关于某些次正规算子的表示

设T是完全非正规的次正规算子。给出了T＝N＋K的充要条件，其中N是正规算子，K是拟正规算子，且NK＝KN。     

A remark on density theorems for Riemann's zeta-function

The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann’s zeta-function which are sufficiently strong to break the density hypothesis for Re s>7/8.Apart from a simple but ingenious idea of Halász the proof uses only classical knowledge about the zeta-function,results known for at least hundred years.     

A remark on two famous theorems concerning polynomials

A relation between Gauss-Lucas Theorem and Laguerre Theorem concerning the zeros of a polynomial in complex domain is discussed.     

Ultradifferentiable extension theorems: A survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney’s classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun–Meise–Taylor classes based on their representation as intersections and unions of Denjoy–Carleman classes.     

Pontriagin空间上算子的交换性

证明了:当Pontrjagin空间上的正规算子A没有零性不变子空间时,Putnam- Fuglede定理成立;当正规算子A有零性不变子空问时,通过构造反例说明此时Putnam- Fuglede定理不成立,并对Π1空间上算子相关的交换性条件进行了讨论,得到了Π1空间上算子代数的二次交换定理．     

A remark on commuting operator exponentials

In a previous paper the author proved that for square matrices with algebraic entries expexpexpexp if and only if . This result is extended here to bounded operators on an arbitrary Banach space.

     

A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.     

Voronovskaya‐type theorems for Urysohn type nonlinear Bernstein operators

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators.     

Browder-Weyl theorems, tensor products and multiplications

A Banach space operator T∈B(X) satisfies Browder's theorem if the complement of the Weyl spectrum σw(T) of T in σ(T) equals the set of Riesz points of T; T is polaroid if the isolated points of σ(T) are poles (no restriction on rank) of the resolvent of T. Let Φ(T) denote the set of Fredholm points of T. Browder's theorem transfers from A,B∈B(X) to S=LARB (resp., S=A⊗B) if and only if A and B^∗ (resp., A and B) have SVEP at points μ∈Φ(A) and ν∈Φ(B) for which λ=μν∉σw(S). If A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to LARB; again, restricting ourselves to the completion of X⊗Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to A⊗B.     

关于近次正常加权移位算子的一个注记

本文给出了双侧加权移位算子的近次正常性的完全刻画.作为主要结果的应用,文章的最后提供了Hilbert空间第160问题的许多新的答案.     

A new criterion for -hyponormality via weak subnormality

In this article we obtain a criterion for -hyponormality via weak subnormality. Using this criterion we recapture Spitkovskii's subnormality criterion and give a simple proof of the main result in Gu's preprint (2001), which describes a gap between -hyponormality and ()-hyponormality for Toeplitz operators. In addition, we notice that the minimal normal extension of a subnormal operator is exactly the inductive limit of its minimal partially normal extensions.

     

Heisenberg群上Folland—Stein定理的注记

该文在G．B．Folland与E．M．Stein研究的算子的基础上．拓展考虑了算子，其中λ＋μ≠0且λ≠α／2n，μ≠—α／2n），证明了：如果，使得有限（其中ψa，b，1（z，t）＝—4（λ＋μ）ab（｜z｜2＋1—it）a-1（｜z｜2＋1＋it）b-1，那么在分布的意义下将有．特别，当λ＝μ＝—1／2时，此结果即原来的Folland－Stein定理．     

Constructive insertion theorems and extension theorems on extremally disconnected frames

Constructive insertion theorems and extensions theorems on extremally disconnected frames are proved. As a consequence, constructive Eilenberg-Steenrod-Taimanov extension theorem is presented. Received March 15, 1999; accepted in final form July 10, 2000.     

A remark on Littlewood-Paley theory for the distorted Fourier transform

We consider the classical theorems of Mikhlin and Littlewood-Paley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator . We show that for such operators which display a zero energy resonance the full range in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to .

     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.