Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

Algebraic functions of normal operators

The solutions ofp(T)=S, whereS is normal andp a polynomial, are described. This work was supported, in part, by Grant GP-3628 of the National Science Foundation.     

Algebraic extensions of semi-hyponormal operators

In this paper, we show that algebraic extensions of semi-hyponormal operators (defined below) are subscalar. As corollaries we get the following:

Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g^∗ related to an algebraic Nijenhuis operator on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on gln.     

Algebraic closure operators and strong amalgamation bases

Axioms for a closure operator are given and it is proved that, for suitable classes, a unique largest closure operator exists. Necessary and sufficient conditions in terms of closure properties are given for structures to be strong amalgamation bases. Research supported by NSF GP 34091X Presented by B. Jónsson     

Fredholm composition operators

Fredholm composition operators on a variety of Hilbert spaces of analytic functions on domains in , are characterized.

     

Equivalent composition operators

If is an analytic function taking the unit disk into itself then the composition operatorC can be defined on the Hardy spaceH ^p(D) byC (f)=fo. In this work, the question of when two of these operators are equivalent in some sense is studied. In some cases, it turns out that the only equivalences are those that are induced by an invertible composition operator. However, other cases are exhibited in which there are equivalences that are not induced by an invertible composition operator.Much of this work comes from the author's thesis, written under the direction of Carl C. Cowen at Purdue University. Final preparations were accomplished with the help of a faculty development grant from the University of Tampa.     

Bloch-to-Hardy composition operators

Let φ be a holomorphic mapping between complex unit balls. We characterize those regular φ for which the composition operators C _φ : f ? f ○ φ map the Bloch space into the Hardy space.     

Algebraic operators and pairs of self-adjoint operators connected by a polynomial relation

Institute of Mathematics, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 72–74, October–December, 1991.     

Algebraic and spectral properties of dual Toeplitz operators

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.

     

Invertible weighted composition operators

A weighted composition operator Cψ,φ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ⋅f°φ where φ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and φ are characterized when Cψ,φ acts on the Hardy-Hilbert space of the unit disc H²(D). Depending upon the nature of the fixed points of φ spectra are then investigated.     

Measure preserving composition operators

We consider composition operators T induced on functional Hilbert spaces H = L₂(S, ∑, μ) byTf(·) = f(h(·)) where h: S → S is a nonsingular transformation. For these mappings T: H → H we give conditions under which they accept invariant Borel probability measures, and we relate the two structures of T, i.e., that of a bounded linear operator to that of a measure preserving transformation.     

Slice regular composition operators

In the article, we study the theory of slice regular compositions. Especially, the Littlewood subordination principle and the Denjoy–Wolff-type theorem are established for slice regular functions.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

Algebraic properties of Toeplitz operators on the Dirichlet space

We study some algebraic properties of Toeplitz operators on the Dirichlet space. We first characterize (semi-)commuting Toeplitz operators with harmonic symbols. Next we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. As an application, we show that the zero product of two Toeplitz operators with harmonic symbol has only a trivial solution. Also, the corresponding compact product problem is studied.     

Algebraic properties of Toeplitz operators on weighted Bergman spaces

Czechoslovak Mathematical Journal - We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property...