Abstract Characteristic Function

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k_S(z,w) = (1 - z [`(w)])^-1{k_S(z,w) = (1 - z {\overline {w}})^{-1}} for |z|, |w| < 1, by means of (1/k _S )(T, T*) ≥ 0, we consider an arbitrary open connected domain Ω in \mathbb Cⁿ{{\mathbb {C}}^n}, a kernel k on Ω so that 1/k is a polynomial and a tuple T = (T ₁, T ₂, . . . , T _n ) of commuting bounded operators on a complex separable Hilbert space H{\mathcal H} such that (1/k)(T, T*) ≥ 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.     

Modular curves and bases for the spaces of cuspidal modular forms

Let Γ⊂SL ₂(ℝ) be a Fuchsian group of the first kind. For a character χ of Γ→ℂ^× of finite order, we define the usual space S _m (Γ,χ) of cuspidal modular forms of weight m≥0. For each ξ in the upper half–plane and m≥3, we construct cuspidal modular forms Δ _k,m,ξ,χ ∈S _m (Γ,χ) (k≥0) which represent the linear functionals f?\fracd^kfdz^k|_z=xf\mapsto\frac{d^{k}f}{dz^{k}}|_{z=\xi} in terms of the Petersson inner product. We write their Fourier expansion and use it to write an expression for the Ramanujan Δ-function. Also, with the aid of the geometry of the Riemann surface attached to Γ, for each non-elliptic point ξ and integer m≥3, we construct a basis of S _m (Γ,χ) out of the modular forms Δ _k,m,ξ ,χ (k≥0). For Γ=Γ ₀(N), we use this to write a matrix realization of the usual Hecke operators T _p for S _m (N,χ).     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.      

Some Closed Range Integral Operators on Spaces of Analytic Functions

Our main result is a characterization of g for which the operator S_g(f)(z) = ò₀^z f￠(w)g(w) dw{S_g(f)(z) = \int_0^z f'(w)g(w)\, dw} is bounded below on the Bloch space. We point out analogous results for the Hardy space H ² and the Bergman spaces A ^p for 1 ≤ p < ∞. We also show the companion operator T_g(f)(z) = ò₀^z f(w)g￠(w)  dw{T_g(f)(z) = \int_0^z f(w)g'(w) \, dw} is never bounded below on H ², Bloch, nor BMOA, but may be bounded below on A ^p .     

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C _t = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C ¹-family of circles in the plane such that lim _t→0+ C _t = {a}, lim _t→1− C _t = {b}, a ≠ b, and |c′(t)|² + |r′(t)|² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation ̅c′(t)w ² + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.     

Value distribution of the Painlevé Transcendents

We consider the solutions of the First Painlevé Differential Equationω″=z+6w ², commonly known as First Painlevé Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z ⁻¹ w(z ²), respectively:w#(z)=O(|z|^3/4) andf#(z)=O(|z|^3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painlevé’s Equations II and IV.     

拟圆周的两个几何性质

§1　IntroductionLetΓbe a Jordan curve of R2 and f∶R2→R2 be a k-quasiconformal mapping,where1≤k<+∞.Γis called a quasicirlce ifΓis the image of the unit circle B2 under f.It is well-known that quasicircles play a very important role in quasiconformalmapping theory,complex dynamics,Fuchsian groups,Teichmuller space theory and lowdimensional topology,( see[1—5] etc.)In1 963 ,Ahlfors obtained the three-point property of quasidisks[6] .Later,Gehring[7] ,Osgood[8] ,Krzyz[9] ,Ch…     

On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ _c and τ _w are introduced on M _a (S)^*. We introduce τ _c and τ _w almost periodic functionals in M _a (S)^*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M _a (S)^*, it is proved that T _f ∈ℬ(M _a (S),M _a (S)^*) is strong almost periodic if and only if f is τ _c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M _a (S)) has the semiright invariant isometry property, it is shown that the set of τ _w -almost periodic functionals has a topological left invariant mean.     

Nonfree actions of countable groups and their characters

The paper studies the region of values of the system {c ₂, c ₃, f(z ₁), f′(z ₁)},where z ₁ is an arbitrary fixed point of the disk |z| < 1; f ∈ T,and the class T consists of all the functions f(z) = z + c ₂ z ² + c ₃z³ + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z ₁) in the subclass of functions f ∈ T with prescribed values c ₂, c ₃, and f(z ₁) is determined. Bibliography: 10 titles.     

To distortion theorems for typically real functions

The paper studies the region of values of the system {f(z ₁), f(z ₂), c ₂},where z _j , j=1, 2, are arbitrary fixed points of the disk |z|<1; f ∈ T, and the class T consists of all functions f(z) = z + c ₂ z ² + ··· regular in the disk |z| < 1 and satisfying the condition Im f(z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f(z ₁) in the subclass of functions f (z) ∈ T with prescribed values c ₂ and f(z ₂) is determined. Bibliography: 8 titles.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

On Operators Satisfying T＊｜T2｜T≥ T＊｜T＊｜2T

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-＊-A, of operators satisfying T＊｜T2｜T≥ T＊｜T＊｜2T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T＊ ∈l-＊-A, then w（f（T）） = f（w（T））, σea（f（T）） = f（σea（T）） for every f C H（σ（T））, where g（σ（T）） denotes the set of all analytic functions on an open neighborhood of σ（T）.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

A pure quantum mechanical theory of parity effect in tunneling and evolution of spins

In recent years, the spin parity effect in magnetic macroscopic quantum tunneling has attracted extensive attention. Using the spin coherent-state path-integral method it is shown that if the HamiltonianH of a single-spin system hasM - fold rotational symmetry around z-axis, the tunneling amplitude 〈−S|e ^Ht |S〉 vanishes when S, the quantum number of spin, is not an integer multiple ofM/2, where |m〉 (m=-S, -S +1, ⋯, S) are the eigenstates of S^z. Not only is a pure quantum mechanical approach adopted to the above result, but also is extended to more general cases where the quantum system consists ofN spins, the quantum numbers of which can take any values, including the single-spin system, ferromagnetic particle and antiferromagnetic particle as particular instances, and where the states involved are not limited to the extreme ones. The extended spin parity effect is that if the Hamiltonian ℋ of the system ofN spins also has the above symmetry, then 〈m′_N⋯m′₂ m′₁|e^{−H t} |m ₁ m ₂⋯m _N vanishes when ∑ _i=1 ^N (m _i−m′₁) not an integer multiple ofM, where |m ₁ m ₂⋯m _N〉=∏ _α=1 ^N |m _a 〉 are the eigenstates of S _a ^z . In addition, it is argued that for large spin the above result, the so-called spin parity effect, does not mean the quenching of spin tunneling from the direction of ⊕-z to that of ±z. Project supported by the National Natural Science Foundation of China (Grant Nos. 19674002, 19677101).     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on