<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Normality concerning shared values

Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry     

Remarks on Thurston’s Construction of Pseudo-Anosov Maps of Riemann Surfaces

It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S／{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston＇s construction.     

Binormal operator and *-Aluthge transformation

Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π＇ be automorphic irreducible cuspidal representations of GLm（QA） and GLm′ （QA）, respectively, and L（s, π×π′） be the Rankin-Selberg L-function attached to π and π＇. Without assuming the Generalized Ramanujan Conjecture （GRC）, the author gives the generalized prime number theorem for L（s, π × π′） when π =π＇. The result generalizes the corresponding result of Liu and Ye in 2007.     

Unconditional ideals of finite rank operators

Let X be a Banach space. We give characterizations of when is a u-ideal in for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when is a u-ideal in for every Banach space Y, when is a u-ideal in for every Banach space Y, and when is a u-ideal in for every Banach space Y.     

On the distribution of the residues of small multiplicative subgroups of $$ \mathbb{F}_p $$

Distributional properties of small multiplicative subgroups of are obtained. In particular, it is shown that if H < is of size larger than polylogarithmic in p, then, letting β < 1 be a fixed exponent, most elements of any coset aH (a ∈ , arbitrary) will not fall into the interval [−p ^β, p ^β] ∈ . The arguments are based on the theory of heights and results from additive combinatoric.     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

Compact solutions to the equation Tx = y in a weakly closed $$ \mathcal{T}(\mathcal{N}) $$-module

Given two vectors x, y in a Hilbert space and a weakly closed -module , we provide a necessary and sufficient condition for the existence of a compact operator T in satisfying Tx = y.     

Bernstein n-widths for classes of convolution functions with kernels satisfying certain oscillation properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

Approximately isometric lifting in quasidiagonal extensions

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of such that ∥a−b∥ < . As an application, it is shown that for any positive numbers ε and in U(A/I) ₀ , there exists u in U(A)₀ which is a lifting of such that cel(u) < cel. This work was supported by National Natural Science Foundation of China (Grant No. 10771161)     

On the relations among best approximation and Fourier coefficients

We discuss the relations among the best approximation E _n (f) and the Fourier coefficients of a function, under the conditions that and , where MVBVS^* is the class of the so-called Strong Mean Value Bounded Variation Sequences. This work was supported by NSERC RCD grant and AARMS of Canada (Yu D S), by NSERC of Canada (Zhou P), and by the National Natural Science Foundation of China (Grant No. 10471130) and the Open Fund (Grant No. PLN0613) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) (Zhou S P)     

Compact corigid objects in triangulated categories and co-<Emphasis Type="Italic">t</Emphasis>-structures

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, , of a triangulated category, , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on whose heart is equivalent to Mod(End()^op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, , of a triangulated category, , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End()^op), and hence an abelian subcategory of .      

Divisors of Bernoulli Sums

Let where are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums , where and is a sequence of positive integers. Received: May 23, 2007. Revised: June 8, 2007.     

Stability of homogeneity almost everywhere

We consider approximately ϕ-homogeneous mappings almost everywhere, that is functions F such that the difference F(αx) − ϕ(α)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of ϕ and F, or there exist a homomorphism and a -homogeneous function , which are almost everywhere equal to ϕ and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.      

On the existence of Fagnano trajectories in convex polygonal billiards

We prove that an n-gon is pedal strong Fagnano trajectory in some convex polygonal billiard table if and only if it is a convex Poncelet n-gon.      

Some functional equations on standard operator algebras

The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let (H) be the algebra of all bounded linear operators on H, and let (H) ⊂ (H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T: (H) → (H) is a linear mapping satisfying T(AA* A) = T(A)A* A − AT(A*)A + AA*T(A) for all A ∈ (H). Then T is of the form T(A) = AB + BA for all A ∈ (H), where B is a fixed operator from (H). A result concerning functional equations related to bicircular projections is proved      

Anisotropic estimates for sub-elliptic operators

In the 1970's,Folland and Stein studied a family of subelliptic scalar operators L_λwhich arise naturally in the(?)_b-complex.They introduced weighted Sobolev spaces as the natural spaces for this complex,and then obtained sharp estimates for(?)b in these spaces using integral kernels and approximate inverses.In the 1990's,Rumin introduced a differential complex for compact contact manifolds,showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator,and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper,we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.