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A Comparison of Two Generalisations of Triplets of Hilbert Spaces

Petru Cojuhari, Aurelian Gheondea

TL;DR

The paper clarifies the relationship between two generalisations of triplets of Hilbert spaces: closely embedded triplets and Berezanskii's generalised triplets. It develops operator-model frameworks ${\mathcal{D}}(T)$ and ${\mathcal{R}}(T)$ for closely embedded spaces, derives duality and density properties, and provides precise criteria for when a closely embedded triplet is a generalised triplet. It further shows how to pass from a generalised triplet to a closely embedded triplet that essentially coincides on a dense core ${\mathcal{D}}$, and gives conditions under which the generalised triplet itself is also closely embedded. The results unify the two formalisms, provide symmetry between the dual sides, and illustrate the theory with weighted $L^2$ and Dirichlet-type spaces, highlighting practical pathways to translate between frameworks in analysis and PDE contexts.

Abstract

We compare the concept of triplet of closely embedded Hilbert spaces with that of generalised triplet of Hilbert spaces in the sense of Berezanskii by showing when they coincide, when they are different, and when starting from one of them one can naturally produce the other one that essentially or fully coincides.

A Comparison of Two Generalisations of Triplets of Hilbert Spaces

TL;DR

The paper clarifies the relationship between two generalisations of triplets of Hilbert spaces: closely embedded triplets and Berezanskii's generalised triplets. It develops operator-model frameworks and for closely embedded spaces, derives duality and density properties, and provides precise criteria for when a closely embedded triplet is a generalised triplet. It further shows how to pass from a generalised triplet to a closely embedded triplet that essentially coincides on a dense core , and gives conditions under which the generalised triplet itself is also closely embedded. The results unify the two formalisms, provide symmetry between the dual sides, and illustrate the theory with weighted and Dirichlet-type spaces, highlighting practical pathways to translate between frameworks in analysis and PDE contexts.

Abstract

We compare the concept of triplet of closely embedded Hilbert spaces with that of generalised triplet of Hilbert spaces in the sense of Berezanskii by showing when they coincide, when they are different, and when starting from one of them one can naturally produce the other one that essentially or fully coincides.

Paper Structure

This paper contains 8 sections, 13 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Triplets of Closely Embedded Hilbert Spaces
  3. The Space ${\mathcal{D}}(T)$.
  4. The Space ${\mathcal{R}}(T)$.
  5. Triplets of Closely Embedded Hilbert Spaces
  6. Generalised Triplets in the Sense of Berezanskii
  7. Starting with a Triplet of Closely Embedded Hilbert Space.
  8. Starting with a Generalised Triplet.

Key Result

Proposition 2.1

Let $T\in{\mathcal{C}}({\mathcal{H}},{\mathcal{G}})$, for two Hilbert spaces ${\mathcal{H}}$ and ${\mathcal{G}}$. (a)${\mathcal{D}}(T)$ is closely embedded in ${\mathcal{H}}$ and $i_T$ is the underlying closed embedding. (b)$\operatorname{Ran}(T^*)\subseteq \operatorname{Dom}(i_T^*)$ and equality ho (d)$(i_Ti_T^*)\operatorname{Ran}(T^*T)\subseteq\operatorname{Dom}(T^*T)$ and equality holds provid

Theorems & Definitions (21)

  • Proposition 2.1: CojGh5, Proposition 3.3
  • Lemma 2.2: CojGh5, Lemma 3.8
  • Theorem 2.3: CojGh3, Theorem 2.10
  • Theorem 2.4: CojGh3, Theorem 3.4
  • Remark 2.5
  • Theorem 2.6: CojGh5, Theorem 4.1
  • Theorem 2.7: CojGh5, Theorem 5.1
  • Theorem 2.8: CojGh5, Theorem 5.2
  • Proposition 2.9: CojGh5, Proposition 5.3
  • Lemma 3.1: Berezanskii, page 58
  • ...and 11 more