Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

拟幂零积分算子的等价刻划

本给出了一娄具有非负核函数的积分算子，并得到了该算子为拟零算子的充必要条件，该算子包括了重要的Hardy算子。     

Nilpotent operators similar to irreducible operators

We give a necessary and sufficient condition for the nilpotent operators to be similiar to irreducible operators, and give an answer to D. A. Herrero Conjectures for nilpotent operators.     

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.

     

Triebel-lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel-Lizorkin space. The operators include Littlewood-Paley operator and Marcinkiewicz operator.     

Normal operators similar to irreducible operators

A complete characterization of normal operators which are similar to irreducible operators and some related results are given.     

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 operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Green's operators

Summary Generailzed Green's functions are kernels of integral operators with certain properties. Solution operators with these properties are calledGreen's operators. Necessary and sufficient conditions for the existence of Green's operators are given for the general case, and for operators on Banach spaces. This work has been supported by Sandia Corporation, a prime contractor to the U. S. Atomic Energy Commission.     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

Multilinear localization operators

In this paper we introduce a notion of multilinear localization operators. By reinterpreting these operators as multilinear Kohn-Nirenberg pseudodifferential operators, we give sufficient conditions for their boundedness on products of modulation spaces. Moreover, we prove that these conditions are also necessary for the boundedness of the operators. Finally, as application, we construct various examples of bounded multilinear pseudodifferential operators.     

Eigenvalues of compact operators with applications to integral operators

We give summability results for the eigenvalues of certain types of compact operators that are then applied to study integral operators.     

Gaps of operators

We construct examples which distinguish clearly the classes of p-hyponormal operators for 0<p?∞. In addition, we show that those examples classify the classes of w-hyponormal, absolute-p-paranormal, and normaloid operators on the complex Hilbert space. Only a few examples of p-hyponormal operators have been examined. Our technique can provide many examples related to the above operators.     

Hankel operators in the set of essential Toeplitz operators

In this paper, we study the Hankel operators belonging toE which denotes all of the essential Toeplitz operators, and show that the class of symbols of these Hankel operators is a Douglas algebra. Meanwhile, using the concept of essential commutant and some knowledge of the theory of function algebras, we solve a problem raised by Josè Barria and P. R. Halmos in [1].     

Compactness properties of operators dominated by AM-compact operators

We study several properties about the problem of domination in the class of positive AM-compact operators, and we obtain some interesting consequences on positive compact operators. Also, we give a sufficient condition under which a Banach lattice is discrete.

     

Essentially subnormal operators

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact'.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

     

A way of constructing translation network operators by Bernstein operators

In the present paper,we provide a way of constructing translation network operators by Bernstein-Durrmeyer operators.     

Uncomplementability of spaces of compact operators in larger spaces of operators

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of c ₀ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results in the paper.     

Factorizations of EP operators

In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators.     

Combinatorics of Maass-Shimura operators

Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.