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$K_{0}$-groups and strongly irreducible decompositions of operator tuples

Jing Xu

Abstract

An operator tuple $\mathbf{T}=(T_{1},\ldots,T_{n})$ is called strongly irreducible (SI), if the joint commutant of $\mathbf{T}$ does not any nontrivial idempotent operator. In this paper, we study the uniqueness of finitely strong irreducible decomposition of operator tuples up to similarity by $K$-theory of operator algebra, and give the algebraically similarity invariants of the Cowen-Douglas tuple with index 1 by using $K_{0}$-group of the commutant of operator tuples. As an application, we calculate $K_{0}$-groups of some multiplier algebras, and describe the similarity of backwards multishifts on Drury-Arveson space by means of inflation theory.

$K_{0}$-groups and strongly irreducible decompositions of operator tuples

Abstract

An operator tuple is called strongly irreducible (SI), if the joint commutant of does not any nontrivial idempotent operator. In this paper, we study the uniqueness of finitely strong irreducible decomposition of operator tuples up to similarity by -theory of operator algebra, and give the algebraically similarity invariants of the Cowen-Douglas tuple with index 1 by using -group of the commutant of operator tuples. As an application, we calculate -groups of some multiplier algebras, and describe the similarity of backwards multishifts on Drury-Arveson space by means of inflation theory.
Paper Structure (7 sections, 31 theorems, 152 equations)

This paper contains 7 sections, 31 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. $K_{0}$-group of a unital Banach algebra
  4. The Cowen-Douglas Class
  5. The Drury-Arveson space $H_{m}^{2}$
  6. The uniqueness of finite strongly irreducible decomposition of operator tuples
  7. The similarity of Cowen-Douglas operator tuples with index one

Key Result

Lemma 2.11

Let $\mathbf{T}=(T_{1},T_{2},\ldots,T_{m})\in\mathcal{B}_{n}^{m}(\Omega)$ and $\mathbf{T}$ be unitary equivalent to the adjoint of $m$-tuple $\mathbf{M}_{z}=(M_{z_{1}},\ldots,M_{z_{m}})$ of multiplication operators on analytic function space $\mathcal{H}_{K}$ with reproducing kernel where $z,w\in\Omega$ and $\widehat{f}(\alpha)>0$ for all $\alpha\in\mathbb{N}^{m}$. Then $\mathbf{T}$ is unitary eq

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 55 more