Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

When do Toeplitz and Hankel operators commute?

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolz, where is the symbol of the Toeplitz operator, and is an affine function of the characteristic function of certain anti-symmetric sets of the unit circle.     

q-subharmonicity and q-convex domains in ℂ n

In this paper we study q-subharmonic and q-plurisubharmonic functions in ? ⁿ . Next as an application, we give the notion of q-convex domains in ? ⁿ which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.     

On fixed points of algebraic actions on ℂ n

It is shown that the action regarded for a rather long time by experts as a possible example disproving the conjecture on the existence of fixed points for reductive algebraic group actions on affine spaces is not an action on an affine variety, and therefore provides no example of this kind. Moreover, it is shown that the actions naturally related to the original one provide no examples of this kind as well. Supported by CRDF grant RM1-206 and INTAS grant INTAS-OPEN-97-1570. Moscow Independent University. Moscow Institute of Electronics and Mathematics. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 41–50, January–March, 2000 Translated by V. L. Popov     

总体紧的Toeplitz和Hankel算子

本文刻画具有总体紧性质的Ｔｏｅｐｌｉｔｚ和Ｈａｎｋｌｅｌ算子序列的符号函数，这是单个算子情形的推广。     

On the injectivity of twisted spherical means on ℂ n

In this paper we study the injectivity and injectivity sets for the twisted spherical means on ℂ ⁿ . These results are then used to study the same problems on the reduced Hiesenberg group.     

Zonal spherical function on the Lie ball in ℂn

In this paper, we give the explicit formula for the zonal spherical function on the Lie ball in ?ⁿ.     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .     

关于Toeplitz+Hankel线性方程组的迭代解法

本文研究Toeplitz+Hankel线性方程组的预处理迭代解法.我们提出了几个新的预条件子,并分析了预处理矩阵的谱性质,当生成函数在Wiener类中时,预处理矩阵的特征值聚集在1附近.数值实验表明该预处理子比文[5]中的预处理子更有效.     

圆环上的Hankel与Toeplitz型算子

令Ω是复平面上的圆环 ,则L2 (Ω) = ∞-∞Ak = ∞-∞ Ak 为其正交直和分解 .本文中我们定义了一类Hankel和Toeplitz型算子 ,研究了它们的紧性和Sp 性质 .     

Research problem: on the norms of infinite Cauchy–Toeplitz-plus-Cauchy–Hankel matrices

We give a conjecture and research problem related to ? _p operator and spectral norms of the matrix T _n ?+?H _n .     

位圆上的Hankel和Toeplitz型算子

令D是复平面上的单位圆, 为其加权的测度, 则 为正交直和分解. 定义了一类Hankel和Toeplitz型的算子, 研究了它们的有界性、紧性和Sp准则.     

单位圆上的Hankel和Toeplitz型算子

令D是复平面上的单位圆,dμβ(z)=β+1/π(1-|z|2)βdm(z)β>-1)为其加权的测度,则L2(D,dμβ(z))= ∞k=00(Aβk -βAk)为正交直和分解.定义了一类Hankel和Toeplitz型的算子,研究了它们的有界性、紧性和Sp准则.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

一类Hankel算子与Toeplitz算子的Sp性质

对于L~(α,2)(D))的两类Moebius不变子空间A~(α,2)(D)和A~(β,2)(D),我们定义了它们之间的Toeplitz算子T_f~s与其乘积空间上的Hankel算子H_f~r,并且研究了它们的有界性、紧性及Schatten-von Neumann性质。     

An Internal Polya Inequality for ℂ-Convex Domains in ℂn

Mathematical Notes - Let K ? ? be a polynomially convex compact set, f be a function analytic in a domain $$\overline{\mathbb{C}} \backslash K$$ with Taylor expansion $$f(z) =...