Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.     

A duality for Boolean algebras with operators

Jónsson and Tarski's extension and representation theorems for Boolean algebras with operators ([7], p. 926 and p. 933) can be extended to homomorphisms between these algebras. The result obtained takes the form of a duality between the category of Boolean algebras with operators and that of the “algebras in the wider sense” (whose subjects are defined in [7]) with a suitable topology. This duality generalizes results of Pierce ([10], p. 38). Moreover, it can be extended to more general objects such as Boolean algebras with non-normal operators and even to arbitrary distributive lattices with operators.     

Lie algebras of unbounded derivations

In this paper some new results on analytic domination of operators and on integrability of Lie algebras of operators are proved and then our methods are applied to the study of Lie algebras of unbounded derivations in $\text{C}{}^1$ algebras.     

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.     

Differential operators on the free algebras

Following the definitions of the algebras of differential operators, ??-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in Lunts and Rosenberg (Selecta Math New Ser 3:335?C359, 1997), we describe these algebras on the free associative algebra. We further study their properties.     

Lie-Yamaguti代数的相对微分算子

本文研究Lie-Yamaguti代数的相对微分算子.首先给出Lie-Yamaguti代数上相对微分算子的概念并给出等价刻画.随后,引入Lie-Yamaguti代数上相对微分算子的上同调.最后,讨论Lie-Yamaguti代数上相对微分算子的无穷小形变.     

Wavelets from Laguerre Polynomials and Toeplitz-Type Operators

We study a parameterized family of Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true poly-analytic Bergman spaces. Isomorphisms between the Calderón-Toeplitz operator algebras and functional algebras are described and their consequences in time-frequency analysis and applications are discussed.     

Interior and closure operators on bounded residuated lattices

Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.     

Modal operators on bounded commutative residuated <Emphasis Type="Italic">ℓ</Emphasis>-monoids

Bounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVá,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras. The first author was supported by the Council of Czech Government, MSM 6198959214.     

Hom-李-Yamaguti代数上的相对罗巴算子

本文引入Hom-李-Yamaguti代数上相对罗巴算子的概念,并利用Nijenhuis算子和图给出相对罗巴算子的等价刻画.随后,引入Hom-李-Yamaguti代数上相对罗巴算子的上同调理论.最后,利用上同调方法探讨相对罗巴算子的形变.     

On Tractability and Ideal Problem in Non-Associative Operator Algebras

The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński’s problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.     

A characterization theorem for weak varieties

In this paper I define the notion of mixed product of partial algebras. It is a common generalization of two operators: weak subalgebras and products of partial algebras. I show that a class of similar partial algebras is a weak variety if and only if it is closed under the operators of mixed products and closed homomorphic images.     

Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

We present here a quite unexpected result: Apart from already known commutative C*-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.     

Invariant Subspaces of Subgraded Lie Algebras of Compact Operators

We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators, in particular for Jordan algebras and Lie triple systems of Volterra operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators. The support received from INTAS project No 06-1000017-8609 is gratefully acknowledged by the third author.     

Complex Powers and Non-compact Manifolds

Abstract

We study the complex powers A ^z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A ^z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).     

Spectral Radius Algebras of Idempotents

We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra.      

Stability of the Blok Theorem

W. Blok proved that varieties of boolean algebras with a single unary operator uniquely determined by their class of perfect algebras (i.e., duals of Kripke frames) are exactly those which are intersections of conjugate varieties of splitting algebras. The remaining ones share their class of perfect algebras with uncountably many other varieties. This theorem is known as the Blok dichotomy or the Blok alternative. We show that the Blok dichotomy holds when perfect algebras in the formulation are replaced by ω-complete algebras, atomic algebras with completely additive operators or algebras admitting residuals. We also generalize the Blok dichotomy for lattices of varieties of boolean algebras with finitely many unary operators. In addition, we answer a question posed by W. Dziobiak and show that classes of lattice-complete algebras or duals of Scott-Montague frames in a given variety are not determined by their subdirectly irreducible members. Received February 14, 2006; accepted in final form March 26, 2007.     

Separability and decidability results for varieties of Jónsson dynamic algebras

Dynamic algebras are algebraic counterparts of dynamic logics: propositional logical systems endowed with a set of modal operators. In [18], B. Jónsson introduced dynamic algebras as Boolean algebras with unary operators, the indices of which range over a given Kleene algebra. On the other hand, V.R. Pratt and D. Kozen proposed a two-sorted approach to dynamic algebras, which was followed in the early papers on the topic, such as Fischer and Ladner [15] and Németi [28]. For a recent overview of the field cf. [4]. In the present paper we investigate connections (as well as diversities) between these two approaches. Our main aim is to transfer (where possible) two-sorted results on separability and decidability to the one-sorted case and to extend them to broad classes of varieties of Jónsson dynamic algebras. In particular, as a consequence of such considerations, we obtain a decidability result on Kleene algebras.     

Norm closure and extension of the symbolic calculus for the cone algebra

We investigate the symbolic structure of an algebra of pseudodifferential operators on manifolds with conical singularities which has been introduced by B.-W. Schulze. Our main objective is the extension of the symbolic calculus of this algebra to its norm closure in an adapted scale of Sobolev spaces. This procedure yields Banach algebras and Fréchet algebras of singular integral operators with continuous principal symbols.