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Direct Sum of Lower Semi-Frames in Hilbert Spaces

Hemalatha M, P. Sam Johnson, Harikrishnan P. K

Abstract

In this paper, structural properties of lower semi-frames in separable Hilbert spaces are explored with a focus on transformations under linear operators (may be unbounded). Also, the direct sum of lower semi-frames, providing necessary and sufficient conditions for the preservation of lower semi-frame structure, is examined.

Direct Sum of Lower Semi-Frames in Hilbert Spaces

Abstract

In this paper, structural properties of lower semi-frames in separable Hilbert spaces are explored with a focus on transformations under linear operators (may be unbounded). Also, the direct sum of lower semi-frames, providing necessary and sufficient conditions for the preservation of lower semi-frame structure, is examined.

Paper Structure

This paper contains 5 sections, 22 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generating sequences using operators
  4. Generating sequences using sum
  5. Generating sequences using direct sum

Key Result

Proposition 2.3

4a Let $A$ be a densely defined closed operator on $H$. Then the following statements are equivalent:

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 38 more