Covariant representations for matrix-valued transfer operators

$\rm{SL}(n)$ covariant function-valued valuations

Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.

Orlicz空间上的SL(n)协变矩阵值赋值

SL(n) covariant matrix-valued valuations on Orlicz spaces

该研究在无对称性假设下完全分类了Orlicz空间上连续的SL(n)协变矩阵值赋值，发现n≥3时仅矩矩阵满足条件，二维则存在新泛函，拓展了赋值理论的相关结论。

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite dimensional $C^*$-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algortihms.

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…

Crossed products of Banach algebras. II

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper the theory is developed further. We consider the dependence of the crossed product on the class $\mathcal{R}$ and its essential uniqueness. Next we study generalized Beurling algebras: weighted Bochner spaces of $A$-valued functions on $G$ with a continuous multiplication. Though not Banach algebras in general, they are isomorphic to a crossed product of a …

Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups

Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^*$-algebras associated to certain $C^*$-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-v…