Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.     

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u^′(t)+B(t,u(t))∋f(t) is studied. In this problem, the operators are of type (M) or type (S₊), which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.     

Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach

The main objective of this paper is to prove the essential self-adjointness of Dirichlet operators in L²(μ) where μ is a Gibbs measure on an infinite volume path space C(R,Rd). This operator can be regarded as a perturbation of the Ornstein-Uhlenbeck operator by a nonlinearity and corresponds to a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. In view of quantum field theory, the solution of this SPDE is called a P₁(?)-time evolution.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

Global strong solution to the Thirring model in critical space

We obtain a strong solution in charge critical space L²(R) of the Thirring system and Federbusch equations in one space dimension by using solution representation of the models. The uniqueness is obtained for the solution Ψ∈L^∞([0,T];L²(R)∩L⁴(R)). A decay of local charge and asymptotic behavior of the field can be shown directly.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators

We adapt recent results of Albrecht and Ricker to obtain conditions under which growth constraints on the left resolvent of a Banach space operator are preserved under suitable perturbations. As an application, we establish Bishop's property (β) for certain generalized Cesàro operators on the classical Hardy spaces Hp, 1<p<∞. Our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

Generalized spherical and simplicial coordinates

Elementary trigonometric quantities are defined in l2,p analogously to that in l2,2, the sine and cosine functions are generalized for each p>0 as functions sinp and cosp such that they satisfy the basic equation p|cosp(φ)|+p|sinp(φ)|=1. The p-generalized radius coordinate of a point ξ∈Rn is defined for each p>0 as . On combining these quantities, ln,p-spherical coordinates are defined. It is shown that these coordinates are nearly related to ln,p-simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on ln,p-spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the ln,p-sphere and with the derivation of certain generalized statistical distributions.     

Exact boundary observability for nonautonomous quasilinear wave equations

By means of a direct and constructive method based on the theory of semiglobal C² solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

The H-covariant strong Picard groupoid

The notion of H-covariant strong Morita equivalence is introduced for ^*-algebras over C=R(i) with an ordered ring R which are equipped with a ^*-action of a Hopf ^*-algebra H. This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant ^*-Morita equivalence with its H-covariant ^*-Picard groupoid. We discuss various groupoid morphisms between the corresponding notions of the Picard groupoids. Moreover, we realize several Morita invariants in this context as arising from actions of the H-covariant strong Picard groupoid. Crossed products and their Morita theory are investigated using a groupoid morphism from the H-covariant strong Picard groupoid into the strong Picard groupoid of the crossed products.     

A noncommutative Brooks-Jewett Theorem

In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive-analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C^*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks-Jewett property if, and only if, it is topologically equivalent to an abelian algebra.     

Fixed point properties of semigroups of nonexpansive mappings

We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak^∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L¹-space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

Rough approximations of vague sets in fuzzy approximation space

The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the β-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on β-(complement) operator. Meantime, λ-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the β-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable.