Let H^2(D) denote the classical Hardy space of the open unit disk D in the complex plane. We obtain descriptions of both the spectrum and essential spectrum of composition operators on H^2(D) whose symbols belong to the class S(2) introduced by Kriete and Moorhouse [Trans. Amer. Math. Soc., 359, 2007]. Our work reveals new possibilities for the shapes of composition-operator spectra, settling a conjecture of Cowen's [J. Operator Th. 9, 1983]. Our results depend on a number of lemmas, perhaps of independent interest, that provide spectral characterizations of sums of elements of a unital algebra over a field when certain pairwise products of the summands are zero.
In this paper we investigate the numerical ranges of composition operators whose symbols are elliptic automorphisms of finite orders, on the Hilbert Hardy space $H^2(D)$.
Previously, spectra of certain weighted composition operators on the Hardy Space were discovered under one of two hypotheses: either the compositional symbol converges under iteration to the Denjoy-Wolff point on all of the open disk rather than compact subsets, or the compositional symbol is "essentially linear fractional". We show that if the symbol is a quadratic self-map of the disk of parabolic type, then the spectrum of the weighted composition operators can be found when these maps exhibit both of the aforementioned properties, and most of them do.
We give a new proof of the existence of a surjective symbol whose associated composition operator on H 2 (D) is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the 1-dimensional case, for N $\ge$ 2, the behavior of the approximation numbers a n = a n (C $Φ$), or rather of $β$ -- N = lim inf n$\rightarrow$$\infty$ [a n ] 1/n 1/N or $β$ + N = lim sup n$\rightarrow$$\infty$ [a n ] 1/n 1/N , of composition operators on H 2 (D N) cannot be determined by the image of the symbol. MSC 2…
We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.
We study topologizability and power boundedness of weigh\-ted composition operators on (certain subspaces of) $\mathscr{D}'(X)$ for an open subset $X$ of $\mathbb{R}^d$. For the former property we derive a characterization in terms of the symbol and the weight of the weighted composition operator, while for the latter property necessary and sufficient conditions on the weight and the symbol are presented. Moreover, for an unweighted composition operator a characterization of power boundedness in terms of the symbol is derived for the special case of a bijective symbol.
We give a complete characterization of the spectrum of composition operators, induced by an automorphism of the open unit disk, acting on a family of Banach spaces of analytic functions that includes the Bloch space and BMOA. We show that for parabolic and hyperbolic automorphisms, the spectrum is the unit circle. For the case of elliptic automorphisms, the spectrum is either the unit circle or a finite cyclic subgroup of the unit circle.
In the present paper, we study the composition operators acting on weighted Hardy spaces of polynomial growth, which are concerned with norms, spectra and (semi-)Fredholmness. Firstly, we estimate the norms of the composition operators with symbols of disk automorphisms. Secondly, we discuss the spectra of the composition operators with symbols of disk automorphisms. In particular, it is proven of that the spectrum of a composition operator with symbol of any parabolic disk automorphism is always the unit circle. Thirdly, we consider the Fredholmness of the composition operator $C_φ$ with symb…
We consider composition operators $\mathscr{C}_φ$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $φ$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $φ$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $φ$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particula…
In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H^2 is cyclic.
Composition operators on weighted Bergman spaces of the polydisc
Laboratoire de Math\'ematiques Blaise Pascal UMR 6620 CNRS, Universit\'e Clermont Auvergne, Campus universitaire des C\'ezeaux, 3 place Vasarely, 63178 Aubi\`ere Cedex, France.
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.
The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $ω:Y\to\mathbb{C}$ and $Φ:Y\to X$ are such that the weighted composition operator $W_{Φ,ω}$ is continuous, when can we guarantee that both $Φ$ and $ω$ are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over (connected) complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven …
In this paper we provide a full characterization of cyclic composition operators defined on the d-dimensional Fock space $\mathcal F(\mathbb C^d)$ in terms of their symbol. Also, we study the supercyclicity and convex-cyclicity of this type of operators. We end this work by computing the approximation numbers of compact composition operators defined on $\mathcal F(\mathbb C^d)$.
The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.
We give sufficient conditions on the Lebesgue exponents for compositions of odd numbers of pseudo-differential operators with symbols in modulation spaces. As a byproduct, we obtain sufficient conditions for twisted convolutions of odd numbers of factors to be bounded on Wiener amalgam spaces.
Composition operators for holomorphic Lipschitz functions
Universidad de San Andrés（圣安德烈斯大学）；CONICET（阿根廷国家科学研究委员会）；Universidad de Zaragoza（萨拉戈萨大学）；Universite Marie et Louis Pasteur, CNRS, LmB (UMR 6623)（玛丽和路易·巴斯德大学，法国国家科学研究中心，LmB实验室）
We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk $\D$. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the ``cusp map'' and the lens maps, acting on weighted Dirichlet spaces.
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_φ$ on $\mathscr{H}^2$ which are generated by symbols of the form $φ(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $φ$, then the operator $\mathscr{C}_φ$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_φ$. Furthermore, in the…
For holomorphic pairs of symbols $(u, ψ)$, we study various structures of the weighted composition operator $ W_{(u,ψ)} f= u \cdot f(ψ)$ defined on the Fock spaces $\mathcal{F}_p$. We have identified operators $W_{(u,ψ)}$ that have power bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values $|u(0)|$ and $|u(\frac{b}{1-a})|$ where $ a$ and $b$ are coefficients from linear expansion of the symbol $ψ$. The spectrum of the operators are also determined and applied further to prove results about uniform…
We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $φ$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_α$. We investigate the optimal $β$ such that the composition operator $\mathcal{C}_φ$ maps $\mathcal{D}_α$ boundedly into $\mathcal{D}_β$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and …
Multivalued linear operators, also known as linear relations, are studied on a specific class of weighted, composition transforms on Fock space. Basic properties of this class of linear relations, such as closed graph, boundedness, complex symmetry, real symmetry, or isometry are characterized in simple algebraic terms, involving their symbols.
