We consider the set 𝒰𝒯(A) consisting of all scalars λ for which operator A-λ I is universal in the sense of Gian-Carlo Rota. We determine 𝒰𝒯(A) in most cases when A is a composition operator on the usual Hardy space H^2 induced by an inner map or the adjoint of that operator. The same problem is considered and solved for select, non-inner, analytic selfmaps of the unit disc and the composition operators induced by them.

Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.

Operators of type acting on function spaces are called weighted composition operators. If the weight function psi is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on the Hilbert Hardy space of a half-plane and study compactness, boundedness, invertibility, normality and spectral properties of such operators.

Hypercyclic operators are operators with dense orbits. A contraction cannot be hypercyclic since its orbits are bounded sets. Nevertheless, by multiplying a contraction with a scalar of absolute value larger than 1, the resulting scaled contraction can occasionally be a hypercyclic operator. In this paper, we investigate which Hilbert space contractions have that property and which don’t. We introduce the set $$\Lambda (T)$$ of all scalars which produce a hypercyclic operator, by scaling the operator T, and determine $$\Lambda (T)$$ in various cases. New properties of hyperciclic operators are discovered in this process. For instance, it is proved that any connected component of the essential spectrum of a hypercyclic operator must meet the unit circle.

In this paper we deal with a scale of reproducing kernel Hilbert spaces H2(n), n≥0, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane C+. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel Kz,n of H2(n), from which we can find the estimate ‖Kz,n‖∼|z|−1/2 for z∈C+. Then composition operators Cφ:H2(n)→H2(n), Cφf=f∘φ, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps φ:C+→C+ to induce bounded composition operators.

Operators of type f → f ○ φ acting on function spaces are called composition operators.We consider composition operators acting on the Hilbert Hardy space on the open unit disc or the right half-plane, study when they are similar to contractions, and obtain results interesting from the point of view of dilation theory of contractions and function theory.

The composition operator induced by a hyperbolic Möbius transform ϕ \phi on the classical Hardy space H 2 {H^2} is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u u in H 2 {H^2} if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u u is continuously extendable at one of the fixed points of ϕ \phi and its value at the point is nonzero, then the cyclic subspace generated by u u is minimal if and only if u u is constant.

This paper contains a collection of open problems related to composition operators and weighted composition operators acting on spaces of analytic functions, predominantly on the Hilbert Hardy space H2 over the open unit disc. Some are related to the invariant subspaces of composition operators, some to the spectra and numerical ranges of such operators, others are related to the connection of certain weighted composition on H2 and the unweighted composition operators on Hardy Smirnov spaces, or the connection of composition operators with asymptotically Toeplitz operators. The problems raised are open to the knowledge of this author, and interesting, in his opinion.

A conjecture posed by Cowen and MacCluer in 1995 states that the spectrum of composition operators acting on the Hardy space H2 induced by analytic selfmaps of the open unit disc having a fixed point in that disc, other than the identity or elliptic automorphisms, is always representable as the union of a connected set containing the origin, the so called, Schröder eigenvalues of the inducing selfmap, and 1. Our first result is proving that the aforementioned conjecture holds. We also consider two related conjectures regarding the spectra of composition operators, proving that one of them holds, and showing that the other one (which is still open) is satisfied in particular cases.

AbstractOperators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

In this paper we consider composition operators C-phi on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps phi. We use the fact that the operator C-phi*C-phi is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of C-phi, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C-phi*C-phi is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by phi. An alternative formula for the essential norm of a composition operator ( valid in select cases), in terms of the aforementioned Nikodym derivative, is established. Estimates of the spectra of adjoints of composition operators are obtained. Based on them, we describe the spectrum of composition operators induced by maps fixing a point, whose iterates exhibit a strong form of attractiveness to that point.

A composition operator is an operator on a space of functions defined on the same set. Its action is by composition to the right with a fixed selfmap of that set. A composition operator followed by a multiplication operator is called a weighted composition operator. In this paper, we study when weighted composition operators on the Hilbert Hardy space of the open unit disc are isometric. We find their Wold decomposition in select cases and apply it to the computation of numerical ranges.

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk \({\mathbb{D}}\) onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces \({L^2_a(\mu _p)}\) on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all \({p\in(-1/3,1)}\). Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

Composition operators on the Hilbert Hardy space H(2) whose symbols are analytic selfmaps of the open unit disk having orthogonal powers are considered. The spectra and essential spectra of such operators are described. In the general case of an arbitrary analytic selfmap of the open unit disk, it is proved that the composition operator induced by that map has essential spectral radius less than 1 if and only if the map under consideration is a non inner map with a fixed point in the unit disk. The canonical decomposition of a non unitary composition contraction is determined.

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces trM^*M^2 and trM^*M^3 are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5x5 nilpotent matrix has circular numerical range are thoroughly discussed.

In this paper we generate upper and lower bounds for the sensitivity to noise of a Boolean function using relaxed assumptions on input choices and noise. The robustness of a Boolean network to noisy inputs is related to the average sensitivity of that function. The average sensitivity measures how sensitive to changes in the inputs the output of the function is. The average sensitivity of Boolean functions can indicate whether a specific random Boolean network constructed from those functions is ordered, chaotic, or in critical phase. We give an exact formula relating the sensitivity to noise and the average sensitivity of a Boolean function. The analytic approach is supplemented by numerical results that illustrate the overall behavior of the sensitivities as various Boolean functions are considered. It is observed that, for certain parameter combinations, the upper estimates in this paper are sharper than other estimates in the literature and that the lower estimates are very close to the actual values of the sensitivity to noise of the selected Boolean functions.

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces.