Functional Analysis

Author
Affiliation

Kai Prince

The University of Manchester

Published

5 September 2025

The foundational knowledge relating to Measure & Ergodic Theory, that has not been covered elsewhere, was provided by Jamneshan and Kreidler (2025).

Definition 1 For a Hilbert space \(\HilbertSpace\),

  1. A linear isometry is a linear map \(U:\HilbertSpace \rightarrow \HilbertSpace\) such that \(||Uf||=||f||\) for all \(f\in \HilbertSpace\) (and thus injective).
  2. A linear isometry, \(U\), is a unitary operator if it is surjective (and thus bijective).
  3. We write \(\mathscr{U}(\HilbertSpace)\) for the set of all unitary operators \(U:\HilbertSpace \rightarrow \HilbertSpace\).
  4. For a group \(\Group\), we call a group homomorphism \(U:\Group\rightarrow\mathscr{U}(\HilbertSpace )\), \(\GroupElement\mapsto U_\GroupElement\) a unitary representation of \(\Group\) on \(\HilbertSpace\).

Definition 2 Let \(J:(Y,S)\rightarrow(X,T)\) be an extension of measure-preserving systems. Then \(E=U_J(\text{L}^2(Y))\) is an invariant Markov sub-lattice of \(\text{L}^2(X)\), i.e.,

  1. \(E\) is a closed linear subspace of \(\text{L}^2(X)\),
  2. \(\boldsymbol{1}\in E\),
  3. \(|f|,\ \text{Re}(f),\ \text{Im}(f)\in E\) for every \(f\in E\), and
  4. \(U_{T_\GroupElement}f\in E\) for every \(f\in E\) and \(\GroupElement\in\Group\).

Definition 3 Let \(\HilbertSpace\) be a Hilbert space. \(\mathscr{L}(\HilbertSpace)\) is the space of all bounded linear operators from \(\HilbertSpace\) to \(\HilbertSpace\). 1

A family \(\mathscr{S}\subseteq\mathscr{L}(\HilbertSpace)\) is a semigroup (of operators) if \(UV\in\mathscr{S}\) for all \(U,V\in\mathscr{S}\). It is a contraction semigroup if, in addition, \(||U||\leq1\) for all \(U\in\mathscr{S}\).

We call \[\text{fix}(\mathscr{S}):=\bigcap_{U\in\mathscr{S}}\text{fix}(U)=\{f\in\HilbertSpace\mid Uf=f\text{ for every }U\in\mathscr{S}\}\] the fixed space of \(\mathscr{S}\).23

The closed convex hull \(\overline{\text{co}}\ A\) is the closure of the set of all convex combinations of elements of \(A\).

The closed linear hull \(\overline{\text{lin}}\ A\) is the closure of the set of all linear combinations of elements of \(A\).

Definition 4 Let \(U:\AmenableGroup\rightarrow\mathscr{U}(\HilbertSpace)\) be a unitary representation of a discrete abelian amenable group \(\AmenableGroup\).

A group homomorphism \(\chi:\AmenableGroup\rightarrow\mathbb{T}\), where \(\mathbb{T}=\{z\in\mathbb{C}\mid|z|=1\}\), is called a character. The dual group \(\Dual{\AmenableGroup}\) of \(\AmenableGroup\) is the set of all such characters equipped with the multiplication given by \((\chi_1\chi_2)(\AmenableGroupElement):=\chi_1(\AmenableGroupElement)\chi_2(\AmenableGroupElement)\) for \(\AmenableGroupElement\in\AmenableGroup\) and \(\chi_1,\chi_2\in\Dual{\AmenableGroup}\).4

Definition 5 Let \[M_f:=\text{lin}\{U_\AmenableGroupElement f\mid\AmenableGroupElement\in\AmenableGroup \}. \] We call a subset \(M\subseteq \HilbertSpace\) an invariant finite-dimensional subspace if \(M_f\) is finite-dimensional and \(U_\AmenableGroupElement f\in M\) for all \(f\in M\) and \(\AmenableGroupElement\in\AmenableGroup\).

The closure \[\HilbertSpace_\text{ds}=\overline{\bigcup\{M\subseteq \HilbertSpace\mid M\text{ invariant finite-dimensional subspace} \}}\subseteq \HilbertSpace \] is called the discrete spectrum part of \(U\).5

Proposition 1 (cf. Jamneshan and Kreidler, 2025, Proposition 5.1.10) The eigenspaces \(\text{ker}(\chi-U)\) for \(\chi\in\Dual{\Group}\) are pairwise orthogonal. For \(M\subseteq \HilbertSpace\), then the following are equivalent:

  1. \(M\) is an irreducible invariant finite-dimensional subspace.
  2. \(M\) is an invariant linear subspace which is at most one-dimensional.
  3. There is \(\chi\in\Dual{\Group}\) and \(f\in\text{ker}(\chi-U)\), such that \(M=\mathbb{C}\cdot f\).

Definition 6 A unitary representation \(U:\AmenableGroup\rightarrow\mathscr{U}(\HilbertSpace)\) of a topological group \(\AmenableGroup\) is strongly continuous if \(\AmenableGroup\rightarrow \HilbertSpace\), \(x\mapsto U_xf\) is continuous for every \(f\in \HilbertSpace\).

Definition 7 For every discrete group \(\AmenableGroup\), the maps \[\begin{align} L:\AmenableGroup\rightarrow\mathscr{U}(\text{L}^2(\AmenableGroup)),&\qquad x\mapsto L_x \\ R:\AmenableGroup\rightarrow\mathscr{U}(\text{L}^2(\AmenableGroup)),&\qquad x\mapsto R_x \end{align}\] with \(L_xf:=f\circ l_{x^{-1}}\) and \(R_xf:=f\circ r_{x^{-1}}\) for \(f\in\text{L}^2(\AmenableGroup)\) and \(x\in\AmenableGroup\) are strongly continuous unitary representations. We call \(L\) and \(R\) the left and right regular representation of \(\AmenableGroup\), respectively.6

Assume that \(U:\AmenableGroup\rightarrow\mathscr{U}(\HilbertSpace)\) is a strongly continuous unitary representation of a discrete group \(\AmenableGroup\). Let further \(P\) be the orthogonal projection onto the fixed space \(\text{fix}(U(\AmenableGroup))\). Then \[(Pf\mid g)=\lim_{N\rightarrow\infty}\frac{1}{|\Folner[N]|}\sum_{\AmenableGroupElement\in\Folner[N]}(U_\AmenableGroupElement f\mid g)\] for all \(f,g\in\HilbertSpace\).

If \(U:\AmenableGroup\rightarrow\mathscr{U}(\HilbertSpace)\) is a strongly continuous unitary representation of a compact group \(\AmenableGroup\), then \(U\) has discrete spectrum.7

Exercise 1 Show that \[\HilbertSpace=\HilbertSpace_\text{ds}\oplus \HilbertSpace_\text{wm} \] and what happens to \(\HilbertSpace_\text{ds}\) and \(\HilbertSpace_\text{wm}\) when it is averaged by the Abstract Mean Ergodic Theorem for amenable groups.8

Proposition 2 Let \((X,T)\) be a measure-preserving system. Then \(\text{L}^2(X)_\text{ds}\) is an invariant Markov sublattice of \(\text{L}^2(X)\).

Exercise 2 (Jamneshan and Kreidler, 2025, Lemma 6.1.12) Assume that \(U:\AmenableGroup\rightarrow\mathscr{U}(\HilbertSpace)\) is a strongly continuous unitary representation of a discrete group \(\AmenableGroup\). Let further \(P\) be the orthogonal projection onto the fixed space \(\text{fix}(U(\AmenableGroup))\). Then \((Pf\mid g)=\lim_{N\rightarrow\infty}\frac{1}{|\Folner[N]|}\sum_{\AmenableGroupElement\in\Folner[N]}(U_\AmenableGroupElement f\mid g)\) for all \(f,g\in\HilbertSpace\).

References

Jamneshan, A. and Kreidler, H. (2025). 'ISem 28: Ergodic structure theory and applications',.

Footnotes

  1. Note, \(\mathscr{U}(\HilbertSpace)\subseteq\mathscr{L}(\HilbertSpace)\) and \(||U||=1\) when \(U\in\mathscr{U}(\HilbertSpace)\), so \(\mathscr{U}(\HilbertSpace)\) is a contraction semigroup.↩︎

  2. \(\text{fix}(\mathscr{S})\subseteq \HilbertSpace_\text{ds}\).↩︎

  3. Meeting Notes: 0-dimensional. Next most simple case would be scaling (1-dimensional), and then rotations on a place (2-dimensional).↩︎

  4. Commutator subgroup and abelianisation?↩︎

  5. Meeting Notes: In the \(\AmenableGroup=\mathbb{Z}\) case, then \(\HilbertSpace_\text{ds}\) can be broken up into one-dimensional objects that corresponds to an eigenfunction.↩︎

  6. As \(\AmenableGroup\) is discrete, then any map \(\AmenableGroup\rightarrow \HilbertSpace\) is continuous for any topological space \(\HilbertSpace\). Thus, \(\AmenableGroup\rightarrow \text{L}^2(\AmenableGroup)\) defined by \(\AmenableGroupElement\mapsto L_\AmenableGroupElement f\) or \(\AmenableGroupElement\mapsto R_\AmenableGroupElement f\) is always continuous. Hence, \(L\) and \(R\) are strongly continuous unitary representations.↩︎

  7. This \(\AmenableGroup\) would be the compact group behind the factor subsystem of the larger system. For example, \(\AmenableGroup=\mathbb{Z}\) for the irrational \(\alpha\) group rotation would give us compact \(G=\{z\in\mathbb{C}\mid|z|=1\}\). The homomorphism from \(\AmenableGroup\rightarrow \Group\) is \(n\mapsto e^{2\pi i\alpha n}\).↩︎

  8. (Proposition 5.3.3)\[\HilbertSpace_\text{wm}=\left\{f\in \HilbertSpace\mid \lim_{N\rightarrow\infty}\frac{1}{|\Folner[N]|}\sum_{\varphi\in\Folner[N]}|(U_\varphi f\mid f)|^2=0 \right\}. \]↩︎