Vector state inequalities related to the Hölder–McCarthy inequality and applications

Weighted least squares solutions of the equation AXB − C = 0

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

On the positivity and stability of non-stationary systems

If X and Y are Banach lattices then there are several spaces of linear operators between them that may be studied. $$ \mathcal{L}^r $$ (X, Y) is the space of all norm bounded operators from X into Y. There is no reason to expect there to be any connection between the order structure of X and Y and that of $$ \mathcal{L} $$ (X, Y). $$ \mathcal{L}^r $$ (X, Y) is the space of regular operators, i.e., the linear span of the positive operators. This at least has the merit that when it is ordered by the cone of positive operators then that cone is generating. $$ \mathcal{L}^b $$ (X, Y) is the space of order bounded operators, which are those that map order bounded sets in X to order bounded sets in Y. We always have $$ \mathcal{L}^r (X,Y) \subseteq \mathcal{L}^b (X,Y) \subseteq \mathcal{L}(X,Y) $$ and both inclusions may be proper.

Different equivalence relations are defined in the set L(H) s of selfadjoint operators of a Hilbert space H in order to extend a very well known relation in the cone of positive operators. As in the positive case, for a ∈ L(H) s the equivalence class C a admits a differential structure, which is compatible with a complete metric defined on C a . This metric coincides with the Thompson metric when a is positive.

In this paper, we present an algorithmic approach to the construction of Lyapunov functions for infinite-dimensional systems. This paper unifies and significantly extends many previous results which have appeared in conference and journal format. The unifying principle is that any linear parametrization of operators in Hilbert space can be used to construct an LMI parametrization of positive operators via squared representations. For linear systems, we get positive linear operators and hence quadratic Lyapunov functions. For nonlinear systems, we get nonlinear operators and hence non-quadratic Lyapunov functions. Special cases of these results include positive operators defined by multipliers and kernels which are: polynomial; piecewise-polynomial; or semi-separable and apply to systems with delay; multiple spatial domains; or mixed boundary conditions. We also introduce a set of efficient software tools for creating these functionals. Finally, we illustrate the approach with numerical examples.

A closed convex cone Kin a finite-dimensional real inner product space Vis said to be symmetric relative to subspace Lof V, or L-symmetric, if x− y∈ Kwhenever x∈ L y∈ L ⊥and x+ y∈ K. Fiedler and Haynsworth have shown that a full pointed cone Kis symmetric relative to a one-dimensional subspace Liff it is “top heavy” relative to some norm ν on L ⊥, i.e., it has the form {x 1 e 1+y| x 1⩾ v(y)}for some e∈ K∩ L. This result is first extended to arbitrary L-symmetric cones using extended seminorms. Cones top heavy relative to vectorial norms are discussed. Finally, it is shown that the cone of positive operators on a given L-symmetric cone is itself symmetric relative to a subspace of operators determined by L.

Lyapunov-like transformation/matrix on a cone appears in the theory of dynamical systems and linear complementarity problems. The set of all Lyapunov-like transformations on a proper cone in a finite dimensional inner product space is the Lie algebra of the automorphism group of that cone. The dimension of this Lie algebra is called the Lyapunov rank. A pair of proper cones is said to be a nuclear pair if one of them is simplicial. In this paper, we find the Lyapunov rank and Lyapunov-like transformations on the tensor product of nuclear pairs of cones. Further, we prove that the space of Lyapunov-like transformations on the tensor product of a nuclear pair is the tensor product of the spaces of Lyapunov-like transformations on the individual cones. As a consequence, given a nuclear pair $(K_1,K_2)$, we describe the space of Lyapunov-like transformations on the cone of positive operators between $K_1$ and $K_2$.

The seminal work of Kubo and Ando (Math Ann 246:205–224, 1979/80) provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavour. On the other hand, it is highly natural to take the geomeric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.

-monotone symplectic operators has a polar decomposition H exp W and a triple decomposition Tau+ HTau-. The aim of the present paper is to give a complete representation which decomposes exp W into the triple decomposition Tau+ HTau-. By the semigroup property of Tau+ HTau-, we obtain a binary operation on the cone of positive operators whose automorphism group is the orthogonal group O(n).

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.

In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Lowner order between operators to the pointwise order between functions, another two turn the spectral order between operators to the same, pointwise order between functions. We investigate which algebraic operations those representations preserve, hence which kind of algebraic structure the representing functions have. We study the differences among the different representing functions of the same positive operator. Finally, we introduce a new complete metric (which corresponds naturally to two of those representations) on the set of all invertible positive operators and formulate a conjecture concerning the corresponding isometry group.

The parallel sum $${A : B}$$ of two bounded positive linear operators A, B on a Hilbert space H is defined to be the positive operator having the quadratic form $$\inf{(A(x - y) | (x - y) + ({By} | {y}) {y \in H}} $$ for fixed $${x \in H}$$ . The purpose of this paper is to provide a factorization of the parallel sum of the form $${J_A PJ_A^*}$$ where $${J_A}$$ is the embedding operator of an auxiliary Hilbert space associated with A and B, and P is an orthogonal projection onto a certain linear subspace of that Hilbert space. We give similar factorizations of the parallel sum of nonnegative Hermitian forms, positive operators of a complex Banach space E into its topological anti-dual $${\bar{E}^{\prime}}$$ , and of representable positive functionals on a $${^*}$$ -algebra.

The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of pos- itive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition un- der which any positive finite-rank operator B can be expressed as a sum of rank-one operators with norms specified by a sequence of positive numbers {ci}. Equivalently, this result proves the existence of a frame with B as it's frame operator and with vector norms { p ci}. We further prove that, given a non-compact positive operator B on an infinite di- mensional separable real or complex Hilbert space, and given an infinite sequence {ci} of positive real numbers which has infinite sum and which has supremum strictly less than the essential norm of B, there is a se- quence of rank-one positive operators, with norms given by {ci}, which sum to B in the strong operator topology. These results generalize results by Casazza, Kovaycevic, Leon, and Tremain, in which the operator is a scalar multiple of the identity op- erator (or equivalently the frame is a tight frame), and also results by Dykema, Freeman, Kornelson, Larson, Ordower, and Weber in which {ci} is a constant sequence.