Positivity

This paper introduces lacunary statistical p-decreasing sequences and lacunary statistical p-convergence in lattice-normed spaces by combining order convergence of vector-valued norms with lacunary density. These notions extend statistical p-convergence to a lacunary setting in ordered vector structures. Fundamental structural properties are established, including linearity, uniqueness of limits, and compatibility with lattice operations in lattice-normed Riesz spaces. Inclusion relations between statistical p-convergence and its lacunary counterpart are characterized in terms of the behavior of lacunary sequences. Moreover, we present a counterexample showing that a natural mean-type order convergence condition does not suffice for lacunary statistical p-convergence in general.

We summarize the existing state of knowledge, which has changed little in recent years, of compact linear operators between Banach lattices and point out various unanswered questions that seem still to be of interest.

Let (Ω ,Σ ,μ ) and (Ψ ,Ξ ,ν ) be finite measure spaces. We study the compactness of the integral operator (Kf)(t)=∫ _Ω K(t,τ )f(τ ) dμ (τ ), t∈Ψ , with a kernel K∈ L^∞ (Ψ×Ω ,ν⊗μ ) , acting from a Banach function space X=X(Ω ,μ ) to a Banach function space Y=Y(Ψ ,ν ) . Let X'=X'(Ω ,μ ) be the associate space of X, let (X')_b and Y_b be the closures of simple functions in X' and Y, respectively, and let (X')_a and Y_a be the subspaces of functions of absolutely continuous norms of X' and Y, respectively. We show that (1) if (X')_a=(X')_b and Y_a=Y_b , then K: X→ Y is compact; (2) if ν is nonatomic and (X')_a (X')_b or μ is nonatomic and Y_a Y_b , then there exists a kernel K∈ L^∞ (Ψ×Ω ,ν⊗μ ) such that K:X→ Y is noncompact.

In a paper of Gupta and Aral, a modified Szász–Mirakyan–Kantorovich operator was introduced in such a way that it reproduces constants and the exponential function e^-x on [0,∞ ) . We observe that the parameter imposed by this reproduction property becomes negative near the origin; consequently, the resulting operator is not positive. We show more generally that within the same Poisson–Kantorovich structure, preservation of constants and of an exponential e^λ x , λ 0 , is incompatible with positivity of the operator. We then discuss two possible remedies. First, the original operator K_n defined by Gupta and Aral maps nonnegative functions to functions which are nonnegative on [δ ,∞ ) for each fixed δ >0 and sufficiently large n; we refer to this property as δ -positivity. Using this, the approximation properties can be recovered rigorously away from the origin. Second, a positive auxiliary modification is obtained by truncating the negative part of the parameter. Finally, corrected proofs of the main approximation results are stated in a self-contained form. At the end, we introduce and investigate positive linear operators that reproduce constants and the exponential function e^λ x, λ 0 , on [0,∞ ) .

This paper investigates extended well-posedness for a class of set optimization problems with an infinite number of constraints. Under suitable continuity assumptions, well-posedness is examined with respect to perturbations generated by sequences of constrained set optimization problems. We propose an appropriate formulation of asymptotically minimizing sequences, wherein both the objective and constraint maps undergo perturbations. Sufficient conditions for the extended well-posedness properties are established in terms of these sequences. As an application, we derive extended well-posedness results for conic vector and set optimization problems.

This paper establishes novel characterizations of the Radon-Nikodým property (RNP) in Banach lattices through the lens of positive linear operators and vector measure theory. By employing operator-theoretic approaches, we demonstrate that a Banach lattice E possesses RNP if and only if every positive Dunford-Pettis operator from L_1[0,1] into E admits Bochner integral representation. Furthermore, we investigate the implications of these results for evolution equations in ordered spaces, proving that the existence of non-constant strong solutions to autonomous equations governed by m-accretive operators fundamentally characterizes RNP in Banach lattices. Our findings refine classical results on differentiability of vector-valued functions and provide new insights into the structural properties of Banach lattices.

Firstly, it is proved that the distributive KS-lattice on a Hilbert space contains no non-trivial reducing projections. Secondly, the necessary and sufficient conditions are given for the projection lattice generated by one-point extensions of finite-nests on a Hilbert space to be a KS-lattice. Finally, all derivations on the projection lattice algebra corresponding to the projection lattice generated by one-point extensions of nests are characterized as inner derivations.

Abstract Let $$\mathbb {L}$$ L be a Dedekind complete unital f -algebra. We prove the Riesz-Kantorovich formulas for order bounded $$\mathbb {L}$$ L -module homomorphisms from a directed partially ordered $$\mathbb {L}$$ L -module with the Riesz Decomposition Property into a Dedekind complete $$\mathbb {L}$$ L -vector lattice satisfying an additional mild condition.

The classical Korovkin approximation theory has recently advanced in two significant directions: convergence towards composition operators in the linear framework, and approximation by monotone sublinear operators in the nonlinear setting. The present study establishes abstract and quantitative Korovkin-type theorems for sequences of weakly nonlinear and monotone operators converging to composition operators. The rate of convergence is quantified by means of the modulus of continuity. Moreover, the results are extended to multidimensional compact spaces, where a refined convergence criterion is established via a finite set of coordinate test functions. To demonstrate the applicability of the main results, novel sequences of sublinear operators are constructed, including a multivariate maximum operator and a Choquet-Chlodovsky-Kantorovich operator, with their quantitative estimates derived explicitly.

This paper is concerned with higher-order optimality conditions for strict minimality in general nonsmooth vector optimization problem with mixed constraints ((GVOPC), for short). Using the higher-order lower and upper set-valued Studniarski derivatives and the higher-order Hadamard differentiability of objective and constraint functions, we establish higher-order necessary optimality conditions for such a problem. Based on these Studniarski derivatives and associated Lagrangian functions, we provide higher-order sufficient optimality conditions for the problem (GVOPC). An application of the result for the twice Fréchet differentiable functions for the second-order strict local efficiency of that problem is presented too. Besides, we provide several methods for checking higher-order sufficient optimality conditions of the problem (GVOPC) using the higher-order lower and upper Studniarski derivatives and Lagrangian functions.

Linearly constrained nonconvex and nonsmooth composite optimization problems are prevalent in practical scenarios such as inverse problems and statistical learning, and hold significant application value. To simplify the iteration architecture, eliminate the need for checkpoint settings, and attain higher operational efficiency, by integrating symmetric updating techniques, the Bregman distance, and a hybrid gradient estimator, a novel stochastic alternating direction method of multipliers (ADMM) is proposed to solve large-scale linearly constrained nonconvex and nonsmooth composite optimization. The method allows larger step sizes to handle objective functions in either expectation or finite-sum form, thereby effectively reducing the number of iterations. At the same time, the introduction of the Bregman distance significantly simplifies the complexity of solving the subproblems. Unlike previous stochastic ADMMs that rely on double-loop structures, our approach employs a hybrid gradient estimator to achieve single-loop and single-sample updates, and maintains the currently optimal oracle complexity 𝒪(ϵ ^-3) . Without relying on the Kurdyka–Łojasiewicz (KL) property and under appropriate mild conditions, we establish global convergence and sublinear convergence rate of the proposed method. Finally, as an application and a numerical experiment, solving a graph-guided fused LASSO problem is given to verify the effectiveness of our algorithm. Furthermore, conclusions on the main results presented in this article are provided, along with a description of future research directions.

We study order ideals in inner product lattices, i.e., normed vector lattices whose norm arises from an inner product. Motivated by the interplay between lattice structure and geometric approximation, we investigate the geometric properties of these ideals. Among other results, we show that an order ideal is Chebyshev if and only if it is a projection band, revealing a close connection between approximation properties and the underlying lattice structure.

This paper investigates sufficient conditions for the existence of solutions to the convex quadratic programming problem with an arbitrary nonempty, closed, and convex constraint set in Hilbert spaces. Additionally, it focuses on the qualitative stability of the solution set and the optimal value function under perturbations of the objective function. Our results contribute to the study of quadratic optimization problems with arbitrary closed convex constraint sets in infinite-dimensional Hilbert spaces.

In this paper we study the following quasilinear nonlocal differential equations with convolution coefficients -M ((a * u^γ )(1) ) (φ _p(u'))' = f(λ , x, u), x ∈ (0,1), where γ >0 , φ _p(u)=|u|^p-2u with p>1 is the p-Laplacian operator, and the reaction term f(λ , x, u) includes linear, eigenvalue, sublinear, and logistic cases. We establish existence results for positive solutions and describe the global structure of the solution set in the degenerate case. The proofs of the main results are based upon fixed point arguments.

Let 𝒯(ℋ) and 𝒯(𝒦) be the Banach spaces of all trace class operators on separable complex Hilbert spaces ℋ and 𝒦, respectively. Our main result reveals that a completely positive map Φ :𝒯(ℋ)→𝒯(𝒦) satisfies ‖Φ (X)‖ _p=‖ X‖ _p for all X∈𝒯(ℋ) if and only if there exist a Hilbert space ℋ_1, an injective positive operator A∈𝒯(ℋ_1) with ‖ A‖ _p=1 and an isometry W from ℋ⊗ℋ_1 into 𝒦 such that Φ (X)=W(X⊗ A)W^* for all X∈𝒯(ℋ). This is equivalent to the existence of a completely positive map Λ : 𝒯(𝒦)→𝒯(ℋ) such that Λ (Φ (X))=X and ‖Λ (Y)‖ _p=‖ Y‖ _p for all X∈𝒯(ℋ) and Y∈Φ (𝒯(ℋ)). Additionally, we characterize the structure of all recovery maps Ψ for a completely positive and trace preserving (CPTP) map Φ , where a recovery map is defined as a CPTP map satisfying Ψ (Φ (X))=X for all X∈𝒯(ℋ).

We use nearly parallel pure states to characterize positive linear functionals ϕ on 𝕄_n as positive multiples of the trace if and only if ϕ(A ♮ B) ≤√(ϕ(A) ϕ(B)) for all positive definite matrices A and B. Here A ♮ B = (A^-1# B)^1/2 A (A^-1# B)^1/2 represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality ϕ(A ♮ B) ≤ ϕ((A+B)/2) for all positive definite matrices A and B. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.

Since the remarkable work of Ando, Li and Mathias on a symmetrization procedure, various types of the multi-variable geometric mean with Ando-Li-Mathias (ALM) properties have been studied. In this paper, we explore on characterization of preservers for two-variable Kubo-Ando’s and non-Kubo-Ando’s operator means and study relationships between mean preservers. Moreover, we characterize the preserver of multi-variable means such as the ALM mean, Karcher mean and Wasserstein mean.

Riesz* homomorphisms on ordered vector spaces are characterized, intrinsically, via a condition on finite sets. As it is not sufficient to reduce to sets with at most two elements, mild Riesz* homomorphisms (of degree 2) were introduced and investigated. In this paper, we introduce mild Riesz* homomorphisms of higher degrees and generalize the results on mild Riesz* homomorphisms of degree 2 to this setting. We focus on order unit spaces and give sufficient conditions such that mild Riesz* homomorphism of degree n and Riesz* homomorphisms coincide.