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Triplets of Closely Embedded Hilbert Spaces

Petru Cojuhari, Aurelian Gheondea

TL;DR

The paper develops a general theory of triplets of closely embedded Hilbert spaces built from a positive selfadjoint Hamiltonian $H$ that may lack a bounded inverse. It constructs an abstract model via a factorization $H=T^*T$ and a kernel operator $A=j_+j_+^*$, establishes existence, uniqueness, and left-right symmetry of such triplets, and demonstrates canonical duality through unitary maps. The authors motivate the framework with diverse examples—including Bessel vs. Riesz potentials, weighted Sobolev spaces, Dirichlet-type spaces on the polydisc, and weighted $L^2$ spaces—and show how closed embeddings accommodate noncoercive settings. They then apply the theory to weak solutions of Dirichlet problems for degenerate elliptic PDEs, providing existence results without requiring coercivity or Poincaré–Sobolev inequalities, thereby broadening applicability to nonstandard contexts.

Abstract

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the system, which is supposed to be one-to-one but may not have a bounded inverse, and for which a model is obtained. From this model we get the abstract concept and show that its basic properties are the same with those of the model. Existence and uniqueness results, as well as left-right symmetry, for these triplets of closely embedded Hilbert spaces are obtained. We motivate this abstract theory by a diversity of problems coming from homogeneous or weighted Sobolev spaces, Hilbert spaces of holomorphic functions, and weighted $L^2$ spaces. An application to weak solutions for a Dirichlet problem associated to a class of degenerate elliptic partial differential equations is presented. In this way, we propose a general method of proving the existence of weak solutions that avoids coercivity conditions and Poincaré-Sobolev type inequalities.

Triplets of Closely Embedded Hilbert Spaces

TL;DR

The paper develops a general theory of triplets of closely embedded Hilbert spaces built from a positive selfadjoint Hamiltonian that may lack a bounded inverse. It constructs an abstract model via a factorization and a kernel operator , establishes existence, uniqueness, and left-right symmetry of such triplets, and demonstrates canonical duality through unitary maps. The authors motivate the framework with diverse examples—including Bessel vs. Riesz potentials, weighted Sobolev spaces, Dirichlet-type spaces on the polydisc, and weighted spaces—and show how closed embeddings accommodate noncoercive settings. They then apply the theory to weak solutions of Dirichlet problems for degenerate elliptic PDEs, providing existence results without requiring coercivity or Poincaré–Sobolev inequalities, thereby broadening applicability to nonstandard contexts.

Abstract

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator , that is called the Hamiltonian of the system, which is supposed to be one-to-one but may not have a bounded inverse, and for which a model is obtained. From this model we get the abstract concept and show that its basic properties are the same with those of the model. Existence and uniqueness results, as well as left-right symmetry, for these triplets of closely embedded Hilbert spaces are obtained. We motivate this abstract theory by a diversity of problems coming from homogeneous or weighted Sobolev spaces, Hilbert spaces of holomorphic functions, and weighted spaces. An application to weak solutions for a Dirichlet problem associated to a class of degenerate elliptic partial differential equations is presented. In this way, we propose a general method of proving the existence of weak solutions that avoids coercivity conditions and Poincaré-Sobolev type inequalities.

Paper Structure

This paper contains 14 sections, 21 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Some Motivations
  3. Bessel Potential versus Riesz Potential.
  4. Weighted Sobolev Spaces.
  5. Dirichlet Type Spaces on the Polydisc.
  6. Weighted $L^2$ Spaces.
  7. Notation and Preliminary Results
  8. The Space ${\mathcal{D}}(T)$.
  9. The Hilbert Space ${\mathcal{R}}(T)$.
  10. A Model of a Triplet of Closely Embedded Hilbert Spaces
  11. Triplets of Closely Embedded Hilbert Spaces
  12. Definition and Basic Properties
  13. Existence and Uniqueness
  14. Weak Solutions for a Class of Dirichlet Problems

Key Result

Theorem 1.1

Let $H$ be a positive selfadjoint operator in a Hilbert space ${\mathcal{H}}$ such that $A=H^{-1}$ is a bounded operator, and let $S\in{\mathcal{B}}({\mathcal{H}})$ be such that $A=S^*S$. With notation as before $({\mathcal{H}}_+;{\mathcal{H}};{\mathcal{H}}_-)$ is a triplet of Hilber spaces. In addi

Theorems & Definitions (35)

  • Theorem 1.1
  • Example 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 25 more