Boundedness Of Composition Operators Research Articles

A CLASS OF SYMBOLS THAT INDUCE BOUNDED COMPOSITION OPERATORS FOR DIRICHLET-TYPE SPACES ON THE DISC

Abstract We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle

We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes S_{p} of the Hardy space \mathcal{H}^{2} of Dirichlet series. For p\geq 2 , these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for \mathcal{H}^{2} and we give examples of bounded composition operators with Dirichlet series symbols on \mathcal{H}^{p} , p>0 .

On composition operators on the Wiener algebra of Dirichlet series

We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions {Φt}t≥0 in the so-called Gordon-Hedenmalm classG, that is, the family of analytic functions Φ:C+→C+ giving rise to bounded composition operators in the Hardy space of Dirichlet series H2. We show that there is a one-to-one correspondence between continuous semigroups {Φt}t≥0 in the class G and strongly continuous semigroups of composition operators {Tt}t≥0, where Tt(f)=f∘Φt, f∈H2. We extend these results for the range p∈[1,∞). For the case p=∞, we prove that there is no non-trivial strongly continuous semigroup of composition operators in H∞. We characterize the infinitesimal generators of continuous semigroups in the class G as those Dirichlet series sending C+ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Boundedness of composition operator on several variable Paley-Wiener space

In this paper we have considered composition operators on the several variable Paley-Wiener space Lπ2(Cn). We have proved that for any continuous function ϕ:Cn→Cn the composition operator Cϕ on Lπ2(Cn) is bounded iff ϕ has the following form:ϕ(z)=Az+b,z∈Cn, where A∈GL(n,R) with ||A||op≤1 and b∈Cn. Our proof is different from the single variable case and mainly depends on a theorem of Malgrange, some techniques from harmonic analysis and certain asymptotic behaviour of Bessel's function.

Complex symmetric weighted composition operators on Bergman spaces and Lebesgue spaces

In the paper, we investigate weighted composition operators on Bergman spaces of a half-plane. We characterize weighted composition operators which are hermitian and those which are complex symmetric with respect to a family of conjugations. As it turns out, weighted composition operators enhanced by a symmetry must be bounded. Hermitian, and unitary weighted composition operators are proven to be complex symmetric with respect to an adapted and highly relevant conjugation. We classify which the linear fractional functions give rise to the complex symmetry of bounded composition operators. We end the paper with a natural link to complex symmetry in Lebesgue space.

Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on Rd. We prove that only affine transforms can do so in a certain large class of RKHS. Our result covers not only the Paley-Wiener space on the real line, studied in previous works, but also much more general RKHSs corresponding to analytic positive definite functions, where existing methods do not work. Our method only relies on intrinsic properties of the RKHSs, and we establish a connection between the behavior of composition operators and asymptotic properties of the greatest zeros of orthogonal polynomials on a weighted L2-space on the real line. We also investigate the compactness of the composition operators and show that any bounded composition operators cannot be compact in our situation.

Coincidence of set functions in quasiconformal analysis

It is known that mappings occurring in quasiconformal analysis can be defined in several equivalent ways: 1) as homeomorphisms inducing bounded composition operators between Sobolev spaces; 2) as Sobolev-class homeomorphisms with bounded distortion whose operator distortion function is integrable; 3) as homeomorphism changing the capacity of the image of a condenser in a controllable way in terms of the weighted capacity of the condenser in the source space; 4) as homeomorphism changing the modulus of the image of a family of curves in a controllable way in terms of the weighted modulus of the family of curves in the source space. A certain set function, defined on open subsets, can be associated with each of these definitions. The main result consists in the fact that all these set functions coincide. Bibliography: 48 titles.

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with BMO-quasiconformal mappings. Using the duality of Hardy spaces and BMO-spaces, we prove that BMO-quasiconformal mappings generate bounded composition operators from Hardy–Sobolev spaces to Sobolev spaces.

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with $\BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $\BMO$-spaces, we prove that $\BMO$-quasiconformal mappings generate bounded composition operators from Hardy--Sobolev spaces to Sobolev spaces.

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

Boundedness of composition operators on Morrey spaces and weak Morrey spaces

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

Adjoint of generalized Cesáro operators on analytic function spaces

Abstract In this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint 𝒜 b , c {\mathcal{A}^{b,c}} for b > 0 {b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces B α + ( c - 1 ) p , q {B_{\alpha+(c-1)}^{p,q}} for c > 1 {c>1} .

Bounded Composition Operators on General Weighted Hardy Spaces

We introduce a general class of weighted Hardy spaces of entire functions, which contains the Fock and Fock-type spaces as well as several other well-known spaces, and study bounded composition operators on these spaces. It is interesting that such characterizations depend closely on the behavior of the weight sequence.

Hilbertian Hardy-Sobolev spaces on a half-plane

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.

Composition Operators on Generalized Fock Spaces of Slice Hyperholomorphic Functions

In this paper, we study composition operators on generalized Fock spaces of slice hyperholomorphic functions. More precisely, some relationship for composition operators between the slice hyperholomorphic setting and the classical setting is explored to characterize the boundedness and compactness of composition operators. Motivated by a question in Kumar (Adv Appl Clifford Algebras 27:1459–1477, 2017), we characterize S-spectra of composition operators. Finally, we characterize connected components of the space of bounded composition operators.

Inhomogeneous multi-parameter Besov and Triebel-Lizorkin spaces associated with different homogeneities and boundedness of composition operators

Inhomogeneous multi-parameter Besov and Triebel-Lizorkin spaces associated with different homogeneities and boundedness of composition operators

Boundedness of composition operators on some analytic function spaces

Boundedness of composition operators on some analytic function spaces

Mean ergodic composition operators on generalized Fock spaces

Every bounded composition operator $$C_{\psi }$$ defined by an analytic symbol $$\psi $$ on the complex plane when acting on generalized Fock spaces $$\mathcal {F}_{\varphi }^{p}, 1 \le p \le \infty $$, and $$p=0$$, is power bounded. Mean ergodic and uniformly mean ergodic bounded composition operators on these spaces are characterized in terms of the symbol. The behaviour for $$p=0$$ and $$p=\infty $$ differs. The set of periodic points of these operators is also determined.