一类新型非正规算子的谱分析

本文引入了一类新型非正规算子,即具复谱的u-标算子.证明了该类算子具有Jordan型分解,讨论了u-标算子与标型谱算子的关系,并通过例子说明了该类算子的构造.     

One-point commuting difference operators of rank 1

One-point commuting difference operators of rank 1 are considered. The coefficients in such operators depend on one functional parameter, and the degrees of shift operators in difference operators are positive. These operators are studied in the case of hyperelliptic spectral curves, where the base point coincides with a point of branching. Examples of operators with polynomial and trigonometric coefficients are constructed. Operators with polynomial coefficients are embedded in differential operators with polynomial coefficients. This construction provides a new method for constructing commutative subalgebras in the first Weyl algebra.     

关于p-弱亚正规算子的一个注记

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Generalized random linear operators on a Hilbert space

In this paper, we are concerned with generalized random linear operators on a separable Hilbert space. Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated. The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Fixed points of generalized e-concave (generalized e-convex) operators and their applications

In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations.     

Estimates on level set integral operators in dimension two

Both oscillatory integral operators and level set operators appear naturally in the study of properties of degenerate Fourier integral operators (such as generalized Randon transforms). The properties of oscillatory integral operators have a longer history and are better understood. On the other hand, level set operators, while sharing many common characteristics with oscillatory integral operators, are easier to handle. We study L²-estimates on level set operators in dimension two and compare them with what is known about oscillatory integral operators. The cases include operators with non-degenerate phase functions and the level set version of Melrose-Taylor transform (as an example of a degenerate phase function). The estimates are formulated in terms of the Newton polyhedra and type conditions.     

Norm Inequalities for Fractional Integral Operators on Generalized Weighted Morrey Spaces

Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds,the fractional integral operators with rough kernels and fractional maximal operators with rough kernels as special cases,we prove that if these operators are bounded on weighted Lebesgue spaces and satisfy some local pointwise control,then these operators and the commutators of these operators with a BMO functions are also bounded on generalized weighted Morrey spaces.     

Norm Inequalities in Generalized Morrey Spaces

We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.     

Super-Ergodic operators

The aim of this work is to study operators naturally connected to Ergodic operators in infinite-dimensional Banach spaces, such as Uniform-Ergodic, Cesaro-bounded and Power-bounded operators, as well as stable and superstable operators. In particular, super-Ergodic operators are introduced and shown to be strictly between Ergodic and Uniform-Ergodic operators, and that any power bounded operator is super-Ergodic in a superreflexive space. New relationships between these operators are shown, others are proven to be optimal or can be ameliorated according to structural properties of the Banach space, such as the superreflexivity or with unconditional basis.