From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Tue, 12 Sep 2023 07:36:33 +0200 Subject: Initial commit --- pages/unbounded-operators/adjoint-operators.md | 15 +++ pages/unbounded-operators/graph-and-closedness.md | 22 ++++ .../hellinger-toeplitz-theorem.md | 116 +++++++++++++++++++++ pages/unbounded-operators/index.md | 7 ++ pages/unbounded-operators/quadratic-forms.md | 23 ++++ 5 files changed, 183 insertions(+) create mode 100644 pages/unbounded-operators/adjoint-operators.md create mode 100644 pages/unbounded-operators/graph-and-closedness.md create mode 100644 pages/unbounded-operators/hellinger-toeplitz-theorem.md create mode 100644 pages/unbounded-operators/index.md create mode 100644 pages/unbounded-operators/quadratic-forms.md (limited to 'pages/unbounded-operators') diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md new file mode 100644 index 0000000..a93e6d4 --- /dev/null +++ b/pages/unbounded-operators/adjoint-operators.md @@ -0,0 +1,15 @@ +--- +title: Adjoint Operators +parent: Unbounded Operators +nav_order: 1 +published: false +description: > + The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric + operator on a Hilbert space is bounded. We give a proof using the Uniform + Boundedness Theorem. We give another proof using the Closed Graph Theorem. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md new file mode 100644 index 0000000..a9bf738 --- /dev/null +++ b/pages/unbounded-operators/graph-and-closedness.md @@ -0,0 +1,22 @@ +--- +title: Graph and Closedness +parent: Unbounded Operators +nav_order: 1 +description: > + The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric + operator on a Hilbert space is bounded. We give a proof using the Uniform + Boundedness Theorem. We give another proof using the Closed Graph Theorem. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + + +{: .definition-title } + +> Definition (Graph of an Operator) +> +> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$ +> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$ +> where $x$ lies in the domain of $T$ and $y=Tx$. diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md new file mode 100644 index 0000000..07d6b81 --- /dev/null +++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md @@ -0,0 +1,116 @@ +--- +title: Hellinger–Toeplitz Theorem +parent: Unbounded Operators +nav_order: 10 +description: > + The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric + operator on a Hilbert space is bounded. We give a proof using the Uniform + Boundedness Theorem. We give another proof using the Closed Graph Theorem. +# cspell:words Hellinger Toeplitz Schwarz Riesz functionals +--- + +# {{ page.title }} + +Conventions: +{: .mb-0 } + +- Hilbert spaces are complex. +- The inner product is anti-linear in its first argument. +- Operators are linear and possibly unbounded. + +Recall that an operator $T : D(T) \to \hilb{H}$ in a Hilbert space $\hilb{H}$ +is called *symmetric*, if is has the property + +$$ +\innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T). +$$ + +{: .theorem-title } +> Hellinger–Toeplitz theorem +> +> An everywhere-defined symmetric operator on a Hilbert space is bounded. + +Consequently, a symmetric Hilbert space operator +that is (truly) unbounded +cannot be defined everywhere. + +--- + +## Proof using the Uniform Boundedness Theorem + +Assume that $T$ is not bounded. +Then there exists a sequence $(x_n)$ of unit vectors in $\hilb{H}$ +such that $\norm{Tx_n} \to \infty$. +Consider the sequence $(f_n)$ of linear functionals on $\hilb{H}$, +defined by + +$$ +f_n(y) = \innerp{Tx_n}{y} = \innerp{x_n}{Ty} \quad y \in \hilb{H}. +$$ + +The second identity is due to the symmetry of $T$. +Apply Cauchy-Schwarz to both expressions to obtain the inequalities + +$$ +\abs{f_n(y)} \le \norm{Tx_n} \norm{y} +\quad \text{and} \quad +\abs{f_n(y)} \le \norm{x_n} \norm{Ty} +$$ + +for each $n \in \NN$ and $y \in \hilb{H}$. +The first inequality shows that the functionals $f_n$ are bounded. +The second one shows that, for fixed $y$, +the sequence $(\abs{f_n(y)})$ is bounded by $\norm{Ty}$, +since $\norm{x_n} = 1$ for all $n$. +By the [Uniform Boundedness Theorem]({% link +pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md +%}), $(\norm{f_n})$ is a bounded sequence. +One has + +$$ +\norm{Tx_n}^2 = \abs{f_n(Tx_n)} \le \norm{f_n} \norm{Tx_n} \quad n \in \NN. +$$ + +Divide by $\norm{Tx_n}$ (if nonzero) +to obtain $\norm{Tx_n} \le \norm{f_n}$ for all but finitely many $n$. +Thus $(\norm{Tx_n})$ is a bounded sequence, +contradicting $\norm{Tx_n} \to \infty$. +{{ site.qed }} + +--- + +## Proof using the Closed Graph Theorem + +By the [Closed Graph Theorem]({% link +pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md %}), +it is sufficient to show that the graph of $T$ is closed. +Let $(x_n)$ be a convergent sequence of vectors in $\hilb{H}$ +such that the image sequence $(Tx_n)$ converges as well. +Naming the limits $x$ and $z$, respectively, we have + +$$ +x_n \to x +\quad \text{and} \quad +Tx_n \to z. +$$ + +Continuity of the inner product implies + +$$ +\innerp{x_n}{Ty} \to \innerp{x}{Ty} +\quad \text{and} \quad +\innerp{Tx_n}{y} \to \innerp{z}{y} +$$ + +for all $y \in \hilb{H}$. +Since $T$ is symmetric, +the first assertion can be rewritten as + +$$ +\innerp{Tx_n}{y} \to \innerp{Tx}{y}. +$$ + +A sequence of complex numbers has at most one limit, +hence $\innerp{Tx}{y} = \innerp{z}{y}$ for all $y$. +By the Riesz representation theorem, $Tx=z$. +{{ site.qed }} diff --git a/pages/unbounded-operators/index.md b/pages/unbounded-operators/index.md new file mode 100644 index 0000000..54ad701 --- /dev/null +++ b/pages/unbounded-operators/index.md @@ -0,0 +1,7 @@ +--- +title: Unbounded Operators +nav_order: 4 +has_children: true +--- + +# {{ page.title }} diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md new file mode 100644 index 0000000..5831b88 --- /dev/null +++ b/pages/unbounded-operators/quadratic-forms.md @@ -0,0 +1,23 @@ +--- +title: Quadratic Forms +parent: Unbounded Operators +nav_order: 5 +published: false +description: > + The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric + operator on a Hilbert space is bounded. We give a proof using the Uniform + Boundedness Theorem. We give another proof using the Closed Graph Theorem. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + + +{: .definition-title } + +> Definition (Graph of an Operator) +> +> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$ +> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$ +> where $x$ lies in the domain of $T$ and $y=Tx$. -- cgit v1.2.3-70-g09d2