RIESZ IDEMPOTENT OF (n,k)-QUASI-*-PARANORMAL OPERATORS

A bounded linear operator T on a complex Hilbert space H is called(n, k)-quasi-*-paranormal if ║T~(1+n)(T~kx) ║~(1/(1+n))║ T~kx║~(n/(1+n))≥║ T*(T~kx)║ for all x ∈ H,where n, k are nonnegative integers. This class of operators has many interesting properties and contains the classes of n-*-paranormal operators and quasi-*-paranormal operators. The aim of this note is to show that every Riesz idempotent E_λ with respect to a non-zero isolated spectral point λ of an(n, k)-quasi-*-paranormal operator T is self-adjoint and satisfies ran E_λ= ker(T- λ) = ker(T- λ)*.     

TIME DECAY RATE OF SOLUTIONS TO THE HYPERBOLIC MHD EQUATIONS IN R~3

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R~3. Then we establish that the solutions with initial data belonging to H~m(R~3) ∩ L~1(R~3) have the following time decay rate:║?~mu(x, t) ║~2+║ ?~mb(x, t)║~ 2+ ║?~(m+1)u(x, t)║~ 2+ ║?~(m+1)b(x, t) ║~2≤ c(1 + t)~(-3/2-m)for large t, where m = 0, 1.     

CONVERGENCE RATE OF MULTIVARIATE K-NEAREST NEIGHBOR DENSITY ESTIMATES

Let be the collection of m-times continuously differentiable probability densities fon R~d such that 丨D~af(x_1)-D~af(x_2)丨≤M‖x_1-x_2‖~β for x_1,x_2∈R~d,[a]=m,where D~adenotes the differential operator defined by D~a=([a])/(x_1~a…x_d~a_d).Under rather weak conditionson K(x),the necessary and sufficient conditions for sup丨_n(x)-f(x)丨=0(((logn/n)~λ/(d+3λ),λ=m+β,f∈ are that ∫x~aK(xi)dx=0 for 0<[a]≤m.Finally the convergenco rate at apoint is given.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

2-CONNECTED k-REGULAR GRAPHS ON AT MOST 3k+3 VERTICES TO BE HAMILTONIAN (CONTINUED)

Let X~* and Y~* be generated by S(?){v_0},where G(S)is Hamiltonian connected and|X~*|=x~*,|Y~*|=Y~* and S_1~*,S_2~*,…,S_(x*)~* be the sets of vertices contained in the opensegments of C between vertices of X~*.Let S_1~*,x_1,S_2~*,x_2,…,S_(x*)~*,x_(?)~* be the segmentsand vertices of X~* in order around C.S_i~* is said to be an X~*(3)-interval if one ofx_(i-1)and x_i belongs to X_j~*—X_2~*.Let S=S_1~*,and S={a_1,c_1,c_2,…,c_1,b_1}.It is easy to see that the statement inLemma 2 can be modified as(?)({a_1,b_1},S_i~*)≤(?)     

Asplund空间的一些几何特征

本文主要讨论Asplund空间的一些几何特征。设 X 为 Banach 空间,本文证明了下述等价:(1)X 是 Asplund 空间;(2)X~*的每个有界范闭子集包含它的ω~*闭凸包的一个端点;(3)X~*的每个有界范闭子集包含它的凸包的一个端点;(4)对 X~*的每个有界范闭子集 A,存在 x_o∈X/{0}和 x_o~*∈A,使得 x_o~*(x_o)=(?)x~*(x_o);(5)对 X~*的每个有界范闭子集 A,集{x∈X,■x_o~*∈A,使得 x_o~*(x)=sup x~*(x)}在 X 中范稠     

On the Weak＊ Drop Property for Polar of Closed Bounded Convex Sets

We define and study the weak* drop property for the polar of a closed bounded convex set in a Banach space which is both a generalization of the weak* drop property for dual norm in a Banach space and a characterization of the sub-differentialmapping x→эp(x) from S(X) into 2^S(X*) that is norm upper semi-continuous and norm compact-valued.     

FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES

Necessary and sufficient conditions are studied that a bounded operator T_x =(x_1~*x, x_2~*x,···) on the space ?_∞, where x_n~*∈ ?_∞~*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x_1~*, x_2~*,···} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of x_n~*= d_nx_(tn)~*,where dn ∈ R and x_(tn)~*≥ 0 are extreme points of the unit ball B_?_∞~*, that is, t_n ∈βN, is considered. In terms of the sequence {t_n}, the conditions of the closedness of the range R(T)are given and the value d(T) is calculated. For example, the condition {n:0 |d_n| δ} = Φ for some δ is sufficient and if for large n points tn are isolated elements of the sequence {t_n},then it is also necessary for the closedness of R(T)(t_(n0) is isolated if there is a neighborhood U of t_(n0) satisfying t_n ■ U for all n ≠ n0). If {n:|d_n| δ} =Φ, then d(T) is equal to the defect δ{_tn} of {t_n}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of N satisfying χ_(A_n)■R(T).     

COMPOSITION OPERATORS ON THE LITTLE BLOCH SPACE IN POLYDISCS

Let Un be the unit polydisc of Cn and φ = (φ1,…,φn) a holomorphic self map of Un. This paper shows that the composition operator Cφ induced by φ is bounded on the little Bloch space β0*(Un) if and only if φ∈β0*(Un) for every l=1,2,…,n, and also gives a sufficient and necessary condition for the composition operator Cφ to be compact on the little Bloch spaceβ0* (Un).     

Convergence of global attractors of a 2D non-Newtonian system to the global attractor of the 2D Navier-Stokes system

This paper discusses the relation between the long-time dynamics of solutions of the two-dimensional(2D) incompressible non-Newtonian fluid system and the 2D Navier-Stokes system.We first show that the solutions of the non-Newtonian fluid system converge to the solutions of the Navier-Stokes system in the energy norm.Then we establish that the global attractors {AHε} 0<ε≤1 of the non-Newtonian fluid system converge to the global attractor AH0 of the Navier-Stokes system as → 0.We also construct the minimal limit AHmin of the global attractors {AHε}0<ε≤1 as ε→ 0 and prove that AHmin is a strictly invariant and connected set.     

A FORMULA OF THE KINEMATIC MEASURE IN NONEUCLIDEAN SPACE AND SOME PROBLEMS ON THE GEOMETRIC PROBABILITIES

Let P_n, be the n-dimensional projective space. Let x_0,x_1,…,x_n be the homogeneous coordinates of point x. Consider the quadric φ(x) =x_0~2+x_1~2+…x_(n-1)~2+εx_n~2, where ε can take the value of +1 or -1. We assume the homogeneous coordinates of the points x that do not belong to φ = 0 are normalized, such that     

Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

We will prove that for 1相似文献   

奇异点处的拟Newton方法

1.引言 假定F是R~m→R~m的可微映射,x~*∈R~m是 F(x)=0 (1.1)的解. 如果在解点处Frechet导数是可逆的,只要F′(x)具有一定的性质,就可保证Newton迭代 x_(i+1)~N=x_i~N-F′(x_i~N)~(-1)F(x_i~N) i=0,1,… (1.2)产生的点列在||x_0-x~*||适当小时二阶收敛于x~*:     

On Universally Left-stability of ε-Isometry

LetX,Y be two real Banach spaces andε≥0.A map f:X→Y is said to be a standardε-isometry if|f(x)f(y)x y|≤εfor all x,y∈X and with f(0)=0.We say that a pair of Banach spaces(X,Y)is stable if there existsγ0 such that,for every suchεand every standardε-isometry f:X→Y,there is a bounded linear operator T:L(f)≡spanf(X)→X so that T f(x)x≤γεfor all x∈X.X(Y)is said to be universally left-stable if(X,Y)is always stable for every Y(X).In this paper,we show that if a dual Banach space X is universally left-stable,then it is isometric to a complemented w-closed subspace of∞(Γ)for some setΓ,hence,an injective space;and that a Banach space is universally left-stable if and only if it is a cardinality injective space;and universally left-stability spaces are invariant.     

More on stability of almost surjective ε-isometries of Banach spaces

Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε = 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the following sharp stability inequality of f:There is a surjective linear operator T:Y→X of norm one so that ||T f(x)-x||= 2ε,for all x∈X.As its application,we prove the following statements are equivalent for a standard ε-isometry f:X→Y:(i)lim inf_(t→∞) dist(ty,f(X))/|t|1/2,for all y∈S_Y;(ii)τ(f)≡sup_(y∈S_Y) lim inf_(t→∞) dist(ty,f(X))/|t|=0;(iii)there is a surjective linear isometry U:X→Y so that || f(x)-Ux||= 2ε,for all x∈X.This gives an affirmative answer to a question proposed by Vestfrid(2004,2015).     

Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities

Let Ω be a smooth bounded domain in R~2, W_0~(1,2)(Ω) be the standard Sobolev space. By the method of energy estimate developed by Malchiodi–Martinazzi(J. Eur. Math. Soc., 16, 893–908(2014)), Mancini–Martinazzi(Calc. Var. Partial Differential Equations, 56, 94(2017)) and Mancini–Thizy(J. Differential Equations, 266, 1051–1072(2019)), we reprove the results of Carleson–Chang(Bull. Sci. Math., 110, 113–127(1986)), Flucher(Comment. Math. Helv., 67, 471–497(1992)), Li(Acta Math. Sin. Engl. Ser., 22, 545–550(2006)) and Su(J. Math. Inequal., in press). Namely, for any real number α≤ 1, the supremum ■ can be achieved by some function v ∈ W_0~(1,2)(Ω) with ║?_v║_2~2≤ 4π.     

WEIGHTED AVERAGE OF DIRICHLET KERNELS FOR VILENKIN-LIKE SYSTEM

For Vilenkin-like system, the authors define a new operator H*f := supn |Hnf|, where Hnf is the weighted average for partial sums, and prove that H* is of type (Hp* (Gm), Lp(Gm)) for all 1/2 < p ≤∞. As a consequence, the authors prove the operator S*f := supn |Snf| is of type (p, p) for 1 < p < ∞, where Snf is the n-partial sum.     

Radial Operators on the Weighted Bergman Spaces over the Polydisk

In this paper, we study radial operators in Toeplitz algebra on the weighted Bergman spaces over the polydisk by the(m, λ)-Berezin transform and find that a radial operator can be approximated in norm by Toeplitz operators without any conditions. We prove that the compactness of a radial operator is equivalent to the property of vanishing of its(0, λ)-Berezin transform on the boundary. In addition, we show that an operator S is radial if and only if its(m, λ)-Berezin transform is a separately radial function.     

Operator Matrix Forms of Positive Operators

If a 3-tuple (A:H1→H1,B:H2→H1,C:H2→H2）of operators on Hibert spaces is given,we proved that the operator ～↑A:=(↑A ↓B^*↑B ↓C) on H=H1 H2 is ≥0 is and only if A≥0,R(B)∪→R（A^1/2) and C≥B^* A^ b,where A^ is the generalized inverse of A.In general,A^ is a closed operator,but since R(B)∪→R(A^1/2,B^* A^ B is bounded yet.     

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.