Composition operators on Hilbert spaces of sequences

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.

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.

The Green's function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green's functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green's functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model.

The paper contains a new proof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic self-map φ of the open unit disk is an inner function if and only if the essential norm of the composition operator of symbol φ is equal to √ (1 + |φ(0)|)/(1 − |φ(0)|). The main ingredient in the proof is a formula for the essential norm of a composition operator in terms of Aleksandrov measures obtained by Cima and Matheson. The necessity was originally proved by Joel Shapiro in 1987. A short proof of the necessity, by Aleksandrov measure techniques, was obtained by Jonathan E. Shapiro in 1998. For each holomorphic self-map φ : U → U of the open unit disk U, the composition operator Cφ of symbol φ is defined as follows Cφf = f ◦ φ, f ∈ H. In this definition H is the Hilbert Hardy space on U, i.e., the set of all analytic functions on U with square summable Taylor coefficients. It is well known that each such φ induces a bounded composition operator Cφ on H. Recently Joel Shapiro obtained the following characterization of inner functions [7], (that is of self-maps φ whose radial limit function is unimodular almost everywhere on the unit circle T). Theorem 1. The function φ : U → U is inner if and only if (1) ‖Cφ‖e = √ 1 + |φ(0)| 1− |φ(0)| , where ‖Cφ‖e denotes the essential norm of Cφ. The necessity in the previous equivalence was originally proved in [6]. Jonathan Shapiro [5] observed that Aleksandrov measures can be used Received by the editors on May 7, 2003, and in revised form on May 30, 2003. Copyright c ©2005 Rocky Mountain Mathematics Consortium 1723

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

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

Complex symmetry of invertible composition operators

Composition operators on Hilbert spaces of entire Dirichlet series

Let l p (1 < p < oo) be the Banach space of all p-summable sequences (bounded sequences for p -oo) of complex numbers under the standard p-norm on it and Cj, be a composition operator on l p induced by a function 4> on N into itself.In this paper we give a characterization of composition operators whose ascent and descent are infinite.DEFINITION 1.1.If there is some integer n > 0 such that N(T n ) = N(T n+1 ), the smallest such integer is called the ascent of T and is denoted by a(T).If no such integer exists we say that a(T) = oo.DEFINITION 1.2.If there is some integer n > 0 such that R(T n+1 ) = R(T n ), the smallest such integer is called the descent of T and is denoted by d(T).If no such integer exists we say that d(T) = oo.THEOREM 1.3.If a(T) is finite and d(T) = 0, then a(T) = 0.

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spacesX, Y. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result forL(X, Y) thereby proving an analogue of the result from [9] forL p (1 <p ≠ 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

The role of the angle in supercyclic behavior

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.

Some Factorization Properties of Composition Operators

Uniformly bounded composition operators in the Banach space of absolutely continuous functions

In this short paper we will discuss recent advances on the problem of characterizing the boundedness of the composition operator acting on the Bergman spaces A β 2 ( D 2 ) ${A}_{\beta }^{2}\left({\mathbb{D}}^{2}\right)$ whenever the self map Φ of the bidisc is induced by Rational Inner Functions. The problem stated here is submitted as part of the Problem List of the Young Researchers Workshop in Complex Analysis and Operator Theory “The Bench Math Session 2025”, organized in Jagiellonian University of Kraków at 10th–11th February 2025.