Composition operators on the polydisc

In this paper we study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc ${\mathbb{D}}^n$. Studying the volume of sublevel sets we show for which n the necessary conditions obtained by Bayart are sufficient. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.     

Composition operators induced by symbols defined on a polydisk

Suppose φ is a holomorphic mapping from the polydisk Dm into the polydisk Dn, or from the polydisk Dm into the unit ball Bn, we consider the action of the associated composition operator Cφ on Hardy and weighted Bergman spaces of Dn or Bn. We first find the optimal range spaces and then characterize compactness. As a special case, we show that if

     

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.     

Characterization of composition operators with closed range for one-dimensional smooth symbols

We give a full characterization of smooth symbols ψ:R→R$\psi :\mathbb{R}\to \mathbb{R}$ for which the composition operator _Cψ:C^∞(R)→C^∞(R)$C_{\psi }:C^{\infty }(\mathbb{R})\to C^{\infty }(\mathbb{R})$, F?F°ψ$F?F°\psi$ has closed range. This generalizes in a special case the result of Kenessey and Wengenroth who gave such a characterization for smooth injective   symbols ψ:R→R^d$\psi :\mathbb{R}\to {\mathbb{R}}^d$.     

Composition operators between Bloch-type spaces in the polydisc

We characterize those holomorphic mappings (?) from the polydisc Dn in Cn to itself for which the induced composition operators C(?) are bounded (or compact) from the Bloch-type space Bωto Bμ(respectively, from the little Bloch-type space Bω,0 to Bμ,0), where ωis a normal function on [0,1) and μis a nonnegative function on [0,1) with μ(tn) > 0 for some sequence {tn}n=1∞(?)[0,1) satisfying limn→∞tn = 1.     

Composition operators on the Newton space

We investigate properties of composition operators C? on the Newton space (the Hilbert space of analytic functions which have the Newton polynomials as an orthonormal basis). We derive a formula for the entries of the matrix of C? with respect to the basis of Newton polynomials in terms of the value of the symbol ? at the non-negative integers. We also establish conditions on the symbol ? for boundedness, compactness, and self-adjointness of the induced composition operator C?. A key technique in obtaining these results is use of an isomorphism between the Newton space and the Hardy space via the Binomial Theorem.     

Composition operators on the polydisk induced by affine maps

We study the continuity of composition operators on the classical Hardy and weighted Bergman spaces of the polydisk. We show that this problem involves some delicate properties of the derivative of the symbol. In particular, we characterize continuity when the symbol is a linear self-map of the polydisk.     

Composition operators on Hardy spaces of a half-plane

We consider composition operators on Hardy spaces of a half-plane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Composition operators on Dirichlet-type spaces

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.

     

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

     

Composition operators with maximal norm on weighted Bergman spaces

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces (in particular, on the space ) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space , where every inner function induces a composition operator with maximal norm.

     

Composition operators on Hardy spaces on Lavrentiev domains

For any simply connected domain , we prove that a Littlewood type inequality is necessary for boundedness of composition operators on , , whenever the symbols are finitely-valent. Moreover, the corresponding ``little-oh' condition is also necessary for the compactness. Nevertheless, it is shown that such an inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Furthermore, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith (2003). Finally, we show a striking link between the geometry of the underlying domain and the symbol inducing the composition operator in , and in this sense, we relate both facts characterizing bounded and compact composition operators whenever is a Lavrentiev domain.

     

Hardy空间到加权Hardy空间上的复合算子

本文研究了从Hardy空间到加权Hardy空间上的复合算子,并给出此复合算子为有界和紧的条件。     

Adjoints of rationally induced composition operators

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H² [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626-639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.     

Composition operators on Lizorkin-Triebel spaces

This paper is devoted to the study of the composition operator Tf(g):=f○g on Lizorkin-Triebel spaces . In case s>1+(1/p), 1<p<∞, and 1?q?∞ we will prove the following: the operator Tf takes to itself if and only if f(0)=0 and f belongs locally to .     

Composition operators on noncommutative Hardy spaces

In this paper we initiate the study of composition operators on the noncommutative Hardy space , which is the Hilbert space of all free holomorphic functions of the form

     

Compact composition operators on the Smirnov class

We show that a composition operator on the Smirnov class is compact if and only if it is compact on some (equivalently: every) Hardy space for . Along the way we show that for composition operators on both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

     

Norm-attaining composition operators on the Bloch spaces

We prove that every composition operator C? on the Bloch space (modulo constant functions) attains its norm and characterize the norm-attaining composition operators on the little Bloch space (modulo constant functions). We also identify the extremal functions for ‖C?‖ in both cases.