Positivity最新文献

In this paper we give necessary and sufficient conditions for the boundedness of the multidimensional Hausdorff operator on weighted Lebesgue spaces. In particular, we establish necessary and sufficient conditions for the boundedness of special type of the multidimensional Hausdorff operator on weighted Lebesgue spaces for monotone radial weight functions. Also, we get similar results for important operators of harmonic analysis which are special cases of the multidimensional Hausdorff operator. The results are illustrated by a number of examples.

Abstract

In this work, we study a comparison of norms in non-commutative spaces of (tau ) -measurable operators associated with a semifinite von Neumann algebra. In particular, we obtain Nazarov–Podkorytov type lemma Nazarov et al. (Complex analysis, operators, and related topics. Operatory theory: advances, vol 113, pp 247–267, 2000) and extend the main results in Astashkin et al. (Math Ann, 2023. https://doi.org/10.1007/s00208-023-02606-w) to non-commutative settings. Moreover, we complete the range of the parameter p for (0<p<1.)

Let ((X, {{mathfrak {A}}},P)) and ((Y, {{mathfrak {B}}},Q)) be two probability spaces and R be their skew product on the product (sigma )-algebra ({{mathfrak {A}}}otimes {{mathfrak {B}}}). Moreover, let ({({{mathfrak {A}}}_y,S_y):yin {Y}}) be a Q-disintegration of R (if ({{mathfrak {A}}}_y={{mathfrak {A}}}) for every (yin {Y}), then we have a regular conditional probability on ({{mathfrak {A}}}) with respect to Q) and let ({{mathfrak {C}}}) be a sub-(sigma )-algebra of ({{mathfrak {A}}}cap bigcap _{yin {Y}}{{mathfrak {A}}}_y). We prove that if (fin {{mathcal {L}}}^{infty }(R)) and ({{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f)) is the conditional expectation of f with respect to ({{mathfrak {C}}}otimes {{mathfrak {B}}}), then for Q-almost every (yin {Y}) the y-section ([{{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f)]^y) is a version of the conditional expectation of (f^y) with respect ({{mathfrak {C}}}) and (S_y). Moreover there exist a lifting (pi ) on ({{mathcal {L}}}^{infty }(widehat{R})) ((widehat{R}) is the completion of R) and liftings (sigma _y) on ({{mathcal {L}}}^{infty }(widehat{S_y})), (yin Y), such that

Both results are generalizations of Strauss et al. (Ann Prob 32:2389–2408, 2004), where ({{mathfrak {A}}}) was assumed to be separable in the Frechet-Nikodým pseudometric and of Macheras et al. (J Math Anal Appl 335:213–224, 2007), where R was assumed to be absolutely continuous with respect to the product measure (Potimes {Q}). Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.

Abstract

Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison’s anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included.

In this paper, we give a new approach to the theory of kernel functions. Our method is based on the structure of Fock spaces. As its applications, two non-Euclidean examples of strictly positive kernel functions are given. Moreover, we give a new proof of the universal approximation theorem for Gaussian type kernels.

In this paper, we introduce two kinds of Hadamard well-posedness for a set optimization problem by taking into consideration perturbations of objective function and a relationship between these two well-posedness is derived. Using the generalized Gerstewitz function, a sequence of scalar optimization problems have been defined and a convergence result is obtained. Sufficient conditions for Hadamard well-posedness for the set optimization problem are established by invoking these scalarization results. Finally, the stability of the convergence of minimal solution sets of the set optimization problem considered is discussed in terms of Painlevé-Kuratowski convergence.

The present paper aims to describe the relationships between the intersection property, introduced and studied in the previous paper by the authors, with other known properties of Riesz spaces, and to prove that every lateral ideal of a Riesz space is a kernel of some positive orthogonally additive operator (it is easy to see that the kernel of every positive orthogonally additive operator is a lateral ideal). We provide examples of Riesz spaces with the principal projection property (and hence, with the intersection property) which fail to be C-complete. The above results give complete answers to problems posed in the first part of the present paper by the authors.

Let E and F be two Archimedean Riesz spaces. An operator (T:Erightarrow F) is said to be unbounded order continuous (uo-continuous), if (u_{alpha }overset{uo}{rightarrow }0) in E implies (Tu_{alpha }overset{uo}{ rightarrow }0) in F. In this study, our main aim is to give the solution to two open problems which are posed by Bahramnezhad and Azar. Using this, we obtain that the space (L_{uo}(E,F)) of order bounded unbounded order continuous operators is an ideal in (L_{b}(E,F)) for Dedekind complete Riesz space F. In general, by giving an example that the space (L_{uo}(E,F)) of order bounded unbounded order continuous operators is not a band in ( L_{b}(E,F)), we obtain the conditions on E or F for the space ( L_{uo}(E,F)) to be a band in (L_{b}(E,F)). Then, we give the extension theorem for uo-continuous operators similar to Veksler’s theorem for order continuous operators.

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.