Linear sums of two composition operators on the Fock space

Linear sums of two composition operators of the multi-dimensional Fock space are studied. We show that such an operator is bounded only when both composition operators in the sum are bounded. So, cancelation phenomenon is not possible on the Fock space, in contrast to what have been known on other well-known function spaces over the unit disk. We also show the analogues for compactness and for membership in the Schatten classes. For linear sums of more than two composition operators the investigation is left open.     

Linear Combinations of Composition Operators on the Fock-Sobolev Spaces

We study linear combinations of composition operators acting on the Fock-Sobolev spaces of several variables. We show that such an operator is bounded only when all the composition operators in the combination are bounded individually. In other words, composition operators on the Fock-Sobolev spaces do not possess the same cancelation properties as composition operators on other well-known function spaces over the unit disk. We also show the analogues for compactness and the membership in the Schatten classes. In particular, compactness and the membership in some/all of the Schatten classes turn out to be the same.     

HardyOrlicz空间之间的加权复合算

该文研究了复平面中单位圆盘上不同Hardy-Orlicz空间之间的加权复合算子,利用Carleson测度不等式给出了有界或紧的加权复合算子ωC_φ:N_p→N_q的特征. 作为推论得到了加权复合算子ωC_φ:N_p→N_q有界(或紧)的充分必要条件是ωC_φ:H_p→H_q是有界(或紧)的. 此外,还给出了Hardy-Orlicz空间上可逆及Fredholm复合算子的特征.     

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.     

μ-Bergman空间到Zygmund型空间的加权复合算子

本文研究了单位圆盘D上的μ-Bergman空间到Zygmund型空间的加权复合算子的有界性和紧性问题.利用泛函分析多复变的方法,获得了单位圆盘上μ-Bergman空间到Zygmund型空间的加权复合算子为有界算子和紧算子的充要条件.     

Integral operators on BMO and Campanato spaces

This paper establishes the mapping properties of integral operators on space of bounded mean oscillation and Campanato spaces. In particular, we have the Hardy’s inequality and the boundedness of the Hadamard fractional integrals on space of bounded mean oscillation and Campanato spaces.     

不同Bers型空间之间的加权复合算子

该文讨论了单位圆盘上不同Bers型空间之间的加权复合算子的有界性、紧性和弱紧性, 给出了一些充分必要的判别条件, 特别地得到不同Bers型空间上加权复合算子的紧性与弱紧性的等价性. 这些推广了经典的复合算子与乘法算子的相关结论. 该文同时也给出了Bers型空间上复合算子的Fredholm性和闭值域问题的刻画, 完善了文献[6]中结论.     

Composition operators between Bergman and Hardy spaces

We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.

     

Two classroom proofs concerning composition operators

We consider composition operators on spaces of analytic functions. First we give an elementary proof for the well-known fact that these operators are bounded on the usual Hardy spaces. Then, for more general spaces, we give two results concerning the question when composition operators are injective or onto, and get a complete answer for a class of spaces, including the Hardy spaces.     

Weighted Composition Operators From H^\infty into the Zygmund Spaces

In this work we characterize the bounded and the compact weighted composition operators from the space $H^\infty $ of bounded analytic functions on the open unit disk into the Zygmund space and the little Zygmund space. We also provide boundedness and compactness criteria of the weighted composition operators from the Bloch space into the little Zygmund space. In particular, we show that the bounded operators between these spaces are necessarily compact.     

Adjoints of linear fractional composition operators on the Dirichlet space

The adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.This work was partially supported by Plan Nacional I+D Ref. BFM2000-0360 and Junta de Andalucía Ref. FQM-260. The first named author was also supported by Plan Propio de la Universidad de Cádiz.     

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.     

Composition Operators on Small Spaces

We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.     

Weighted composition operators between Dirichlet spaces

In this article, we study the boundedness of weighted composition operators between different vector-valued Dirichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued Dirichlet spaces are different counterparts of the classical scalar-valued Dirichlet space and characterize the boundedness of multiplication operators between these different spaces.     

A priori estimates for solution operators of diffusion equations

The diffusion equation [d]=Au is considered, where u=u(t,x), t>0, and A is a second order uniformly elliptic differential operator in R^m Whose coefficients are bounded. Other conditions are prescribed on A to generate known soiution operators. We derive growth estimates for these solution operators in certain function spaces together with estimates for their derivatives in t and also estimates on the products of the first two spatial derivatives with these solution operators. Bounds on the solution operators are given which depmd only upon the i.u.b.'s for the ternination coefficients of A and the formal adjoint A_ * of A : These estimates are best with respect to each function space considered in the sense that equality holds for a particular solution operator     

U-ideals of factorable operators

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons's construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

Difference of Differentiation Composition Operators on the Fractional Cauchy Transforms Spaces

Abstract

In this paper, we consider the boundedness and compactness of the differences of differentiation composition operators from the space of fractional Cauchy transforms to the Bloch-type spaces and the weighted Dirichlet spaces. Surprisingly, these characterizations are free from pseudo-hyperbolic metric, which is a common feature of all the preceding characterizations of difference of differentiation composition operators on various spaces of holomorphic functions.     

AN UPPER BOUND OF THE ESSENTIAL NORM OF COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

In this paper, we define the generalized counting functions in the higher dimensional case and give an upper bound of the essential norms of composition operators between the weighted Bergman spaces on the unit ball in terms of these counting functions. The sufficient condition for such operators to be bounded or compact is also given.