Cones In Hilbert Spaces Research Articles

Asymptotics of the sticky particles evolution

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.

Operator Hilbert Systems

The present note deals with operator Hilbert systems, which are quantizations of unital cones in Hilbert spaces. One central result of the note is that the Pisier operator Hilbert space is an operator system whose quantum cone of positive elements is described in terms of the quantum ball of the relevant conjugate Hilbert space. Finally, we obtain a solution to the problem of Paulsen, Todorov and Tomforde on separable morphisms between operator systems and characterize minmax- completely positive maps between Archimedean order unit spaces.

Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all

Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low-dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the 1-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). In this paper, we establish generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees are illustrated on selected examples. For block structured sparsity in the infinite-dimensional setting, we use the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art.

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].

Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x +. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.

Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems

Conditions for the non-existence of a regular exceptional family of elements with respect to an isotone projection cone in a Hilbert space will be presented. The obtained results will be used for generating existence theorems for a complementarity problem with respect to an isotone projection cone in a Hilbert space.

Normality and modulability indices. Part II: Convex cones in Hilbert spaces

Let K be a closed convex cone in a Hilbert space X. Let B X be the closed unit ball of X and K • = ( B X + K ) ∩ ( B X − K ) . The normality index ν ( K ) = sup { r ⩾ 0 : r K • ⊂ B X } is a coefficient that measures to which extent the cone K is normal. We establish a formula that relates ν ( K ) to the maximal angle of K. A concept dual to normality is that of modulability. As a by-product one obtains a formula for computing the modulability index μ ( K ) = sup { r ⩾ 0 : r B X ⊂ K • } of K. The symbol K • stands for the absolutely convex hull of K ∩ B X . We show that μ ( K ) can be expressed in terms of the smallest critical angle of K.

Sharp bounds for expectations of spacings from decreasing density and failure rate families

We apply the method of projecting functions onto convex cones in Hilbert spaces to derive sharp upper bounds for the expectations of spacings from i.i.d. samples coming from restricted families of distributions. Two families are considered: distributions

Sharp bounds for expectations of spacings from DDA and DFRA families

We present sharp upper bounds for the expectations of spacings from i.i.d. samples coming from restricted families of distributions. Two families are considered: distributions with decreasing density on the average and with decreasing failure rate on the average. We also characterize the distributions attaining the bounds. The results are obtained by projecting functions onto convex cones in Hilbert spaces.

Description for All Regular Cones in Hilbert Space

Description for All Regular Cones in Hilbert Space

For cones in Hilbert spaces, regularity is equivalent to self-duality

The theorem stating that a cone in a Hilbert space is regular if and only if it is self-dual is proved and applied to obtain new proofs of earlier results.

Proximal and Fréchet normal formulae for some small normal cones in Hilbert space

Proximal and Fréchet normal formulae for some small normal cones in Hilbert space

Projections onto convex cones in Hilbert space

Given a closed convex cone C in a Hilbert space H, we investigate the function which assigns to each point x in H the nearest point of C to x. We call this function the projection of H onto C and we give an algebraic characterization of this function which generalizes the well-known characterization of a projection onto a closed subspace as an idempotent, symmetric linear operator.

Minimization of quadratic functionals on cones in Hilbert spaces

This paper concerns a problem of minimization of a quadratic functional on a cone in a Hilbert space. First, a simple auxiliary minimization in R 2, is solved. The obtained results allow the construction of many different algorithms that solve the primary minimization problem. However, only two algorithms are considered in the paper. The first one concerns a general case, i.e., the case when the only restrictions imposed on the cone are its closeness and convexity. This algorithm uses a projection on the cone technique. The second one can be applied if it is assumed, additionally, that the considered Hilbert space is separable and has Riesz basis. The cone is, in this case, defined as a collection of points whose expansions with respect to this basis have nonnegative coefficients. Weak convergence of both algorithms is proved. The rates of their convergence are estimated by the convergence of certain series of reals. Possible applications to an optimal control problem are suggested in the introduction and in the conclusion.

A CONDITION OF JACOBI TYPE FOR A QUADRATIC FORM TO BE NONNEGATIVE ON A FINITE-FACED CONE

The author considers the problem of nonnegativity and positive definiteness of a Legendre quadratic form on a family of cones in Hilbert space that are the intersection of a fixed finite-faced cone with a monotonically increasing family of subspaces. This problem arises in the investigation to second order of an extremal in the Lagrange-Mayer-Bolza problem of classical variational calculus in the presence of finitely many additional constraints in the form of end and integral inequalities when there is only one set of Lagrange multipliers for the given extremal. It is shown that a complete answer can be given in terms of solutions of the corresponding Euler-Jacobi equation. Bibliography: 5 titles.

Quasinormality of cones in Hilbert spaces

We prove that every cone in a Hilbert space satisfies an optimal quasinormality condition and obtain an eigenvalue theorem for k-set contractions on cones.

Obtuse cones in Hilbert spaces and applications to partial differential equations

The positive cone K in a partially ordered Hilbert space is said to be obtuse with respect to the inner product if the dual cone K ∗ ⊂ K . Obtuseness of cones with respect to non-symmetric bilinear forms is also defined and characterized. These results are applied to the generalized Sobolev space associated with an elliptic boundary value problem, in particular to the question of determining the non-negativity of the Green's function. A notion of strict obtuseness is defined, characterized and applied to the question of strict positivity of the Green's function. Applications to positivity preserving semi-groups are also given.

A classification for selfdual cones in Hilbert space

A classification for selfdual cones in Hilbert space

A penalty method for the approximation of projections over cones in Hilbert spaces

Given a closed convex coneK in a Hilbert spaceH and a vectoru 0 ∈H, a penalty method is built up in order to approximate the projection ofu 0 over the polar coneK * ofK, without making use of the inverse transform of the canonical mapping ofH into its dual spaceH′. Such method is outlined in n0 1, 2. In n03 a complete analysis of the errors of the method is explained. In n04 the method is applied to find error bounds for the numerical approximation of the projection ofu 0 onK.

Direct integrals on selfdual cones and standard forms of von Neumann algebras

Connes [6] and Haagerup [11] established the relation between von Neumann algebras (abbreviate: VNA) and selfdual, homogenous and orientated cones in Hilbert space. The ideal center ofa selfdual cone in a Hilbert space is the hermitian part of an abelian VNA [3]. This gives rise to direct integral decompositions of selfdual cones with respect to von Neumann subalgebras of the VNA generated by the ideal center (see w Theorem 3.17). The existence of a "central" decomposition has been shown in [18], too. The purpose of the present paper is the study of the properties of such decompositions (w 3). Especially, we determine the facial structure of a direct integral of selfdual cones (w 3, Proposition 3.5, Lemma 3.6, 3.7, Corollary 3.8). It follows rough ly spokenthat all properties of selfdual cones, which are expressed by means of facial projections, hold for a direct integral of selfdual cones iff they hold for the fibres (w 3, Corollary 3.9, 3.10, Remark 3.11). Further, this is used to study the connection between direct integrals of VNA and the associated selfdual, homogenous and orientated cones, i.e. the associated standard forms (w Proposition 3.14, Theorem 3.15). Another consequence is the decomposibility of certain linear maps between direct integrals of VNA (w Theorem 4.1). Direct integrals of standard forms of VNA have also been considered in [13, 15] and [21]. But the approach to direct integrals of standard forms of VNA given there is based on the decomposition theory for left Hilbert algebras, which isn't used in this paper. The paper is subdivided into four sections. In the preliminaries (w 1) we recall basis facts on selfdual cones in Hilbert space and the terminology needed here. w 2 is devoted to the theory of direct sums. In w 3 and w 4 we develop the decomposition theory for selfdual cones and present as mentioned above applications to von Neumann algebra theory.