Given two quasi-Banach ideals \oid{A}{}{} and \oid{B}{}{} we investigate the regular hull of their composition - $(\oid{A}{}{} \circ \oid{B}{}{})^{reg}$. In concrete situations this regular hull appears more often than the composition itself. As a first example we obtain a description for the regular hull of the nuclear operators which is a "reflected" Grothendieck representation:\\ $\oid{N}{}{reg} \stackrel{1}{=} \oid{I}{}{} \circ \oid{W}{}{}$ (theorem 2.1). Further we recognize that the class of such ideals leads to interesting relations concerning the question of the accessibility of (injective) operator ideals.
This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the hull-minimal ideals.
Given two metric spaces $M$ and $N$ we study, motivated by a question of N. Weaver, conditions under which an isometric composition operator $C_ϕ:\mathrm{Lip}_0(M)\longrightarrow \mathrm{Lip}_0(N)$ is isometric depending on the properties of $ϕ$. We obtain a complete characterisation of those operators $C_ϕ$ in terms of a property of the function $ϕ$ in the case that $B_{\mathcal F(M)}$ is the closed convex hull of its preserved extreme points. Also, we obtain necessary and sufficient conditions for $C_ϕ$ being isometric in the case that $M$ is geodesic.
We study in this paper some property of Lipschitz mappings which admit factorization through an operator ideal. We try to construct Lipschitz cross-norms from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of p-summing linear operators we introduce a new concpet in the the category of Lipschitz mappings that is called strictly Lipschitz p-summing.
We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.
We introduce and develop the notion of hyper-ideals of multilinear operators between Banach spaces. While the well studied notion of ideals of multilinear operators (multi-ideals) relies on the composition with linear operators, the notion we propose, by considering the composition with multilinear operators, explores more deeply the nonlinear feature of the subject. The results we prove show that, although more restrictive a priori, hyper-ideals enjoy nice general properties and its theory is rich enough to provide distinguished examples in every situation where the corresponding multi-ideal …
Let $K_θ$ be a model space generated by an inner function $θ$. We study the Schatten class membership of embeddings $I : K_θ\to L^2(μ)$, $μ$ a positive measure, and of composition operators $C_ϕ:K_θ\to H^2(\mathbb D)$ with a holomprphic function $ϕ:\mathbb D\rightarrow \mathbb D$. In the case of one-component inner functions $θ$ we show that the problem can be reduced to the study of natural extensions of $I$ and $C_ϕ$ to the Hardy-Smirnov space $E^2(D)$ in some domain $D\supset \mathbb D$. In particular, we obtain a characterization of Schatten membership of $C_ϕ$ in terms of Nevanlinna count…
Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual ideal of $\mathcal{I}$ is introduced and its elements are characterized as those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded holomorphic mappings from an open subset $U\subseteq E$ to $F$ such as $p$-integral holomorphic mappings and $p$-nuclear holomorphic mappings with…
We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the $\A$-approximation property if the identity map is uniformly approximable on $\A$-compact sets by finite rank operators. The Grothendieck's classic approximation property is the $\K$-approximation property for $\K$ the ideal of compact operators and the $p$-approximation property is obtained as the $\mathcal N…
We study the space of $p$-compact operators $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We a…
Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the so-called ideal property: if a polynomial P belongs to a class Q, then the composition u o P o v of P with linear operators u and v belongs to Q as well. In an attempt to explore the nonlinearity of the subject in a more consistent way, and taking into account recent results in the field, in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. Some importante classes justify the stud…
For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell_{\infty}$-extension and $\ell_{1}$-lifting properties for an operator ideal $\mathcal A$ and st…
Let $φ$ be a linear-fractional, non-automorphism self-map of $\mathbb{D}$ that fixes $ζ\in \mathbb{T}$ and satisfies $φ^{\prime}(ζ) \neq 1$ and consider the composition operator $C_φ$ acting on the Hardy space $H^2(\mathbb{D}).$ We determine which linear-fractionally-induced composition operators are contained in the unital C$^*$-algebra generated by $C_φ$ and the ideal $\mathcal{K}$ of compact operators. We apply these results to show that $C^*(C_φ, \mathcal{K})$ and $C^*(\mathcal{F}_ζ)$, the unital C$^*$-algebra generated by all composition operators induced by linear-fractional, non-automor…
Let $ϕ$ be an analytic map taking the unit disk $\mathbb{D}$ into itself. We establish that the class of composition operators $f \mapsto C_ϕ(f) = f \circ ϕ$ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space $H^p$, for $1\le p < \infty$ and $p \neq 2$. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) $C_ϕ$ is a compact operator $H^p \to H^p$, (ii) $C_ϕ$ fixes a (linearly isomorphic) copy of $\ell^p$ in $H^p$, but $C_ϕ$ does not fix any copies of $\ell^2$ in $H^p$, (iii) $C_ϕ$ fixes a copy of $\ell^2$…
For 0 < s < 1, let phi_s(z)=sz+(1-s). We investigate the unital C*-algebra generated by the semigroup {C_{phi_s} : 0 < s < 1} of composition operators acting on the Hardy space of the unit disk. We determine the joint approximate point spectrum of a related collection of operators and show that the quotient of the C*-algebra by its commutator ideal is isomorphic to the direct sum of the complex numbers and the algebra of almost periodic functions on the real line. In addition, we show that the C*-algebra is irreducible.
In this paper we consider the C*-algebra $C^{*}(\{C_φ\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.
It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_α(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.
Given a strictly increasing sequence $Λ=(λ_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{λ_n}<\infty$, the Müntz spaces $M_Λ^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{λ_n}$. We discuss how properties of the embedding $M_Λ^2\subset L^2(μ)$, where $μ$ is a finite positive Borel measure on the interval $[0,1]$, have immediate consequences for composition operators on $M^2_Λ$. We give criteria for composition operators to be bounded, compact, or to belong to the Schatten--von Neumann ideals.
Let $ϕ$ be a linear-fractional self map of the open unit disk, not an automorphism, such that $ϕ(ζ)=η$ for distinct points $ζ,η$ in the unit circle. We consider the question of which composition operators lie in the unital C*-algebra generated by the composition operator $C_ϕ$ and the ideal of compact operators, acting on the Hardy space $H^2$.
A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α<ω_1$) satisfying $I_{α+1}/I_α\cong K(\ell_2)$ for each $α<ω_…
Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite.
Let R be a finite Blaschke product. We study the C*-algebra TC_R generated by both the composition operator C_R and the Toeplitz operator T_z on the Hardy space. We show that the simplicity of the quotient algebra OC_R by the ideal of the compact operators can be characterized by the dynamics near the Denjoy-Wolff point of R if the degree of R is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of OC_R such that R is a finite Blaschke product of degree at least two and the Julia set of R is the unit circle, using the …
