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On Simultaneous Triangularization of Matrices and Quasinilpotency of Commutator of Compact Operators

Sasmita Patnaik, Rahul Sethi

Abstract

In this paper we determine a sufficient condition for the quasinilpotency of a commutator of compact operators via block-tridiagonal matrix form associated with a compact operator. We also prove that every compact operator is unitarily equivalent to the sum of a compact quasinilpotent operator and a triangularizable compact operator.

On Simultaneous Triangularization of Matrices and Quasinilpotency of Commutator of Compact Operators

Abstract

In this paper we determine a sufficient condition for the quasinilpotency of a commutator of compact operators via block-tridiagonal matrix form associated with a compact operator. We also prove that every compact operator is unitarily equivalent to the sum of a compact quasinilpotent operator and a triangularizable compact operator.
Paper Structure (3 sections, 3 theorems, 26 equations)

This paper contains 3 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. On quasinilpotency of a commutator of compact operators
  3. On the structure of a compact operator via its block-tridiagonal matrix form

Key Result

Theorem 1

For $A,B \in \mathcal{B}(\mathcal{H})$, there is an orthonormal basis $\{f_n\}$ with respect to which both $A$ and $B$ have matrices in the block-tridiagonal form. That is, there is a unitary operator $U$ such that for each of them $U^*AU$ and $U^*BU$ simultaneously the central blocks have block siz The central blocks $C_n$ and $Z_n$ are of size $k_n \times k_n$, where $k_1 = 1$ and $k_n = 4(5^{n-

Theorems & Definitions (7)

  • Theorem
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 3.1