Positivity最新文献

We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor (mathcal {M}) as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function (p mapsto alpha (tau (p))) for projections (p in mathcal {M}), which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type, respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions (alpha ) are concave. For the algebras of compact operators and factors of type (textrm{II}), we completely determine the condition that the associated weighted (L^p)-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions (alpha ) for any (0< p < infty ). We also show that any non-linear trace of the Sugeno type yields a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors.

The concept of weak orthogonality of order p ((1 le p le infty )) in Banach lattices is introduced in order to obtain spaces with the weak fixed point property of order p. Moreover, various connections between a number of Banach space properties to imply the weak fixed point property, such as Opial condition, weak normal structure and property (M) are investigated. In particular, it is established that for each Banach space X and a suitable Banach lattice F, a Banach lattice (mathcal {M}subset K(X,F)) has the weak fixed point property of order p, if each evaluation operator (psi _{y^*}) on (mathcal {M}) is a p-convergent operator for (y^*in F^*).

In this paper we investigate the polynomial-like iterative equation on Riesz spaces. Since a Riesz space does not need to have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using the Knaster–Tarski fixed point theorem, we first obtain the existence and uniqueness of order-preserving solutions on convex complete sublattices of Riesz spaces. Then, restricting to (mathbb {R}) and (mathbb {R}^n), special cases of Riesz space, we obtain semi-continuous solutions and integrable solutions, respectively. Finally, we present more special cases of Riesz space in which solutions to the iterative equation can be discussed.

The primary aim of this paper is to establish bounds on the joint spectral radius for a finite set of nonnegative matrices based on their diagonal elements. The efficacy of this approach is evaluated in comparison to existing and related results in the field. In particular, let (Sigma ) be any finite set of (Dtimes D) nonnegative matrices with the largest value U and the smallest value V over all positive entries. For each (i=1,ldots ,D), let (m_i) be any number so that there exist (A_1,ldots ,A_{m_i}in Sigma ) satisfying ((A_1ldots A_{m_i})_{i,i} > 0), or let (m_i=1) if there are no such matrices. We prove that the joint spectral radius (rho (Sigma )) is bounded by

Let C(X) be the algebra of all real-valued continuous functions on a Tychonoff space X, and (C^*(X)) the subalgebra of bounded functions. We prove that if B is any subalgebra of C(X) containing (C^*(X)), then no maximal solid subspace of B contains (C^*(X)), and we derive from this that the maximal solid subspaces of B are exactly the real maximal ideals of B. Then we extend the above to the case of intermediate algebras in A, where A is a (varPhi )-algebra with bounded inversion.

Let A be a positive semidefinite matrix. It is known that the Hadamard exponential of A is positive semidefinite; it is positive definite if and only if no two columns of A are identical. We give an alternative proof of the latter part with an application to Hadamard inverses.

We shall give a characterization for the strong and weak type Adams type boundedness of the fractional maximal operator (M_{alpha }) on total Morrey spaces (L^{p,lambda ,mu }(mathbb {R}^n)), respectively. Also we give necessary and sufficient conditions for the boundedness of the fractional maximal commutator operator (M_{b,alpha }) and commutator of fractional maximal operator ([b,M_{alpha }]) on (L^{p,lambda ,mu }(mathbb {R}^n)) when b belongs to (BMO(mathbb {R}^n)) spaces, whereby some new characterizations for certain subclasses of (BMO(mathbb {R}^n)) spaces are obtained.

We define a free uniformly complete vector lattice (text {FUCVL}(A)) over a non-empty set A and give its representation as a sublattice of the space (H(mathbb {R}^A)) of continuous in the product topology positively homogeneous functions on (mathbb {R}^A).

This paper deals with a certain class of multiobjective semi-infinite programming problems with switching constraints (in short, MSIPSC) in the framework of Hadamard manifolds. We introduce Abadie constraint qualification (in short, ACQ) for MSIPSC in the Hadamard manifold setting. Necessary criteria of weak Pareto efficiency for MSIPSC are derived by employing ACQ. Further, sufficient criteria of weak Pareto efficiency for MSIPSC are deduced by using geodesic quasiconvexity and pseudoconvexity assumptions. Subsequently, Mond–Weir type and Wolfe type dual models are formulated related to the primal problem MSIPSC, and thereafter, several duality results are established that relate MSIPSC and the corresponding dual models. Several non-trivial examples are furnished in the framework of well-known Hadamard manifolds, such as the set consisting of all symmetric positive definite matrices and the Poincaré half plane, to illustrate the importance of the results derived in this article. To the best of our knowledge, this is the first time that optimality conditions and duality results for MSIPSC have been studied in the setting of Hadamard manifolds.

This paper investigates second-order optimality conditions for general constrained set-valued optimization problems in normed vector spaces under the set criterion. To this aim we introduce several new concepts of second-order directional derivatives for set-valued maps by means of excess from a set to another one, and discuss some of their properties. By virtue of these directional derivatives and by adopting the notion of set criterion intoduced by Kuroiwa, we obtain second-order necessary and sufficient optimality conditions in the primal form. Moreover, under some additional assumptions we obtain dual second-order necessary optimality conditions in terms of Lagrange–Fritz–John and in terms of Lagrange–Karush–Kuhn–Tucker multipliers.