On Two Natural Extensions of n-normality

A bounded linear operator T on a complex Hilbert space H is called n-normal if T^*Tⁿ=TⁿT^*.By Fuglede’s theorem T is n-normal if and only if Tⁿ is normal.Let k,n∈ N.Then a bounded linear operator T is said to be of type Ⅰ k-quasi-n-normal if T^*k{T^*Tⁿ-TⁿT^*}T^k=0,and T is said to be of type Ⅱ k-quasi-n-normal if T^*k{T^*nTⁿ-TⁿT^*n...     

Log-ψω亚正规算子(英文)

Let T be an operator on a separable Hilbert space H and T=U|T|bethe polaf decomposition.T is said to be log-ω-hyponormal if log|T|≥log|T|≥log|(T)*|In this paper we prove that the point spectrum of T is equal to its joint point spectrum if T is log-ω-hyponormal.We also prove that a log-ω-hyponormal operator is normaloid,i.e.,r(T)=‖T‖.Finally,we obtain Putnam's theorem for log-ω-hyponormal Operators.     

RIESZ IDEMPOTENT AND BROWDER＇S THEOREM FOR ABSOLUTE-（p,r ）-PARANORMAL OPERATORS

An operator T is said to be paranormal if ||T 2x|| ≥ ||T x||2 holds for every unit vector x.Several extensions of paranormal operators are considered until now,for example absolute-k-paranormal and p-paranormal introduced in [10],[14],respectively.Yamazaki and Yanagida [38] introduced the class of absolute-(p,r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators.An operator T ∈ B(H) is called absolute-(p,r)-paranormal operator if |||T |p|T |r x||r ≥ |||T |rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0.The famous result of Browder,that self adjoint operators satisfy Browder’s theorem,is extended to several classes of operators.In this paper we show that for any absolute-(p,r)paranormal operator T,T satisfies Browder’s theorem and a-Browder’s theorem.It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p,r)-paranormal operator T,then E is self-adjoint if and only if the null space of T μ,N(T μ) N(T μ).     

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHR?DINGER PROPAGATOR ON NONCOMPACT MANIFOLDS

Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥ 2),complete,noncompact Riemannian manifold M,we prove that if 0 <α <1,s> α/2 and f ∈ H^s(M),then the fractional Schr?dinger propagator e⁽it|L|^α/2)(f)(x)→f(x) a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.     

THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES

In this paper,we prove that(X,p)is separable if and only if there exists a w^*-lower semicontinuous norm sequence{p_n}_n=1^∞of(X^*,p)such that(1)there exists a dense subset G_nof X^*such that p_nis Gateaux differentiable on G_nand dp_n(Gn_n)■X for all n∈N;(2)p_n≤p and p_n→p uniformly on each bounded subset of X^*;(3)for anyα∈(0,1),there exists a ball-covering{B(x^*i,n,Ti,n)}∞i=1 of(X^*,p_n)such that it isα-off the origin and x_i,n^*∈Gn_n.Moreover,we also prove that if Xi is a Gateaux differentiability space,then there exist a real numberα>0 and a ball-covering(B)i of Xi such that(B)i isα-off the origin if and only if there exist a real numberα>0 and a ball-covering B of l^∞(X_i)such that(B)isα-off the origin.     

Property(ω) and Its Perturbations

A Hilbert space operator T is said to have property(ω1) if σa(T)\σaw(T) ? π00(T), where σa(T) and σaw(T) denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and π00(T) = {λ∈ iso σ(T), 0 dim N(T- λI) ∞}. If σa(T)\σaw(T) = π00(T), we say T satisfies property(ω). In this note, we investigate the stability of the property(ω1) and the property(ω) under compact perturbations, and we characterize those operators for which the property(ω1) and the property(ω) are stable under compact perturbations.     

Ambarzumyan's Theorem for the Dirac Operator on Equilateral Tree Graphs

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the SturmLiouville operator-d²/dx²+q with an integrable real-valued potential q on [0,π] are {n²:n≥0},then q=0 for almost all x∈ [0,π].In this work,the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs.We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case,then the potential is identically zero.     

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).     

A multidimensional central limit theorem with speed of convergence for axiom a diffeomorphisms

Let T:X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f:X → R^d is a Hlder continuous function with ∫_X^fdm = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ²:=σ² （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ² . Moreover, there exists a real number A > 0 such that, for any integer n ≥ 1,Π( m*（ 1√ nS f n ）,N （0,σ² ） ≤A√n, where m*（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.     

加权Baouendi-Grushin型算子的基本解与一类加权Hardy不等式(英文)

In this paper we obtain the fundamental solution for a class of weighted BaouendiGrushin type operator L_p,γ,αu = ▽_γ·(|▽_γu|^p-2ρ^α▽_γu) on R^m+n with singularity at the origin,where ▽_γ is the gradient operator defined by ▽_γ =（▽_x,|x|^γ▽_y） and ρ is the distance function.As an application,we get some Hardy type inequalities associated with ▽_γ.