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Triangular factorization of operators and reconstruction of systems

M. I. Belishev

Abstract

The paper provides a coherent presentation of an operator scheme, which is used in an approach to inverse problems of mathematical physics (the boundary control method). The scheme is based on the triangular factorization of operators. It not only solves inverse problems but provides the functional models of a class of symmetric semi-bounded operators. The class is characterized in the terms of an evolutionary dynamical system associated with the operator.

Triangular factorization of operators and reconstruction of systems

Abstract

The paper provides a coherent presentation of an operator scheme, which is used in an approach to inverse problems of mathematical physics (the boundary control method). The scheme is based on the triangular factorization of operators. It not only solves inverse problems but provides the functional models of a class of symmetric semi-bounded operators. The class is characterized in the terms of an evolutionary dynamical system associated with the operator.
Paper Structure (4 sections, 14 theorems, 79 equations)

This paper contains 4 sections, 14 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Finite dimension case
  3. Infinite dimension case
  4. Factorization and models of operators

Key Result

Lemma 1

Let $C=\{c_{ij}\}_{i,j=1}^n>O_n$ be a positive-definite matrix. Then there is an (upper) triangular matrix $V=\{v_{ij}\}_{i,j=1}^n:\,\, v_{ij}=0,\,\,i>j$ such that holds. Assuming $v_{ii}>0,\,\,i=1,\dots,n$, such a matrix $V$ is unique.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 14 more