Notes on Lewin Ch3: Self-adjointness Criteria: Rellich, Kato and Friedrichs

Rellich–Kato

• Let be a self-adjoint operator and . We say that an operator defined on is -bounded with relative bound if there exists such that

• is infinitesimally -bounded if it is A bounded with relative bound for all .

• Theorem (Rellich–Kato) Suppose that is -bounded with relative bound . Then is self-adjoint on .

  Moreover, if is a dense subspace such that , then we also have , i.e. is essentially self-adjoint.

  Finally, if for some , then .

  • in the above theorem is called the core of the operator .

  • Proof uses the identity

    and then exploits to get invertibility of for large enough .

• Theorem Let where

  Then is infinitesimally -bounded, and

• As a corollary, we have that is self-adjoint on with spectrum that is bounded below.

Friedrichs

Let be a sesquilinear form on a dense domain . Let denote the associated real-valued quadratic form.

• Definition (Coercivity and Closure) We say an are bounded from below, or semi-bounded, if there is an such that .

  If we can take , we say and are coercive.

  We say that and are closed if they are coercive and is a Hilbert space.

  We say that and are closable in when they admit a closed extension, in which case there is a minimal closed extension which we denote and call the closure.

• Lemma Let be a coercive sesquilinear form on . Then is closable iff for any Cauchy sequence satisfying in addition in , we have .

Symmetric Operators

• Let be a symmetric operator. The quadratic form associated with is the form

The associated sesquilinear form is .

• A symmetric operator is bounded from below/semi-bounded, or coercive, if the associated form is, and in which case we will write .

• Theorem Let be a coercive symmetric operator. Then is closable to a form defined on

  Furthermore, we have

  • is dense in with respect to the norm
  • is dense in
  • For all and all ,
  • the embedding is continuous, i.e. for all .
  • If is closed, then the embedding is also continuous, i.e. for all ,

  If is only bounded below we can apply the above with to get a closed form.

  Self-adjoint Operators

  • Theorem Let and let be the associated closed form defined in the previous Theorem. For any , the following are equivalent:

    1. and for all .
    2. and .

    In particular we can reconstruct from the closure of its quadratic form.

    • The equation is the weak formulation of the equation . It is “weak” because we only suppose that belongs to . is the subset

  • As a self-adjoint operator is fully characterized by the closure , we will abusively write , .

  Friedrichs Realisation

  • Theorem 3.14 showed that every semi-bounded self-adjoint operator is given by a closed quadratic form. Below we show that every closed quadratic form that is bounded below is given by a self-adjoint operator.

    • this allows us to define the Friedrichs extension of any bounded below symmetric operator : from compute . Since it is bounded below, we can close it. Then the Theorem gives us a self-adjoint operator that extends .
  • Theorem Let be Hilbert spaces such that is dense and continuously embedded in . Then there is a unique self-adjoint operator on , with

  such that , and .

  Moreover, if is a self-adjoint operator on such that is dense on and on , then .

  Theorem (Kato-Lions-Milgram-Nelson) Let be a coercive self-adjoint operator and let be its associated closed quadratic form with domain . Let be another quadratic form on such that for some , and for all ,

  Then is closed and coercive on , and hence it is associated with a unique self-adjoint operator , with .

  If in particular is the quadratic form of a symmetric operator defined on , then is the unique self-adjoint extension of defined on , such that .

Theres more on specific examples and exercises which I didn’t get to.