order Research Articles

Fine spectra and compactness of generalized Cesàro operators in Banach lattices in [formula omitted

The generalized Cesàro operators Ct, for t∈[0,1), introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro averaging operator C1 and act in various Banach sequence spaces X⊆CN0. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invariant sequence spaces, various weighted c0 and ℓp spaces and many others. In such Banach lattices X the operators Ct, for t∈[0,1), are always compact (unlike C1) and a full description of their point, continuous and residual spectrum is given. Estimates for the operator norm of Ct are also presented.

Order spectrum of the Cesàro operator in Banach lattice sequence spaces

The discrete Cesaro operator C acts continuously in various classical Banach sequence spaces within $$ {\mathbb {C}}^{{\mathbb {N}}}.$$ For the coordinatewise order, many such sequence spaces X are also complex Banach lattices [eg. $$c_0, \ell ^p $$ for $$ 1 < p \le \infty , $$ and $$ {{\text {ces}}}(p)$$ for $$ p \in \{ 0 \} \cup ( 1, \infty )$$]. In such Banach lattice sequence spaces, C is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on X. The purpose of this note is to show, for every X belonging to the above list of Banach lattice sequence spaces, that the order spectrum $$ \sigma _\mathrm{o} (C)$$ of Ccoincides with its usual spectrum $$ \sigma ( C)$$ when C is considered as a continuous linear operator on the Banach space X.

Martingale-like sequences in Banach lattices

Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437--456]. In these frameworks, a collection of bounded $X$-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a normed lattice but not necessarily a Banach space under the supremum norm.

Isotonic regression and isotonic projection

The note describes the cones in the Euclidean space admitting isotonic metric projection with respect to the coordinate-wise ordering. As a consequence it is shown that the metric projection onto the isotonic regression cone (the cone defined by the general isotonic regression problem) admits a projection which is isotonic with respect to the coordinate-wise ordering.

Solid Extensions of the Cesàro Operator on ℓ p and c 0

We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of \({\mathbb{C}^\mathbb{N}}\) and to which the classical Cesaro operator \({\mathcal{C}\colon\ell^p \to \ell^p}\) (a positive operator) can be continuously extended while still maintaining its values in lp. Properties of this optimal Banach lattice \({[\mathcal{C}, \ell^p]_s}\) are presented. In addition, all continuous convolution operators of \({[\mathcal{C}, \ell^p]_s}\) into itself are identified and the spectrum of \({\mathcal{C}\colon[\mathcal{C}, \ell^p]_s \to[\mathcal{C}, \ell^p]_s}\) is determined. A similar investigation is undertaken for the Cesaro operator \({\mathcal{C}\colon c_0\to c_0}\).

Real Dimension Groups

AbstractDimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Random Partial Orders Defined by Angular Domains

The d-dimensional random partial order is the intersection of d independently and uniformly chosen (with replacement) linear orders on the set [n] = {1, 2, . . . , n}. This is equivalent to picking n points uniformly at random in the d-dimensional unit cube \(Q_d=[0,1]^d\) with the coordinate-wise ordering. If d = 2, then this can be rephrased by declaring that for any pair P1, P2 ∈ Q2 we have P1 ≺ P2 if and only if P2 lies in the positive upper quadrant defined by the two axis-parallel lines crossing at P1. In this paper we study the random partial order with parameter α (0 ≤ α ≤ π) which is generated by picking n points uniformly at random from Q2 equipped with the same partial order as above but with the quadrant replaced by an angular domain of angle α.

Multiobjective problems of convex geometry

UDC 514.172:517.982:519.81:519.855 Abstract: Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Pareto-optimal solutions of isoperimetric-type vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes. The birth of the theory of extremal problems is usually tied with the mythical Phoenician Princess Dido. Virgil told about the escape of Dido from her treacherous brother in the first chapter of The Aeneid. Dido had to decide about the choice of a tract of land near the future city of Carthage, while satisfying the famous constraint of selecting "a space of ground, which (Byrsa call'd, from the bull's hide) they first inclos'd." By the legend, Phoenicians cut the oxhide into thin strips and enclosed a large expanse. Now it is customary to think that the decision by Dido was reduced to the isoperimetric problem of finding a figure of greatest area among those surrounded by a curve whose length is given. It is not excluded that Dido and her subjects solved the practical versions of the problem when the tower was to be located at the sea coast and part of the boundary coastline of the tract was somehow prescribed in advance. The foundation of Carthage is usually dated to the ninth century bce when there was no hint of the Euclidean geometry, the cadastral surveying was the job of harpedonaptae, and measuring the tracts of land was used in decision making. Rope-stretching around stakes leads to convex figures. The Dido problem has a unique solution in the class of convex figures provided that the fixed nonempty part of the boundary is a convex polygonal line. Decision making has become a science in the twentieth century. The presence of many contradictory conditions and conflicting interests is the main particularity of the social situations under control of today. Management by objectives is an exceptional instance of the stock of rather complicated humanitarian problems of goal agreement which has no candidates for a unique solution. The extremal problems of optimizing several parameters simultaneously are collected nowadays un- der the auspices of vector or multiobjective optimization. Search for control in these circumstances is multiple criteria decision making. The mathematical apparatus of these areas of research is not rather sophisticated at present (see (1, 2) and the references therein). The today's research deals mostly with the concept of Pareto optimality (e.g., (3-6)). Let us explain this approach by the example of a bunch of economic agents each of which intends to maximize his own income. The Pareto efficiency principle asserts that as an effective agreement of the conflicting goals it is reasonable to take any state in which nobody can increase his income in any way other than diminishing the income of at least one of the other fellow members. Formally speaking, this implies the search of maximal elements of the set comprising the tuples of incomes of the agents at every state; i.e., some vectors of a finite-dimensional arithmetic space endowed with the coordinatewise order. Clearly, the concept of Pareto optimality was already abstracted to arbitrary ordered vector spaces (for more detail see (7-10)).

Lattice of Keychains

In this paper we consider the set of all n + 1-tuples of real numbers, not necessarily all distinct, in the decreasing order from the unit interval under the usual ordering of real numbers, always including 1. Such n+1-tuples inherently arise as the membership values of fuzzy subsets and are called keychains. An natural equivalence relation is introduced on this set and the equivalence classes of keychains are studied here. The number of such keychains is finite and the set of all keychains is a lattice under the coordinate-wise ordering. Thus keychains are subchains of a finite chain of real numbers in the unit interval. We study some of their properties and give some applications to counting fuzzy subsets of finite sets.

Fields of quotients of lattice-ordered domains

It is not known whether the field of fractions of an integral domain with a compatible lattice order has a compatible lattice order that extends the given order on the integral domain. The polynomial ring $$\mathbb{R}[x,x^{ - 1} ]$$ over the real numbers $$\mathbb{R}$$ has a natural compatible lattice order, viz, the coordinatewise order ≥. We describe circumstances in which the field of fractions of $$(\mathbb{R}[x,x^{ - 1} ], + , \cdot , \geq )$$ has no archimedean lattice order that extends ≥.

Shortest Paths Through Pseudo-Random Points in the d-Cube

A lower bound for the length of the shortest path through n points in [0, Ild is given in terms of the discrepancy function of the n points. This bound is applied to obtain an analogue for several pseudorandom sequences to the known limit behavior of the length of the shortest path through n independent uniformly distributed random observations from [0, lJd. Introduction For a sequence {z,: 1 1, to be considered as a candidate for a sequence of pseudorandom observations, it must at least be uniformly distributed in the sense that the empirical frequency of any subcube must tend to the volume of that subcube [5, pp. 127-157]. For such a minimally pseudorandom sequence, it seems of interest to determine the extent to which additional features of independent uniformly distributed random variables must also hold. The additional feature which is considered in this note is the asymptotic growth rate of the shortest path through the initial sample { Z1, Z2, . . .* Zn)} For a sequence {x,, 1 0. While a result of such precision is too much to expect of a general pseudorandom sequence, a somewhat weaker version of (1) can be obtained in sufficient generality to cover a variety of classical cases. To make this precise, first suppose [0, I]d is partially ordered (<) by the usual coordinatewise ordering, and for each x E [0, j]d ,define the empirical distribution function Fn(x) = (I/n)#{1 < i < n: z, < x} of the sequence {z,: 1 < i < oo). The discrepancy function Dn is then defined by D = sup IF (x)F(x)I x E [O, ]d Received by the editors June 29, 1979 and, in revised form, November 8, 1979. 1980 Mathematics Subject Classification. Primary 65C10, 90B10, 1OF40, 60D05.

A divergent, two-parameter, bounded martingale

An example is given of a divergent, uniformly bounded martingale X = { X t : t ∈ T } X = \{ {X_t}:t \in T\} where the index t ranges over the set T of pairs of positive integers with the usual coordinatewise ordering.