Showing papers in "Positivity in 2015"

2-Local derivations on von Neumann algebras

Abstract: The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type \(III\) von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation.

A generalization of two refined Young inequalities

Abstract: We prove that if $$a,b>0$$ and $$0\le u \le 1$$ , then for $$m=1,2,3,\ldots $$ , we have $$\begin{aligned} \left( a^{ u }b^{1- u }\right) ^{m}+r_{0}^{m}\left( a^{\frac{m}{2}}-b^{\frac{m}{2}}\right) ^{2}\le \Big ( u a+(1- u )b\Big ) ^{m}, \end{aligned}$$ where $$r_{0}=\min \left\{ u ,1- u \right\} $$ . This is a considerable generalization of two refinements of the classical Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases $$m=1$$ and $$m=2$$ , respectively. As applications of this inequality, we give refined Young-type inequalities for the traces, determinants, and norms of positive definite matrices.

On the R-boundedness of stochastic convolution operators

Abstract: The \(R\)-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal \(L^p\)-regularity, \(2

Statistical approximation by double sequences of positive linear operators on modular spaces

Abstract: In this paper, using the concept of statistical convergence, which is stronger than the Pringsheim convergence, we study the problem of approximation to a function by means of double sequences of positive linear operators defined on a modular space. Also, a non-trivial application is presented.

The Gould integral in Banach lattices

Abstract: In this paper, we study a Gould type integral in a new frame of Banach lattices We consider the Gould integral of real functions relative to a non-additive set function taking values in a Banach lattice Some continuity properties of this integral and relationships between integrability and total measurability are presented

On the continuous Cesàro operator in certain function spaces

Abstract: Various properties of the (continuous) Cesaro operator \(\mathsf {C}\), acting on Banach and Frechet spaces of continuous functions and \(L^p\)-spaces, are investigated. For instance, the spectrum and point spectrum of \(\mathsf {C}\) are completely determined and a study of certain dynamics of \(\mathsf {C}\) is undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of \(\mathsf {C}\) acting in the various spaces is identified.

Normality of spaces of operators and quasi-lattices

Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces

Abstract: In this paper, via a modification of the notion of weak upper gradients, we introduce and investigate properties of the Newton–Besov spaces \(\textit{NB}^s_{p,q}(X)\) and the Newton–Triebel–Lizorkin spaces \(\textit{NF}^s_{p,q}(X)\), with \(s\in [0,1]\), \(1\le p<\infty \) and \(q\in (0,\infty ]\), of functions on a metric measure space \(X\) and prove that, when \(1

Gaussian lower bound for the Neumann Green function of a general parabolic operator

Abstract: Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish a Gaussian lower bound for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for Gaussian two-sided bounds for the fundamental solution.

Some new results on infinite series and Fourier series

Abstract: In this paper, we generalize some known theorems by using a general class of quasi power increasing sequences, which is a wider class of sequences, instead of an almost increasing sequence These theorems also include some known and new results

Bands in partially ordered vector spaces with order unit

Abstract: In an Archimedean directed partially ordered vector space X, one can define the concept of a band in terms of disjointness Bands can be studied by using a vector lattice cover Y of X If X has an order unit, Y can be represented as a subspace of $$C(\Omega )$$ , where $$\Omega $$ is a compact Hausdorff space We characterize bands in X, and their disjoint complements, in terms of subsets of $$\Omega $$ We also analyze two methods to extend bands in X to $$C(\Omega )$$ and show how the carriers of a band and its extensions are related We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $$\frac{1}{4}2^{2^n}$$ for $$n\ge 2$$ We also construct examples of $$(n+1)$$ -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n\\ n\end{array}}\right) +2$$ bands This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$

Positive Schur properties in spaces of regular operators

Abstract: Properties of Schur type for Banach lattices of regular operators and tensor products are analyzed. It is shown that the dual positive Schur property behaves well with respect to Fremlin’s projective tensor product, which allows us to construct new examples of spaces with this property. Similar results concerning the positive Grothendieck property are also presented.

Convergence in Riesz spaces with conditional expectation operators

Abstract: A conditional expectation, \(T\), on a Dedekind complete Riesz space with weak order unit is a positive order continuous projection which maps weak order units to weak order units and has \(R(T)\) a Dedekind complete Riesz subspace of \(E\). The concepts of strong convergence and convergence in probability are extended to this setting as \(T\)-strongly convergence and convergence in \(T\)-conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as \(T\)-uniformity and \(T\)-boundedness. Here we show that if a net is \(T\)-uniform and convergent in \(T\)-conditional probability then it is \(T\)-strongly convergent, and if a net is \(T\)-strongly convergent then it is convergent in \(T\)-conditional probability. For sequences we have the equivalence that a sequence is \(T\)-uniform and convergent in \(T\)-conditional probability if and only if it is \(T\)-strongly convergent. These results are applied to Riesz space martingales and are applicable to stochastic processes having random variables with ill-defined or infinite expectation.

Some classes of weighted conditional type operators and their spectra

Abstract: In this paper, some \(*\)-classes of weighted conditional expectation type operators, such as \(A\)-class, \(*\)-\(A\)-class and quasi-\(*\)-\(A\) classes on \(L^{2}(\Sigma )\) are investigated. Also, the spectrum, point spectrum and spectral radius of these operators are computed.

Domination problem on Banach lattices and almost weak compactness

Abstract: In this paper we give several new results concerning domination problem in the setting of positive operators between Banach lattices. Mainly, it is proved that every positive operator \(R\) on a Banach lattice \(E\) dominated by an almost weakly compact operator \(T\) satisfies that the \(R^2\) is almost weakly compact. Domination by strictly singular operators is also considered. Moreover, we present some interesting connections between strictly singular, disjointly strictly singular and almost weakly compact operators.

Order-preserving nonautonomous discrete dynamics: attractors and entire solutions

Abstract: The concept of pullback convergence turned out to be a central idea to describe the long-term behavior of nonautonomous dynamical systems. This paper provides a general framework for the existence and structure of pullback attractors capturing the asymptotics of nonautonomous and order-preserving difference equations in Banach spaces. Furthermore we obtain criteria for the convergence to bounded entire solutions and additionally discuss various applications.

On fixed point theorems for monotone increasing vector valued mappings via scalarizing

Abstract: In this paper, first we prove some lemma, then by using the nonlinear scalarization mapping, we present some fixed point theorems for a vector valued mapping. The main result obtained can be viewed as an extension, improvement and repairment of the main theorem given in Kostrykin and Oleynik (Fixed Point Theory Appl 2012:211, 2012).

Existence and uniqueness of positive mild solutions for nonlocal evolution equations

Abstract: This paper deals with the existence and uniqueness of positive mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. The existence and uniqueness of mild solution for the associated linear evolution equation nonlocal problem is established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness of positive mild solutions for nonlinear evolution equation nonlocal problem are obtained by using the monotone iterative method without the assumption of lower and upper solutions. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. An example is also given to illustrate that our results are valuable.

Submajorization inequalities of \tau -measurable operators for concave and convex functions

Abstract: Let \(M\) be a von Neumann algebra with a normal faithful semifinite trace \(\tau \). Let \(x_1 ,\ldots ,x_n \) be \(n\, \tau \)-measurable positive operators with respect to \(( {M,\tau })\), and let \(z_1 ,\ldots ,z_n \) be \(n\) expansive operators in \(M.\) We prove that for a concave function \(f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) ,\, f(\sum olimits _{{k = 1}}^{n} {z_{k}^{*} x_{k} z_{k} } ) \) is submajorised by \(\sum olimits _{{k = 1}}^{n} {z_{k}^{*} f(x_{k} )z_{k} }\) and the reverse submajorisation holds if \(f\) is a positive convex function with \(f( 0)=0.\)

On some positive definite functions

Abstract: We study the function \((1 - \Vert x\Vert )/ (1 - \Vert x\Vert ^r),\) and its reciprocal, on the Euclidean space \(\mathbb {R}^n,\) with respect to properties like being positive definite, conditionally positive definite, and infinitely divisible.

On Fuglede–Putnam properties

Abstract: In this paper, we study the Fuglede–Putnam property. We give a necessary and sufficient condition for which $$(\oplus _{i=1}^{n}A_i,\oplus _{i=1}^{n}B_i)$$ satisfies the Fuglede–Putnam property. We also study the local spectral theory associated with the Fuglede–Putnam property. Finally, we define the weak Fuglede–Putnam property and we investigate several cases which satisfy the weak Fuglede–Putnam property.

Fully symmetric function spaces without an equivalent Fatou norm

Abstract: We present examples of fully symmetric function spaces without an equivalent Fatou norm and examples of symmetric function spaces, which do not admit an equivalent strongly symmetric norm.

A Liouville theorem for p-harmonic functions on exterior domains

Abstract: We prove Liouville type theorems for $$p$$ -harmonic functions on exterior domains of $${\mathbb {R}}^{d}$$ , where $$1

The noncommutative \(H^{(r,s)}_{p}({\mathcal A};\ell _{\infty })\) and \(H_{p}({\mathcal A};\ell _{1})\) spaces

Abstract: In this paper we introduce the noncommutative $$H^{(r,s)}_{p}({\mathcal A};\ell _{\infty })$$ and $$H_{p}({\mathcal A};\ell _{1})$$ spaces, and prove the contractivity of the underlying conditional expectation $$\varPhi $$ on these spaces. We also give results on duality and complex interpolation.

Strict vector variational inequalities and strict Pareto efficiency in nonconvex vector optimization

Abstract: In this paper, we consider a vector optimization problem involving the difference of two cone convex functions in Banach spaces. In terms of Clarke subdifferential, we formulate strict Minty vector variational inequality problem and strict Stampacchia vector variational inequality problem. We mainly study the relationships between the solutions of these vector variational inequality problems and the strict Pareto efficient solution of the vector optimization problem.

Swartz type results for nuclear and multiple 1-summing bilinear operators on $$c_{0}\left( \mathcal {X}\right) \times c_{0}\left( \mathcal {Y}\right) $$

Abstract: In this paper we investigate the bilinear versions of Swartz’s theorem Thus we characterize the nuclear and the multiple 1-summing operators on a cartesian product of \(c_{0}\left( \mathcal {X}\right) \) As applications we give the necessary and sufficient conditions for some natural operators on a cartesian product of \(c_{0}\left( \mathcal {X}\right) \) to be multiple 1-summing and nuclear operators By an example, we show that the natural bilinear version of Swartz’s theorem is not necessarily true

Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

Abstract: We give a short proof of a recent result of Drury on the positivity of a \(3\times 3\) matrix of the form \((\Vert R_i^*R_j\Vert _{\mathop {\mathrm{tr}}\,})_{1 \le i, j \le 3}\) for any rectangular complex (or real) matrices \(R_1, R_2, R_3\) so that the multiplication \(R_i^*R_j\) is compatible for all \(i, j,\) where \(\Vert \cdot \Vert _{\mathop {\mathrm{tr}}\,}\) denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.

First and second-order optimality conditions without differentiability in multivalued vector optimization

Abstract: By developing and using approximations as generalized derivatives of set-valued mappings, we establish both necessary conditions and sufficient conditions of order one and two for various kinds of solutions to a multivalued vector optimization problem with general inequality constraints and high level of nonsmoothness. An emphasis was paid on the important (but still not widely known) so-called envelope-like effect. Our results are more applicable than a number of recent existing ones in many situations as illustrated in detail by examples.

Some Positivstellensätze for polynomial matrices

Abstract: In this paper we give a version of Krivine–Stengle’s Positivstellensatz, Schweighofer’s Positivstellensatz, Scheiderer’s local-global principle, Scheiderer’s Hessian criterion and Marshall’s boundary Hessian conditions for polynomial matrices, i.e. matrices with entries from the ring of polynomials in the variables \(x_1,\ldots ,x_d\) with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in \(\mathbb R^{d}\) with respect to a subset \(\mathcal {G}\) of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by \(\mathcal {G}\).

Weakly compact composition operators on spaces of Lipschitz functions

Abstract: Let \(X\) be a pointed compact metric space such that \({\mathrm {lip}}_0(X)\) has the uniform separation property. We prove that every weakly compact composition operator on spaces of Lipschitz functions \({\mathrm {lip}}_0(X)\) and \({\mathrm {Lip}}_0(X)\) is compact.