Composition Operators on the Space of Bounded Harmonic Functions

We study composition operators on the space of bounded harmonic functions on the open unit disk. The principal goal of this paper is to provide criteria for determining the essential norm of difference of two composition operators. The third author is partially supported by Grant-in-Aid for Scientific Research (No.17540169), Japan Society for the Promotion of Science.     

A Uniqueness Theorem for Bounded Analytic Functions

Suppose that K is a linear space of functions analytic in somedomain D in the complex plane. A sequence = (_k) of distinctpoints from D is said to be a set of uniqueness for K if fKand f(_k) = 0 for all k imply f0. Depending on the dispersionand the density of on the one hand, and the growth of the functionsin K on the other, one may often require only |f(_k)| a_k forsome sequence of positive numbers a_k, and still conclude thatf0 for fK. Of particular interest are sharp conditions on thedecay of a_k, which reflect the interplay between growth anddecay of analytic functions. 1991 Mathematics Subject Classification30A99, 31A05.     

On Bounded Variation Functions by General MKZD Operators

In this paper, we study the rate of convergence for functions of bounded variation for the recently introduced Bzier variant of the Meyer-Knig-Zeller-Durrmeyer operators.     

Sufficient Conditions for Bounded Turning of Analytic Functions

Ukrainian Mathematical Journal - Consider a function f analytic in the open unit disk and normalized so that f(0) = f′(0) ? 1 = 0. The methods of the theory of first-order differential...     

Weighted Composition Operators on the Space of Bounded Harmonic Functions

We determine the norm and the essential norm of the difference of weighted composition operators on the space of bounded harmonic functions on the open unit disk. The argument is done on the boundary.     

A Uniqueness Theorem for Bounded Analytic Functions on the Polydisc

Given n, N ≥ 1 we construct a set of points ${\lambda_1,{\ldots},\lambda_{N^n}\in{\mathbb D}^n}$ such that for each rational inner function f on ${{\mathbb D}^n}$ of degree less than N the Pick problem on ${{\mathbb D}^n}$ with data ${\lambda_1,{\ldots},\lambda_{N^n}}$ and ${f(\lambda_1),{\ldots},f(\lambda_{N^n})}$ has a unique solution. In particular, we construct a 1-dimensional inner variety V and show that the points ${\lambda_1,{\ldots},\lambda_{N^n}}$ may be chosen almost arbitrarily in ${V\cap{\mathbb D}^n}$ . Our results state that f is uniquely determined in the Schur class of ${{\mathbb D}^n}$ by its values on ${\lambda_1,{\ldots},\lambda_{N^n}}$ .     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

Optimal Recovery of the Derivative of Bounded Analytic Functions

Given (–1, 0), n N, we discuss the optimal recoveryof (), for analytic and bounded in < 1, from the knowledge of the values of at n points z1,.zm[0,l),where these points are chosen to produce the least possibleintrinsic error. The optimal algorithms are explicitly determined.     

Third-Order Hankel Determinant for Transforms of the Reciprocal of Bounded Turning Functions

Mathematical Notes - In this paper, we make an attempt to introduce a new subclass of analytic functions. Using the Toeplitz determinants, we obtain the best possible upper bound for the...     

Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

The essential norm of weighted composition operators on weightedBanach spaces of analytic functions is computed in terms ofthe weights and the inducing symbols. As a consequence the boundednessand compactness of these operators is characterized. As anotherconsequence the essential norm of composition operators on weightedBloch spaces is obtained and, consequently, the boundednessand compactness of composition operators on these spaces isalso characterized. Particular instances of weighted Bloch spacesare the Lipschitz spaces. The method used allows a unified treatmentof the problem of boundedness and compactness on these spaces.     

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 .     

Uniformly Bounded Composition Operators on the Space of Bounded Variation Functions in the Sense of Waterman

In this paper, we demonstrate under some general assumptions, that a generator of any uniformly bounded composition operator, mapping spaces of bounded variation (Waterman) functions into other spaces of this type, must be an affine function in the functional variable.

     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

加权BMO函数空间上的Littlewood-Paley算子

Littlewood—Paley算子(g—函数,s—函数与_λ^*—函数,λ>3)作为BMO_w或(BMO)_w上的算子都是“有界的”,确切地说,我们证明了:若f∈BMO_w或(BMO)_w且|{x:Tf(x)≠∞}|>0,则Tf也属于BMO_w或(BMO)_w并且存在与f无关的常数C使‖Tf‖_BMOw<     

Tensor Products and Operators in Spaces of Analytic Functions

Let X be an infinite dimensional Banach space. The paper provesthe non-coincidence of the vector-valued Hardy space H^p(T, X)with neither the projective nor the injective tensor productof H^p(T) and X, for 1 < p < . The same result is provedfor some other subspaces of L^p. A characterization is givenof when every approximable operator from X into a Banach spaceof measurable functions F(S) is representable by a functionF:S X* as x F(·), x. As a consequence the existenceis proved of compact operators from X into H^p(T) (1 p <) which are not representable. An analytic Pettis integrablefunction F:T X is constructed whose Poisson integral does notconverge pointwise.     

Operators of Hankel type

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. The present note provides a parametric labeling of all the Hankel symbols of a given Hankel operator X by means of Schur class functions. The result includes uniqueness criteria and a Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to the data of the problem a suitable isometry V so that there is a bijective correspondence between the symbols of X and the minimal unitary extensions of V.     

Hankel Operators on Weighted Fock Spaces

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations.     

广义Fock空间上的Hankel算子

利用有界(消失)平均振荡函数的性质,本文刻画了一类广义Fock空间上的Hankel算子的有界性(紧性),同时,还刻画了换位子[M_f,P]的有界性和紧性,其中P是一个Toeplitz投影算子,而M_f表示符号为f的乘子.最后,应用Berezin变换来研究了BMO空间和VMO空间的几何性质.