Difference of composition operators on Hardy space

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.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

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 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.

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C

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.

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

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

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.

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

A fractional linear transformation, mapping the unit disc into itself, gives rise to a weighted composition operator on the Hardy space H 2 .Such operators have been recently used in [11] in connection with an extremal problem from operator theory.In this paper, we investigate the basic properties of these operators and determine their spectra.The results can be compared to those for unweighted composition operators on various spaces [1], [3], [5].

A necessary and sufficient condition is obtained for M‐quasi‐hyponormal composition operators. It has also been proved that the class of M‐quasi‐hyponormal composition operators coincides with the class of M‐paranormal composition operators. Existence of M‐hyponormal composition operators which are not hyponormal; and M‐quasihyponormal composition operators which are not M‐hyponormal and quasi‐hyponormal are also shown.

The adjoint of the classic composition operator on the Hardy space of the unit disc determined by a holomorphic self map of the unit disc is well known to send the Szegő kernel function associated to a point in the unit disc to the Szegő kernel associated to the image of that point under the self map. The purpose of this paper is to show that a constructive proof that holomorphic functions that extend past the boundary can be well approximated by complex linear combinations of the Szegő kernel function gives an explicit formula for the adjoint of a composition operator that yields a new way of looking at these objects and provides inspiration for new ways of thinking about operators that act on linear spans of the Szegő kernel. Composition operators associated to multivalued self mappings will arise naturally, and out of necessity. A parallel set of ideas will be applied to composition operators on the Bergman space.

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).