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

Rellich–Kato

• Let $(𝐴,𝐷(𝐴))$ be a self-adjoint operator and $0\le 𝛼<1$. We say that an operator $𝐵$ defined on $𝐷(𝐴)$ is $𝐴$-bounded with relative bound $𝛼$ if there exists $𝐶$ such that

• $𝐵$ is infinitesimally $𝐴$-bounded if it is $𝐴$-bounded with relative bound $𝛼$ for all $𝛼>0$.

• Theorem (Rellich–Kato) Suppose that $𝐵$ is $𝐴$-bounded with relative bound $𝛼$. Then $𝐴+𝐵$ is self-adjoint on $𝐷(𝐴)$.

  Moreover, if $\mathrm{𝒟︀}\subset 𝐷(𝐴)$ is a dense subspace such that $\overline{𝐴{|}_{\mathrm{𝒟︀}}}=𝐴$, then we also have $𝐴+𝐵=\overline{(𝐴+𝐵){|}_{\mathrm{𝒟︀}}}$, i.e. $(𝐴+𝐵){|}_{\mathrm{𝒟︀}}$ is essentially self-adjoint.

  Finally, if $𝜎(𝐴)\subset \left[𝑎,\infty \right)$ for some $𝑎\in ℝ$, then $𝜎(𝐴+𝐵)\subset \left[𝑎-\frac{𝐶}{1-𝛼},\infty \right)$.

  • $\mathrm{𝒟︀}$ in the above theorem is called the core of the operator $𝐴$.

  • Proof uses the identity

    $$𝐴+𝐵+𝑖𝜇=(1+{𝐵(𝐴+𝑖𝜇)}^{-1})(𝐴+𝑖𝜇)$$

    and then exploits $𝛼<1$ to get invertibility of $(1+{𝐵(𝐴+𝑖𝜇)}^{-1})$ for large enough $𝜇$.

• Theorem Let $𝑉\in {𝐿}^{𝑝}+{𝐿}^{\infty }$ where

  $$\{\begin{array}{c}𝑝=2\text{if}𝑝<4\\ 𝑝>2\text{if}𝑝=4\\ 𝑝=\frac{𝑑}{2}\text{otherwise}\end{array}$$

  Then $𝑉$ is infinitesimally $(-\mathrm{\Delta })$-bounded, and

  $$\left|{\int }_{{ℝ}^{𝑑}}𝑉|𝑓{|}^2\right|\le 𝜀{\int }_{{ℝ}^{𝑑}}|\nabla 𝑓{|}^2+{𝐶}_{𝜀}{\int }_{{ℝ}^{𝑑}}|𝑓{|}^2$$
• As a corollary, we have that $-\mathrm{\Delta }+𝑉$ is self-adjoint on $𝐷(\mathrm{\Delta })={𝐻}^2({ℝ}^{𝑑})$ with spectrum that is bounded below.

Friedrichs

Let $𝜑(\cdot ,\cdot )$ be a sesquilinear form on a dense domain $\mathrm{𝒬︀}\subset ℌ$. Let $𝑞$ denote the associated real-valued quadratic form.

• Definition (Coercivity and Closure) We say $𝑞$ an $𝜑$ are bounded from below, or semi-bounded, if there is an $𝛼\in ℝ$ such that $𝑞(𝑢)\ge 𝛼{\Vert 𝑢\Vert }^2$.

  If we can take $𝛼>0$, we say $𝑞$ and $𝜑$ are coercive.

  We say that $𝑞$ and $𝜑$ are closed if they are coercive and $(\mathrm{𝒬︀},𝜑)$ is a Hilbert space.

  We say that $𝑞$ and $𝜑$ are closable in $ℌ$ when they admit a closed extension, in which case there is a minimal closed extension which we denote $(\overline{\mathrm{𝒬︀}},\overline{𝜑})$ and call the closure.

• Lemma Let $𝜑$ be a coercive sesquilinear form on $\mathrm{𝒬︀}\subset ℌ$. Then $𝜑$ is closable iff for any Cauchy sequence ${𝑣}_{𝑛}\in \mathrm{𝒬︀}$ satisfying in addition ${𝑣}_{𝑛}\to 0$ in $ℌ$, we have $𝑞({𝑣}_{𝑛})\to 0$.

Symmetric Operators

• Let $(𝐴,𝐷(𝐴))$ be a symmetric operator. The quadratic form associated with $𝐴$ is the form

  $${𝑞}_{𝐴(𝑣)}≔⟨𝑣\text{,}𝐴𝑣⟩,\mathrm{𝒬︀}≔𝐷(𝐴).$$

The associated sesquilinear form is ${𝜑}_{𝐴(𝑣,𝑤)}≔⟨𝑣\text{,}𝐴𝑤⟩$.

• A symmetric operator $(𝐴,𝐷(𝐴))$ is bounded from below/semi-bounded, or coercive, if the associated form ${𝑞}_{𝐴}$ is, and in which case we will write $𝐴\ge 𝛼$.

• Theorem Let $𝐴\ge 𝛼>0$ be a coercive symmetric operator. Then ${𝑞}_{𝐴}$ is closable to a form $\overline{{𝑞}_{𝐴}}$ defined on

  $${\mathrm{𝒬︀}}_{𝐴}≔\left\{𝑣\in ℌ:\exists {𝑣}_{𝑛}\in 𝐷(𝐴)\text{such that}{𝑣}_{𝑛}\to 𝑣,\underset{𝑛,𝑚\to \infty }{lim}𝑞({𝑣}_{𝑛}-{𝑣}_{𝑚})=0\right\}$$

  Furthermore, we have

  • $𝐷(𝐴)$ is dense in ${\mathrm{𝒬︀}}_{𝐴}$ with respect to the norm $\sqrt{\overline{{𝑞}_{𝐴}}}$
  • ${\mathrm{𝒬︀}}_{𝐴}$ is dense in $ℌ$
  • For all $𝑢\in 𝐷(𝐴)$ and all $𝑣\in {\mathrm{𝒬︀}}_{𝐴}$, $\overline{{𝜑}_{𝐴}}(𝑣,𝑢)=⟨𝑣\text{,}𝐴𝑢⟩$
  • the embedding ${\mathrm{𝒬︀}}_{𝐴}\hookrightarrow ︎ℌ$ is continuous, i.e. $𝛼{\Vert 𝑣\Vert }^2\le \overline{{𝑞}_{𝐴}}(𝑣)$ for all $𝑣\in {\mathrm{𝒬︀}}_{𝐴}$.
  • If $𝐴$ is closed, then the embedding $𝐷(𝐴)\hookrightarrow ︎{\mathrm{𝒬︀}}_{𝐴}$ is also continuous, i.e. for all $𝑣\in 𝐷(𝐴)$,
  $$\overline{{𝑞}_{𝐴}}(𝑣)=⟨𝑣\text{,}𝐴𝑣⟩\le \frac{1}{2}{\Vert 𝑣\Vert }_{𝐷(𝐴)}^2=\frac{{\Vert 𝑣\Vert }^2}{2}+\frac{{\Vert 𝐴𝑣\Vert }^2}{2}.$$

  If $𝐴\ge 𝛼$ is only bounded below we can apply the above with $𝐴+𝛼+𝜀$ to get a closed form.

  Self-adjoint Operators

  • Theorem Let $𝐴\ge 𝛼$ and let $\overline{{𝜑}_{𝐴}}$ be the associated closed form defined in the previous Theorem. For any $𝑣,𝑧\in ℌ$, the following are equivalent:

    1. $𝑣\in {\mathrm{𝒬︀}}_{𝐴}$ and $\overline{{𝜑}_{𝐴}}(𝑣,ℎ)=⟨𝑧\text{,}ℎ⟩$ for all $ℎ\in {\mathrm{𝒬︀}}_{𝐴}$.
    2. $𝑣\in 𝐷(𝐴)$ and $𝐴𝑣=𝑧$.

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

    • The equation $\overline{{𝜑}_{𝐴}}(𝑣,ℎ)=⟨𝑧\text{,}ℎ⟩$ is the weak formulation of the equation $𝐴𝑣=𝑧$. It is “weak” because we only suppose that $𝑣$ belongs to ${\mathrm{𝒬︀}}_{𝐴}$. $𝐷(𝐴)$ is the subset
    $$𝐷(𝐴)=\{𝑣\in {\mathrm{𝒬︀}}_{𝐴}:\exists 𝑧\in ℌ:\overline{{𝜑}_{𝐴}}(𝑣,ℎ)=⟨𝑧\text{,}ℎ⟩,\forall ℎ\in {\mathrm{𝒬︀}}_{𝐴}\}.$$
  • As a self-adjoint operator is fully characterized by the closure $\overline{{𝑞}_{𝐴}}$, we will abusively write ${𝑞}_{𝐴}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\overline{{𝑞}_{𝐴}}$, ${𝜑}_{𝐴}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\overline{{𝜑}_{𝐴}},$ $𝑄(𝐴)\stackrel{\scriptscriptstyle\mathrm{def}}{=}{\mathrm{𝒬︀}}_{𝐴}$.

  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 $𝐴$: from $𝐴$ compute ${𝑞}_{𝐴}$. Since it is bounded below, we can close it. Then the Theorem gives us a self-adjoint operator that extends $𝐴$.
  • Theorem Let $\mathrm{𝒬︀}\subset ℌ$ be Hilbert spaces such that $\mathrm{𝒬︀}$ is dense and continuously embedded in $ℌ$. Then there is a unique self-adjoint operator $𝐴$ on $𝐷(𝐴)\subset ℌ$, with
  $$𝐷(𝐴)=\{𝑣\in \mathrm{𝒬︀}:\exists 𝑣\in ℌ:{⟨𝑣\text{,}ℎ⟩}_{\mathrm{𝒬︀}}={⟨𝑧\text{,}ℎ⟩}_{ℌ},\forall ℎ\in ℌ\}$$

  such that ${𝑞}_{𝐴}={\Vert \cdot \Vert }_{\mathrm{𝒬︀}}^2,$ ${𝜑}_{𝐴}={⟨\cdot \text{,}\cdot ⟩}_{\mathrm{𝒬︀}}$, and $\mathrm{𝒬︀}=𝑄(𝐴)$.

  Moreover, if $𝐵$ is a self-adjoint operator on $𝐷(𝐵)\subset \mathrm{𝒬︀}$ such that $𝐷(𝐵)$ is dense on $\mathrm{𝒬︀}$ and ${𝑞}_{𝐵}=𝑞$ on $𝐷(𝐵)$, then $𝐵=𝐴$.

  Theorem (Kato-Lions-Milgram-Nelson) Let $𝐴$ be a coercive self-adjoint operator and let ${𝑞}_{𝐴}$ be its associated closed quadratic form with domain $𝑄(𝐴)$. Let $𝑏$ be another quadratic form on $𝑄(𝐴)$ such that for some $𝜂\in \left[0,1\right)$, $𝜅>0$ and for all $𝑣\in 𝑄(𝐴)$,

  $$|𝑏(𝑣)|\le 𝜂{𝑞}_{𝐴}(𝑣)+𝜅{\Vert 𝑣\Vert }^2$$

  Then ${𝑞}_{𝐴}+𝑏$ is closed and coercive on $𝑄(𝐴)$, and hence it is associated with a unique self-adjoint operator $𝐶$, with $𝑄(𝐶)=𝑄(𝐴)$.

  If in particular $𝑏$ is the quadratic form of a symmetric operator $𝐵$ defined on $𝐷(𝐴)$, then $𝐶$ is the unique self-adjoint extension of $𝐴+𝐵$ defined on $𝐷(𝐴+𝐵)=𝐷(𝐴)$, such that $𝐷(𝐴)\subset 𝐷(𝐶)\subset 𝑄(𝐴)$.

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