Thomas Kriete, Jennifer Moorhouse
We consider this and related questions: When is a finite linear combination of composition operators, acting on the Hardy space or the standard weighted Bergman spaces on the unit disk, a compact operator?
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In this paper, we give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property.
For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $λI + K$. We prove that $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ is a Calkin algbra i.e., there exists a Banach space $\mathcal{Y}_X$ so that the Calkin algebra of $\mathcal{Y}_X$ is isomorphic as a Banach algebra to $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$. Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed wi…
Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may …
We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989).
For every well founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a $\mathcal{L}_\infty$-space $X_{\mathcal{T}}$. We prove that for each such tree $\mathcal{T}$, the Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$, equipped with the usual topology. We use this fact to conclude that for every countable compact metric space $K$ there exists a $\mathcal{L}_\infty$-space whose Calkin algebra is isomorphic, as a Ban…
Let $Φ$ be a $C^*$-subalgebra of $L^\infty(\mathbb{R})$ and $SO_{X(\mathbb{R})}^\diamond$ be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space $X(\mathbb{R})$. We show that the intersection of the Calkin image of the algebra generated by the operators of multiplication $aI$ by functions $a\inΦ$ and the Calkin image of the algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in $SO_{X(\mathbb{R})}^\diamond$ coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.
A Unital Banach Algebra Which Is Not a Calkin Algebra
本文构造了一个密度特征为2^{\aleph_0}的单位Banach代数A，其具有唯一非零闭两面理想，并且在两种常规定义下都不是Calkin代数。
The Rademacher quotient of $L^1$ embeds into $L^1$ and its weak Calkin algebra is uniformly incompressible
本文证明 $L^1(0,1)$ 关于 Rademacher 函数的商空间可同构嵌入 $L^1(0,1)$，并由此推出其弱 Calkin 代数具有均匀不可压缩性。
Assuming the continuum hypothesis, we show that the Calkin algebra has 2^{aleph_1} outer automorphisms.
Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces
该研究证明了Calkin代数在$C(K)$空间中的本质范数为极小代数范数，并通过非紧致算子的因子分解方法得出相关代数的唯一范数性质。
It is proved that for every compact metric space $K$ there exists a Banach space $X$ whose Calkin algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is homomorphically isometric to $C(K)$. This is achieved by appropriately modifying the Bourgain-Delbaen $\mathscr{L}_\infty$-space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded.
Ball Covering Property on Operators and Calkin Algebra
本文研究了算子上的球覆盖性质与卡林代数，证明了特定条件下卡林代数不满足BCP，并展示了在某些条件下B(X)具有UBCP。
Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients
本文通过oracle相对的双类型有限扩张构造，证明每个可分Banach空间的对偶可实现为可分Banach空间的Calkin代数，且存在泛型理想商覆盖所有可分空间。
This paper has been withdrawn by the author, due to a recent article by Kriete and Moorhouse (Linear relations in the Calkin algebra for composition operators, Trans. AMS, 359 (2007), 2915-2944) which includes more general results.
Read produced the first example of a Banach space $E_{\text{R}}$ such that the associated Banach algebra $\mathscr{B}(E_{\text{R}})$ of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalise Read's main theorem about $\mathscr{B}(E_{\text{R}})$ from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence {0} --> $\mathscr{W}(E_{\text{R}})$ --> $\mathscr{B}(E_{\text{R}})$--> $\tilde{\ell_2}$-->{0}, where $\mathscr{W}(E_{\text{R}})$ denotes the ideal of weakly compact o…
The compact operators on $c_0$ as a Calkin algebra
本研究证明$c_0$上紧算子代数的酉化同构于某Banach空间的Calkin代数，该空间由Argyros-Haydon空间拷贝的Argyros-Haydon和构造，内外参数取自不相交集。
Embedding C*-algebras into the Calkin algebra of $\ell^{p}$
本研究针对p∈(1,∞)的情形，通过构造保Fredholm指标的同构映射，证明任意可分C*-代数均可嵌入ℓᵖ空间的Calkin代数，还得到ℓᵖ上存在具任意Fredholm指标的类本质正规算子的推论。
Automatic continuity of operator semigroups in the Calkin algebra
本文研究Calkin代数中的算子半群，利用BDF理论将正规$C_0$-半群与扩张关联，并证明在自然情形下，强算子拓扑连续性蕴含均匀连续性。
Inverse-Limit Detection Envelopes as Calkin Algebras
本文引入有界检测包络作为 Calkin 代数，通过构造可分 Banach 空间实现任意此类包络，并刻画其结构，应用于多种算子代数与卷积代数。
Small masas of the Calkin algebra in the Cohen model
本文在 Cohen 模型下研究 Calkin 代数的极大交换子代数，证明其密度可小于连续统，并刻画了由布尔代数构造的 masa 的同构与酉等价类型，同时给出 ZFC 中不存在密度小于 𝔭 的 masa 的结论。
Reflexive Calkin algebras
本研究证明Banach空间的Calkin代数可以是无穷维自反空间甚至同构于Hilbert空间，通过构造特定Banach空间及相互零化投影列，得到Calkin代数与对应空间酉化的Banach代数同构关系，并列举了多类满足条件的Banach空间。
Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal…
Products of C*-algebras that do not embed into the Calkin algebra
本文在Cohen模型下证明阿贝尔C*-代数$c_0(2^ω)$的无穷乘积$(c_0(2^ω))^{\mathbb{N}}$不嵌入Calkin代数，扩充了已知反例，且不依赖刚性现象。
On masas of the Calkin algebra generated by projections
在连续假设下，研究了由投影生成的卡拉金代数中masas的*同构分类，展示了其可能具有与已知类型不同的意外性质。
Operator Algebras of Bourgain Delbaen Spaces: Realization, Rigidity, and Ideal Structure
本文研究了Calkin代数及其作为紧算子商的结构，通过保持几何和拓扑信息实现交换C*代数。证明了对于每个紧度量空间K，存在反射Banach空间X_C(K)，其Calkin代数同构于C(K)。同时建立了稳定性、局部化原理、Hölder连续性、刚性定理及理想结构分类等贡献。
Disjointness-preserving mappings on Calkin operator spaces and positive isometries
该研究在半有限冯·诺依曼代数框架下，证明了保不交映射恒为零的条件，给出正等距映射保不交的结论，得到其一般形式，推广并统一了相关经典结果。
In this paper we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally we show that there is no surjective *-homomorphism from a SAW*-algebra onto C*-tensor product of two infinite dimensional C*-algebras.
Volterra-type inner derivations on Hardy spaces
该研究针对巴拿赫空间上不满足Calkin经典内导子结论的情况，借助广义面积算子与算子紧缠绕关系，刻画了Hardy空间上的Volterra型内导子及Hardy与Bergman空间间算子的紧缠绕关系，补充了Calkin的结果。
A Three-Dimensional Operator System without the Smith--Ward Property
研究通过将哈里斯论证中相关部分替换，把四维算子系统换成三维超刚性算子系统，得到反例说明三维算子系统不具有史密斯 - 沃德性质，此系统还非精确，是首个三维非精确算子系统的例子。
We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $δ> 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $δ$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $\ell_1$ dual. The first class of spaces are HI and possess few, but not very few …
We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in $\ell_1(\mathbb{N}_0)$ is open. By appealing to ultraproduct techniques, we demonstrate that neither in $\ell_1(\mathbb{Z})$ nor in $\ell_1(\mathbb Q)$ convolution is uniform…
On Classes of Fredholm Type Operators
本文在 Banach 空间的 Calkin 代数中研究了 p-可逆性及广义 Drazin 可逆性，进而定义并刻画了 P-Fredholm、伪 B-Fredholm 和弱 B-Fredholm 等算子类。
In 1976, D. Voiculescu proved that every separable unital sub-C*-algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see reference), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a $σ$-unital C*-algebra. In this note, we answer Pedersen's question for a stable $σ$-unital C*-algebra.
Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectr…
The algebraic numerical range as a spectral set in Banach algebras
本文研究 Banach 代数中代数数值域成为 $C$-谱集的条件，通过反例、矩阵代数及无限维情形的正面结果刻画其性质。
We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.
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.
We show that the generalised composition of generalised integral operators is well defined on the space Colombeau algebras of tempered generalised functions.
The ball-covering property of non-commutative spaces of operators on Banach spaces
该研究针对Banach空间上算子非交换空间的球覆盖性质展开，通过构造等价范数证明一类商Banach代数不具BCP，并得出特定空间上算子空间具有UBCP的结论。
In this paper, we study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. We study the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
We study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. The central issue is to understand the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
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.
For each $S \in L(E)$ (with $E$ a Banach space) the operator $R(S) \in L(E^{**}/E)$ is defined by $R(S)(x^{**}+E) = S^{**}x^{**}+E$ \quad ($x^{**}\in E^{**}$). We study mapping properties of the correspondence $S\to R(S),$ which provides a representation $R$ of the weak Calkin algebra $L(E)/W(E)$ (here $W(E)$ denotes the weakly compact operators on $E$). Our results display strongly varying behaviour of $R.$ For instance, there are no non--zero compact operators in Im$(R)$ in the case of $L^1$ and $C(0,1),$ but $R(L(E)/W(E))$ identifies isometrically with the class of lattice regular operators…
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.
We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do su…
Spectrum of Weighted Composition Operators. Part X. The spectrum and essential spectra of weighted automorphisms of the polydisc algebra
本文研究了多圆盘代数加权自同构的谱与本质谱，并在二维情形下给出了这些谱的详细且多数情况下完整的描述。
Quantifying (non-)weak compactness of operators on $AL$- and $C(K)$-spaces
研究$AL$-空间间非弱紧算子的表示，证明每个算子在弱紧算子理想中有最佳近似，并通过对偶方法扩展至极不连通$C(L)$-空间。进一步用$\ell_1$上的恒等式因子化刻画$AL$-空间间算子的弱本质范数，得出弱Calkin代数有唯一范数。
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 $H^2$.
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 $φ$ 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…
An operator $A$ on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of $A$ in the Calkin algebra determines, for example, the Fredholmness of $A$, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of $A$. This coset can be identified with the collection of all so-called limit operators of $A$. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent no…
Let $R$ be a rational function of degree at least two, let $J_R$ be the Julia set of $R$ and let $μ^L$ be the Lyubich measure of $R$. We study the C$^*$-algebra $\mathcal{MC}_R$ generated by all multiplication operators by continuous functions in $C(J_R)$ and the composition operator $C_R$ induced by $R$ on $L^2(J_R, μ^L)$. We show that the C$^*$-algebra $\mathcal{MC}_R$ is isomorphic to the C$^*$-algebra $\mathcal{O}_R (J_R)$ associated with the complex dynamical system $\{R^{\circ n} \}_{n=1} ^\infty$.
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 …
The C$^*$-algebra of a composition reflection
本文研究单位圆盘Hardy空间上由复合反射算子$C_a$生成的C$^*$代数，并考察相关算子，探讨其结构性质。
Finite Blaschke Symbols and the $K$-Theory of Scalar Odometer $C^*$-Algebras
研究标量里程表\(C^*\) - 代数\(\mathcal A_\xi\)，证明其含紧算子，等距标量符号下\(W_\xi\)是Fredholm的条件，以及其在Calkin代数中像与里程表边界商同构等，得出不同次数有限Blaschke符号生成非同构\(C^*\) - 代数的结论。
Let $K$ be a compact metric space and let $γ= (γ_1, \dots, γ_n)$ be a system of proper contractions on $K$. We study a C*-algebra $\mathcal{MC}_{γ_1, \dots, γ_n}$ generated by all multiplication operators by continuous functions on $K$ and composition operators $C_{γ_i}$ induced by $γ_i$ for $i=1, \dots, n$ on a certain $L^2$ space. Suppose that $K$ is self-similar. We consider the Hutchinson measure $μ^H$ of $γ$ and the $L^2$ space $L^2(K, μ^H)$. Then we show that the C*-algebra $\mathcal{MC}_{γ_1, \dots, γ_n}$ is isomorphic to the Cuntz algebra $\mathcal{O}_n$ under some conditions.
Let $K$ be a compact metric space and let $φ: K \to K$ be continuous. We study C*-algebra $\mathcal{MC}_φ$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_φ$ induced by $φ$ on a certain $L^2$ space. Let $γ= (γ_1, \dots, γ_n)$ be a system of proper contractions on $K$. Suppose that $γ_1, \dots, γ_n$ are inverse branches of $φ$ and $K$ is self-similar. We consider the Hutchinson measure $μ^H$ of $γ$ and the $L^2$ space $L^2(K, μ^H)$. Then we show that the C*-algebra $\mathcal{MC}_φ$ is isomorphic to the C*-algebra $\mathcal{O}_γ(K)$ associat…
We study lifting problems for operator semigroups in the Calkin algebra $\mathscr{Q}(\mathcal{H})$, our approach being mainly based on the Brown--Douglas--Fillmore theory. With any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension $Γ\in\mathrm{Ext}(Δ)$, where $Δ$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $σ(A)$ of the generator of $(q(t))_{t\geq 0}$. By using Milnor's exact sequence, we show that if each $q(t)$ has a normal lift, then the question whether $Γ$ is trivial reduces to the question w…
In this paper we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application we prove that for "quasi-parabolic" composition operators the spectra and the essential spectra are equal.
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_…
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