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Hyers-Ulam Stability of Unbounded Closable Operators in Hilbert Spaces

Arup Majumdar, P. Sam Johnson, Ram N. Mohapatra

Abstract

In this paper, we discuss the Hyers-Ulam stability of closable (unbounded) operators with several interesting examples. We also present results pertaining to the Hyers-Ulam stability of the sum and product of closable operators to have the Hyers-Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of $2 \times 2$ block matrix $\mathcal A$ in order to have the Hyers-Ulam stability.

Hyers-Ulam Stability of Unbounded Closable Operators in Hilbert Spaces

Abstract

In this paper, we discuss the Hyers-Ulam stability of closable (unbounded) operators with several interesting examples. We also present results pertaining to the Hyers-Ulam stability of the sum and product of closable operators to have the Hyers-Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of block matrix in order to have the Hyers-Ulam stability.
Paper Structure (4 sections, 37 theorems, 26 equations)

This paper contains 4 sections, 37 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Characterizations of Hyers-Ulam Stable Operators
  3. Sum and Product of Hyers-Ulam Stable Operators
  4. An Illustrative Example of Closable (Non-closed) Hyers-Ulam Stable Operator

Key Result

Theorem 1.3

MR0200692 Let $T$ be a closed operator from $D(T) \subset H$ to $K$. The range of $T$ is closed iff $\gamma(T) >0$.

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Example 2.1
  • Example 2.2
  • ...and 48 more