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PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.RealAnalytic

Criteria for essential self-adjointness

i. Overview

A densely defined symmetric operator is essentially self-adjoint when both deficiency spaces are trivial. In particular a symmetric operator whose domain is the span of a Hilbert basis of eigenvectors with real eigenvalues is essentially self-adjoint.

ii. Key results

- `DefectIndexCertificate` : a densely defined symmetric operator with trivial deficiency spaces. - `DefectIndexCertificate.essentiallySelfAdjoint` : such an operator is essentially self-adjoint. - `isEssentiallySelfAdjoint_of_hilbertBasis_eigenvectors` : an operator diagonal in a Hilbert basis with real eigenvalues is essentially self-adjoint.

iii. Table of contents

  • A. Von Neumann's defect criterion
  • B. Essential self-adjointness from a Hilbert basis of eigenvectors

A. Von Neumann's defect criterion

B. Essential self-adjointness from a Hilbert basis of eigenvectors

A symmetric operator whose domain is *exactly* the (algebraic) span of a Hilbert basis of eigenvectors with real eigenvalues is essentially self-adjoint. This is the standard "eigenbasis density" criterion for essential self-adjointness (see e.g. Reed–Simon, *Methods of Modern Mathematical Physics I: Functional Analysis*, Theorem VIII.3): reality of the eigenvalues makes `T ∓ i` bounded below by `1` on the eigenbasis span, and the Hilbert-basis expansion of an arbitrary `y : H` (with coefficients divided by `λ - z`) produces, in the limit of its partial sums, an explicit preimage of `y` under `T.closure ∓ i`.

4 declarations

theorem

A symmetric operator with trivial deficiency indices is essentially self-adjoint

Let HH be a complex Hilbert space and T:D(T)⊆H→HT: D(T) \subseteq H \to H be a partially defined linear operator. If TT possesses a `DefectIndexCertificate` (meaning TT is a densely defined symmetric operator with trivial deficiency spaces), then TT is essentially self-adjoint.

theorem

The closure T‾\overline{T} is self-adjoint if TT has a defect index certificate

Let HH be a complex Hilbert space and TT be a densely defined linear operator on HH. If TT has a defect index certificate (signifying that TT is a symmetric operator whose deficiency spaces are both trivial), then the closure T‾\overline{T} is a self-adjoint operator.

theorem

A Self-Adjoint Operator is Essentially Self-Adjoint

Let HH be a complex Hilbert space. If TT is a self-adjoint partially defined linear operator on HH (i.e., T=T∗T = T^*), then TT is essentially self-adjoint.

theorem

An operator diagonal in a Hilbert basis with real eigenvalues is essentially self-adjoint

Let HH be a complex Hilbert space and e={ei}i∈ιe = \{e_i\}_{i \in \iota} be a Hilbert basis for HH. Let TT be a partially defined linear operator on HH whose domain D(T)D(T) is the algebraic span of the set {ei∣i∈ι}\{e_i \mid i \in \iota\}. If for every i∈ιi \in \iota, eie_i is an eigenvector of TT with a real eigenvalue λi∈R\lambda_i \in \mathbb{R} (that is, Tei=λieiT e_i = \lambda_i e_i), then TT is essentially self-adjoint. An operator is essentially self-adjoint if it is densely defined and its closure is a self-adjoint operator.