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On invariant subspaces of a linear operator

M. I. Belishev, S. A. Simonov

Abstract

We discuss the concept of invariant subspaces for unbounded linear operators, point out some shortcomings of known definitions, and propose our own.

On invariant subspaces of a linear operator

Abstract

We discuss the concept of invariant subspaces for unbounded linear operators, point out some shortcomings of known definitions, and propose our own.
Paper Structure (4 sections, 1 theorem, 7 equations)

This paper contains 4 sections, 1 theorem, 7 equations.

Table of Contents

  1. About the work
  2. Bounded operators
  3. Unbounded operators
  4. Our proposals

Key Result

Theorem 1

If the operator $A$ is symmetric (i.e., $A\subset A^*$), $\mathscr G$ is its invariant subspace, and its part in $\mathscr G$ is self-adjoint (i.e., $A_\mathscr G=A_\mathscr G^*$ in $\mathscr G$), then $\mathscr G$ reduces $A$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Definition 5