Schur Class Operator Functions and Automorphisms of Hardy Algebras

解析自同构群与解析函数的相似表示

Analytic automorphism group and similar representation of analytic functions

本文受 Gromov 多项式增长定理启发，研究多项式增长的加权 Hardy 空间，证明解析自同构诱导的复合算子有界且 M_z 与 M_φ 相似，并肯定回答 Douglas 问题的广义版本，同时给出中间增长情形下的无界反例及 Jordan 表示与相似分类。

Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting

Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy alge…

Reflexivity of non commutative Hardy Algebras

Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-correspondence $E$. These algebras were introduced in 2004, ~\cite{MuS3}, by P. Muhly and B. Solel, and generalize the classical Hardy algebra of the unit disc $H^{\infty}(\mathbb{D})$. As a special case one obtains also the algebra $\mathcal{F}^{\infty}$ of Popescu, which is $H^{\infty}(\mathbb{C}^n)$ in our setting. In this paper we view the algebra $H^\infty(E)$ as acting on a Hilbert space via an induced representation $ρ(H^{\infty}(E))$, and we study the reflexivity of $ρ(H^{\infty}(E))$. This question was stu…

Toeplitz-Composition C*-Algebras

Let $ζ$ and $η$ be distinct points on the unit circle and suppose that $ϕ$ is a linear-fractional self-map of the unit disk D, not an automorphism, with $ϕ(ζ)=η$. We describe the C*-algebra generated by the associated composition operator $C_ϕ$ and the shift operator, acting on the Hardy space on D.

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of aut…

Majorization and a Schur-Horn Theorem for positive compact operators, the nonzero kernel case

Schur-Horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes, formally described as the range of the canonical conditional expectation of its unitary orbit. Following a brief background survey, we prove an infinite dimensional Schur-Horn theorem for positive compact operators with infinite dimensional kernel, one of the two open cases posed recently by Kaftal- Weiss. There, they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators. Here we show how the characteriza…

The C*-algebra generated by irreducible Toeplitz and composition operators

We describe the C*-algebra generated by an irreducible Toeplitz operator $T_ψ$, with continuous symbol $ψ$ on the unit circle $\mathbb{T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear-fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For composition operators with automorphism symbols, we show that this algebra is not isomorphic to the one generated by the shift and composition operators.

Composition operators within singly generated composition C*-algebras

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 $H^2$.

Homology for operator algebras I: Spectral homology for reflexive algebras

A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In particular it is shown that if the first spectral homology group is trivial then Schur automorphisms are automatically quasispatial. This motivates the introduction of essential Hochschild cohomology which we define by using …

Fixed Point Composition and Toeplitz-Composition C*-algebras

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…

Beurling type invariant subspaces of composition operators

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $φ: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_φ$ on $H^2$ is defined by \[ (C_φ f)(z)=f(φ(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}). \] Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is \[ \mathcal{S}(\mathbb{D}) = \{ψ\in H^\infty(\mathbb{D}): \|ψ\|_{\infty} := \sup_{z \in \mathbb{D}} |ψ(z)| \leq 1\}. \] The elements of $\mathca…

Noncommutative maximal ergodic inequalities associated with doubling conditions

This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $α$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by \begin{equation*} A_{n}x= \frac{1}{m(V^{n})} \int _{V^{n}}α_{g}x\,dm(g),\quad x\in L_{p}( \mathcal{M}),n\in \mathbb{N},1\leq p\leq \infty , \end{equation*} are of weak type $(1,1)$ and of strong type $(p,p)$ for …