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Squares of symmetric operators

Yury Arlinskii

Abstract

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal T}_z\}_{{\rm Im\,} z>0}$ of closed densely defined symmetric operators with the properties: (I) the domain of ${\mathcal T}_z^2$ is a core of ${\mathcal T}_z$, (II) the domain of ${\mathcal T}_z^2$ is dense but note a core of ${\mathcal T}_z$, (III) the domain of ${\mathcal T}_z^2$ is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case ${\rm dom\,} {\mathcal T}_z^2=\{0\}$ has been considered in [5]. Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with ${\rm ran\,} (S-λI)$ and the projection of the domain of the adjoint $S^*$ on ${\rm ran\,} (S-λI)$, $λ\in{\mathbb C}\setminus{\mathbb R}$, necessary and sufficient conditions for the cases (I)--(III) related to the domain of $S^2$, are obtained.

Squares of symmetric operators

Abstract

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families of closed densely defined symmetric operators with the properties: (I) the domain of is a core of , (II) the domain of is dense but note a core of , (III) the domain of is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case has been considered in [5]. Given a densely defined closed symmetric operator , in terms of the intersection of the domain of with and the projection of the domain of the adjoint on , , necessary and sufficient conditions for the cases (I)--(III) related to the domain of , are obtained.
Paper Structure (20 sections, 26 theorems, 240 equations)

This paper contains 20 sections, 26 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Symmetric operators
  3. Maximal dissipative operators and their Cayley transforms
  4. The Cayley transform
  5. The linear manifold ${\mathcal{L}}_T$ and its representations
  6. Hermitian domains and Hermitian parts of maximal dissipative operators
  7. Boundary pairs associated with a maximal dissipative operator
  8. Rigged Hilbert space associated with a closed operator
  9. The Green identity and boundary pairs
  10. Maximal dissipative operators of the class ${\mathbf D}_{{\rm{sing}}}$
  11. The Shtraus extensions of symmetric operators and their matrix representations
  12. Maximal dissipative operators from the class ${\mathbf D}_{{\rm{sing}}}$ whose domains intersect domains of their adjoint only at zero
  13. Operators from the class ${\mathbf D}_{{\rm{sing}}}$ with non-trivial non-dense Hermitian domains
  14. The shorted operator
  15. Proof of Theorem \ref{['ceotcn']}
  16. ...and 5 more sections

Key Result

Lemma 1.1

(1) Let $S$ be a linear operator. Then for any complex number $\lambda$ the relation holds and the following statements are equivalent: (2) Let the operator $S$ be closed and let $\lambda$ be a point of regular type of $S$A complex number $\lambda$ is called a point of regular type for a linear operator $S$ in a Banach space ${\mathcal{X}}$ if there exists a positive number $c$ such that $||(S-\

Theorems & Definitions (50)

  • Lemma 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 40 more