A class of weighted composition operators on $H^2$

The role of the angle in supercyclic behavior

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

(Communicated by G. Sinnamon) Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic func- tion on D. Then a weighted composition operator induced by ϕ with weight w is given by (Ww,ϕ f)(z )= w(z)f(ϕ(z)) ,f orz ∈ D and f analytic on D.W efind a sufficient condition un- der which two composition operators lie in the same path component of C(H 2 ) ,a nd wefind as ufficient condition for the difference of such operators to be compact on H 2 (D) .T hen we provide another example that answers a question raised by Shapiro and Sundberg (18 )n ega- tively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H2(D).

The aim of the paper is to obtain suitable state-space models of continuous biotechnological processes (CBTP) in the framework of Linear Fractional Transformation (LFT). The LFT models are starting point in most of the advanced robust control design and analysis methods. Therefore, a linearized process model in the state-space is used whose elements are supposed to vary within certain bounds to represent the nonlinear behaviour of the real plant. The performance specifications are defined in the frequency domain through weighting functions. Two LFT models of CBTP are obtained ready for controller design aimed to optimize robust stability margins and robust performance, respectively.

This is an introductory print on the basic material in the topic of fractional linear maps. It begins with the classical complex analysis material in the complex plane and in the unit disc and segues into a few such topics in advanced linear algebra, geometric function theory, etc. The efficacy of the material and its inroads into many other aspects of classical complex analysis, from operator theory (composition operators on spaces of analytic functions) to automorphisms of unit balls (in spaces of dimension one to those of infinite dimension) and other topics, is extremely useful and ramified. We hope the reader will find it of interest and possibly pursue one of these topics.

Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism.This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra W 1,2 [0, ∞) that induces a bijection between the lattice of the composition operator and the closed ideals of W 1,2 [0, ∞).In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions.As a consequence, such composition operators have no non-trivial reducing subspaces.For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra W 1,2 [0, ∞).Although such a characterization is known, the proof we provide here is somehow different.

We prove that the norm of a weighted composition operator on the Hardy space H 2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space as- sociated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H 2 , and recover the standard upper bound for the norm. Sim- ilar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.

We show that every non-compact weighted composition operator $f \mapsto u\cdot (f\circ\phi)$ acting on a Hardy space $H^p$ for $1 \leq p < \infty$ fixes an isomorphic copy of the sequence space $\ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a copy of the Hilbert space $\ell^2$. These results extend earlier ones obtained for unweighted composition operators.

Linear Fractional Transformations (LFTs) are a widely used modelling framework in modern control and identification. Deriving such models from physical first principles is a non-trivial and often error-prone process, if carried out manually. In this paper a new approach to reduced-order LFT modelling and identification starting from equation-based, object-oriented (O-O) descriptions of the plant (formulated using the Modelica language) and experimental data are presented. This approach allows to reduce the gap between user-friendly model representations, based on object diagrams with physical connections, block diagrams with signal connections, and generic differential-algebraic models, and the use of advanced LFT based identification and control techniques.

In this paper, we study stability and performance properties of linear fractional transformation (LFT) parameter-dependent systems using duality theory and tools from convex analysis. A pair of conjugate functions, the convex hull and the maximum of a family of quadratic functions, are used for analysis and synthesis of LFT systems. Sufficient synthesis conditions for both robust state feedback and gain-scheduling output feedback control problems are formulated as a set of linear matrix inequalities (LMIs) with linear search over scalar variables. Finally, a numerical example is used to demonstrate the advantages of the proposed approaches.

Linear combinations of composition operators induced by linear fractional maps on the infinite polydisk

The Invariant Subspace Problem (“ISP”) for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H 2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results: If φ is a hyperbolic disc automorphism with fixed points at a and β (both necessarily on the unit circle), and C φ the composition operator it induces on H 2, then for every \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H 2, the doubly C φ-cyclic subspace generated by f contains many independent eigenvectors; more precisely, the point spectrum of C φ’s restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C φ is hypercyclic (some forward orbit is dense). Under the stronger restriction \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H p for some p > 2, the point spectrum of the restricted operator contains an open annulus centered at the origin.KeywordsComposition operatorhyperbolic automorphismInvariant Subspace ProblemMathematics Subject Classification (2000)Primary 47B33Secondary 47A15

This paper is concerned with the gain-scheduled H∞ filtering problem for a class of parameter-varying continuous systems with time delays. The systems under consideration are represented by nonlinear fractional transformation (NFT) which is a generalization of linear fractional transformation (LFT). Attention is focused on the design of a stable filter guaranteeing a prescribed disturbance attenuation level in an H∞ sense. Sufficient solvability conditions of this problem are obtained based on Lyapunov function approach. A gain-scheduled filter can be constructed in terms of a set of linear matrix inequalities (LMIs). A numerical example is provided to demonstrate the applicability of the proposed approach.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.