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Example 15.1 (Regular Representations). Recall the definition of the operators \( {L}_{x} \) and \( {R}_{x}, x \in G \), from (10.3) and (10.4). For \( f \in {\mathrm{L}}^{1}\left( G\right) \) and \( x \in G \) we define\n\n\[ \n\left( {{\tau }_{x}f}\right) \left( y\right) \mathrel{\text{:=}} f\left( {{x}^{-1}y}\right)... | Clearly, every strongly continuous representation of \( G \) is weakly continuous. We shall prove below that, actually, weak and strong continuity are equivalent properties. | No |
Lemma 15.2. In the setting described above, one has \( {\pi }_{f}u \in E \) for all \( f \in {\mathrm{L}}^{1}\left( G\right) \) and for all \( u \in E \) . | Proof. It suffices to consider a real-valued function \( f \) . Furthermore, by (15.3) and by density, we may suppose that \( f \in {\mathrm{L}}^{\infty }\left( G\right) \) and, after adding a multiple of 1 and multiplying by a scalar, even that \( f \geq 0 \) and \( {\int }_{G}f = 1 \) . By Krein’s Theorem C. 11 the s... | Yes |
Lemma 15.5. For each \( u \in E \) one has \( u \in \operatorname{cl}\left\{ {{\pi }_{f}u : f \in \mathrm{C}\left( G\right) }\right\} \) . In particular,\n\n\[ E = \overline{\operatorname{lin}}\left\{ {{\pi }_{f}u : f \in \mathrm{C}\left( G\right), u \in E}\right\} . \] | Proof. Let \( {u}^{\prime } \in {E}^{\prime } \) such that for all \( f \in \mathrm{C}\left( G\right) \) one has \( \left\langle {{\pi }_{f}u,{u}^{\prime }}\right\rangle = 0 \) . This means that\n\n\[ {\int }_{G}f\left( x\right) \left\langle {{\pi }_{x}u,{u}^{\prime }}\right\rangle \mathrm{d}x = 0\;\text{ for all }f \i... | Yes |
Theorem 15.6. For a representation \( \pi : G \rightarrow \mathcal{L}\left( E\right) \) of a compact group \( G \) on a Banach space \( E \) the following assertions are equivalent:\n\n(i) \( \pi \) is weakly continuous.\n\n(ii) \( \pi \) is strongly continuous.\n\n(iii) The mapping\n\n\[ G \times E \rightarrow E,\;\le... | Proof. (i) \( \Rightarrow \) (ii): We define\n\n\[ F \mathrel{\text{:=}} \left\{ {u \in E : \text{ the mapping }G \rightarrow E, x \mapsto {\pi }_{x}u\text{ is continuous }}\right\} .\n\]\nThen \( F \) is a closed subspace of \( E \) by the uniform boundedness of the operators \( {\pi }_{x} \) for \( x \in G \) . If \(... | Yes |
Lemma 15.7. For \( f, g, h \in {\mathrm{L}}^{1}\left( G\right) \) and \( x \in G \) we have\na) \( \parallel f * g{\parallel }_{p} \leq \parallel f{\parallel }_{1}\parallel g{\parallel }_{p} \) for \( 1 \leq p \leq \infty \) (Young’s Inequality).\nb) \( {\left( f * g\right) }^{ * } = {g}^{ * } * {f}^{ * } \).\nc) \( {\... | Proof. a) follows, for \( 1 \leq p < \infty \), from (15.4) and the fact that the left regular representation is isometric. For \( p = \infty \) one computes\n\[ \langle f * g, h\rangle = {\int }_{G}f\left( x\right) \left\langle {{\tau }_{x}g, h}\right\rangle \mathrm{d}x = {\int }_{G}f\left( x\right) \left\langle {g,{\... | Yes |
Theorem 15.8. Let \( G \) be a compact group. Then the linear span of irreducible coordinate functions is dense in \( \mathrm{C}\left( G\right) \) . | The proof of this theorem is rather lengthy. First, we recall from the above that any finite-dimensional unitary representation decomposes orthogonally into irreducible representations. By picking an orthonormal basis subordinate to this decomposition, we see that any coordinate function is a linear combination of irre... | No |
Lemma 15.9. The product of two coordinate functions is again a coordinate function. The pointwise conjugate of a coordinate function is a coordinate function. | Proof. Let \( \pi : G \rightarrow \mathrm{U}\left( n\right) \) and \( \rho : G \rightarrow \mathrm{U}\left( m\right) \) be two unitary representations of the compact group \( G \) . Denote \( {\pi }_{x} = {\left( {\pi }_{ij}\left( x\right) \right) }_{i, j} \) and \( {\rho }_{x} = {\left( {\rho }_{kl}\left( x\right) \ri... | No |
Lemma 15.13. Let \( \pi : G \rightarrow \mathcal{L}\left( E\right) \) be a continuous representation of the compact group \( G \) on some Banach space \( E \) . a) Let \( \chi : G \rightarrow \mathrm{U}\left( n\right) \) be a finite-dimensional unitary representation of \( G \) and \( u \in E \) . Then the finite-dimen... | Proof. a) We note first that \[ \left( {{\tau }_{x}{\chi }_{ij}}\right) \left( y\right) = {\chi }_{ij}\left( {{x}^{-1}y}\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{\chi }_{ik}\left( {x}^{-1}\right) {\chi }_{kj}\left( y\right) \;\left( {x, y \in G}\right) . \] Hence \( {\pi }_{x}{\pi }_{{\chi }_{ij}}u = {\pi }_{{\tau ... | Yes |
Theorem 15.14. Let \( \pi : G \rightarrow \mathcal{L}\left( E\right) \) be a continuous representation of the compact group \( G \) on some Banach space \( E \) . Then\n\n\[ E = \operatorname{cl}\bigcup \left\{ {F : F\text{ is a }{\pi }_{G}\text{-invariant subspace of }E,\dim F < \infty }\right\} \]\n\n\[ = \overline{\... | Proof. By Lemma 15.13 each \( {\pi }_{G} \) -invariant finite-dimensional subspace \( F \) of \( E \) is the span of a unitary system. Since one can decompose each unitary representation orthogonally into irreducible subspaces, the second equality is clear. To prove the first, note that by Theorem 15.8 the linear span ... | Yes |
Theorem 15.21. There is a natural correspondence between topological \( \Gamma \) -actions \( K \times \Gamma \rightarrow K \) and representations \( \pi : \Gamma \rightarrow \mathcal{L}\left( {\mathrm{C}\left( K\right) }\right) \) by one-preserving lattice homomorphisms on \( \mathrm{C}\left( K\right) \) . Moreover, i... | Proof. The first part has already been proved. The second part is an immediate consequence of Theorem 4.17. | No |
Every homogeneous \( \Gamma \) -system \( \left( {H \smallsetminus \Gamma ;\Gamma }\right), H \) a cocompact subgroup of \( \Gamma \), is minimal. | The kernel of this representation is \( \mathop{\bigcap }\limits_{{g \in \Gamma }}{g}^{-1}{Hg} \), i.e., the so-called normal core of the subgroup \( H \) . | No |
Theorem 15.24. Let \( G \) be a compact group and let \( K \times G \rightarrow K \) be a continuous action of \( G \) on the compact space \( K \) with associated Koopman representation \( \kappa \) : \( G \rightarrow \mathcal{L}\left( {\mathrm{C}\left( K\right) }\right) \) . Then the following assertions are equivale... | Proof. (i) \( \Leftrightarrow \) (iii): This has been shown above.\n\n(i) \( \Rightarrow \) (ii): Let \( f \in \operatorname{fix}\left( {\kappa }_{g}\right) \) and fix \( z \in K \) . Then \( f\left( {z \cdot g}\right) = f\left( z\right) \) for every \( g \in G \) . Since \( z \cdot G = K \) by minimality, it follows t... | Yes |
Corollary 15.28. For a continuous Markov representation \( \pi : G \rightarrow \operatorname{Aut}\left( \mathrm{X}\right) \) of a compact group \( G \) on the probability space \( \mathrm{X} \) the following assertions are equivalent:\n\n(i) The system \( \left( {\mathrm{X};G}\right) \) is ergodic.\n\n(ii) The system \... | Proof. We leave the implication (ii) \( \Rightarrow \) (i) as Exercise 10 and prove the converse (i) \( \Rightarrow \) (ii). It follows from Theorem 15.27 that we can assume without loss of generality that the given Markov representation is the Koopman representation associated with a continuous action \( K \times G \r... | No |
Theorem 15.31. Let \( \pi : \left( {K;\varphi }\right) \rightarrow \left( {L;\psi }\right) \) be an abstract group extension by the compact group \( G \subseteq \operatorname{Aut}\left( {K;\varphi }\right) \) . Let \( v \) be any \( \varphi \) -invariant probability measure on \( K \) and suppose that the Haar lift \( ... | Proof. Let us write \( \mu \mathrel{\text{:=}} {\pi }_{ * }v \) . Then, by hypothesis, \( \mu * {\mathrm{\;m}}_{G} = {P}^{\prime }\mu \) is an ergodic measure. For \( f \in \mathrm{C}\left( K\right) \) we have\n\n\[ \left\langle {f,\mu * {\mathrm{\;m}}_{G}}\right\rangle = {\int }_{L}{\int }_{G}{\kappa }_{g}f\mathrm{\;d... | Yes |
Lemma 16.1 (Minimal Idempotents). Let \( S \) be a semigroup, and let \( e \in S \) be an idempotent. Then the following assertions are equivalent:\n\n(i) eS (or Se) is a minimal right ideal (or left ideal).\n\n(ii) \( e \) is contained in some minimal right ideal (left ideal).\n\n(iii) eSe is a group (with neutral ele... | Proof. (i) \( \Rightarrow \) (ii) is trivial. For the proof of (ii) \( \Rightarrow \) (iii) let \( R \) be a minimal right ideal with \( e \in R \), and let \( G \mathrel{\text{:=}} {eSe} \) . Clearly, \( e \) is a neutral element in \( G \) . Let \( s \in S \) be arbitrary. Then ese \( R \subseteq R \) is also a right... | Yes |
Theorem 16.3 (Ellis). In a compact left-topological semigroup every right ideal contains a minimal right ideal. Every minimal right ideal is closed and contains at least one idempotent. In particular, every compact left-topological semigroup contains an idempotent. | Proof. Note that \( {sS} \) is a closed right ideal for any \( s \in S \) . If \( R \) is a right ideal and \( x \in R \), then \( {xS} \subseteq R \), and hence any right ideal contains a closed one. If \( R \) is minimal, then we must have \( R = {xS} \), and \( R \) itself is closed.\n\nNow if \( {J}_{0} \) is a giv... | Yes |
Lemma 16.4. In a compact left-topological semigroup the Sushkevich kernel satisfies\n\n\[ K\left( S\right) = \bigcup \{ R : R\text{ minimal right ideal }\} . \]\n\n(16.1)\n\nIn particular, \( K\left( S\right) \neq \varnothing \) . Moreover, an idempotent of \( S \) is minimal if and only if it is contained in \( K\left... | Proof. To prove \ | No |
Theorem 16.5. Let \( S \) be a compact semitopological semigroup. Then the following assertions are equivalent:\n\n(i) Every minimal right ideal is a minimal left ideal.\n\n(ii) There is a unique minimal right ideal and a unique minimal left ideal.\n\n(iii) There is a unique minimal idempotent.\n\n(iv) The Sushkevich k... | Proof. Clearly, (ii) implies (i) by Lemma 16.4.\n\n(i) \( \Rightarrow \) (iv): Let \( R \) be a minimal right ideal and let \( e \in R \) be an idempotent, which exists by Theorem 16.3. By hypothesis, \( R \) is also a left ideal, i.e., an ideal. Hence, \( K\left( S\right) \subseteq R \subseteq K\left( S\right) \) (by ... | Yes |
Proposition 16.8. Let \( Z, Y \) be metric spaces, \( X \) a Baire space, and let \( f : X \times Y \rightarrow \) \( Z \) be a separately continuous function. For every \( b \in Y \), the set\n\n\[ \n\{ x \in X : f\text{ is not continuous at }\left( {x, b}\right) \}\n\]\n\nis of first category in \( X \) . | Proof. Let \( b \in Y \) be fixed. For \( n, k \in \mathbb{N} \) define\n\n\[ \n{X}_{b, n, k} \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{y \in \mathrm{B}\left( {b,\frac{1}{k}}\right) }}\left\{ {x \in X : d\left( {f\left( {x, y}\right), f\left( {x, b}\right) }\right) \leq \frac{1}{n}}\right\} .\n\]\n\nBy the continu... | Yes |
Theorem 16.11. Let \( E \) be a Hilbert space, or, more generally, a reflexive Banach space. Then for each \( M \geq 0 \) the closed norm-ball \[ \{ T \in \mathcal{L}\left( E\right) : \parallel T\parallel \leq M\} \] is compact in the weak operator topology. In particular, the set \( \operatorname{Con}\left( E\right) \... | Proof. It is clear that it suffices to consider the case \( M = 1 \), i.e., the set \( \operatorname{Con}\left( E\right) \) . If \( E \) is a Hilbert space, this is simply Theorem D. 7 from Appendix D. The proof for the case that \( E \) is reflexive is completely analogous and is left as Exercise 7. | No |
Theorem 16.12. Let \( \mathcal{S} \subseteq \operatorname{Con}\left( H\right) \) be a weakly closed subsemigroup of contractions on a Hilbert space \( H \) . If \( \mathcal{S} \) is algebraically a group, then it is a compact topological group, and the strong and weak operator topologies coincide on \( \mathcal{S} \) . | Proof. Let \( P \) be the unit element of \( \mathcal{S} \) . Then \( {P}^{2} = P \) and \( \parallel P\parallel \leq 1 \), hence \( P \) is an orthogonal projection onto the closed subspace \( K \mathrel{\text{:=}} \operatorname{ran}\left( P\right) \) . Since \( S = {PS} \) for each \( S \in \mathcal{S} \), the space ... | Yes |
Lemma 16.13. Let \( \mathrm{X} \) be a measure space, and let \( E \mathrel{\text{:=}} {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) . Then the semigroup \[ \mathcal{S} \mathrel{\text{:=}} \left\{ {T \in \mathcal{L}\left( E\right) : \parallel T\parallel \leq 1\;\text{ and }\;\parallel {Tf}{\parallel }_{2} \leq \parallel ... | Proof. We can consider \( \mathcal{S} \) as a subset of \( \operatorname{Con}\left( {\mathrm{L}}^{2}\right) \) . As such, it is weakly closed since the \( {\mathrm{L}}^{1} \) -contractivity of an \( {\mathrm{L}}^{2} \) -contraction \( T \) is expressible in weak terms by \[ \left| {{\int }_{\mathrm{X}}\left( {Tf}\right... | Yes |
Corollary 16.14. Let \( \mathrm{X} \) be a finite measure space. Then the following sets are weakly compact subsemigroups of \( \mathcal{L}\left( {\mathrm{L}}^{1}\right) \) :\n\n1) the set of all Dunford-Schwartz operators,\n\n2) the set of all positive Dunford-Schwartz operators,\n\n3) the set \( \mathrm{M}\left( \mat... | Proof. Combine Exercises 8 and 9 and the interpolation Theorem 8.23, cf. also the proof of Theorem 13.8. | No |
Corollary 16.15. Let \( \mathcal{S} \subseteq \mathrm{M}\left( \mathrm{X}\right) \) be a weakly compact subsemigroup of Markov operators on some probability space \( \mathrm{X} \) . If \( \mathcal{S} \) is algebraically a group, then it is a compact topological group, and the weak and strong operator topologies coincid... | Let us look, in the situation of Corollary 16.15, at the proof of Theorem 16.12. The unit element \( Q \) of \( \mathcal{S} \) is a Markov projection, and we can choose a model for\n\n\( F \mathrel{\text{:=}} \operatorname{ran}\left( Q\right) \), i.e., a probability space \( \mathrm{Y} \) and a Markov embedding \( J \i... | Yes |
Lemma 16.16. Let \( E \) be a Banach space, let \( \mathcal{T} \subseteq \mathcal{L}\left( E\right) \), and let \( D \subseteq E \) be a subset such that \( \operatorname{lin}D \) is dense in \( E \) . Then the following assertions are equivalent:\n\n(i) \( \mathcal{T} \) is relatively weakly compact, i.e., relatively ... | Proof. (i) \( \Rightarrow \) (ii): For fixed \( x \in E \) the mapping \( {\mathcal{L}}_{\mathrm{w}}\left( E\right) \rightarrow \left( {E,\sigma \left( {E,{E}^{\prime }}\right) }\right), T \mapsto {Tx} \) , is continuous and hence carries relatively compact subsets of \( {\mathcal{L}}_{\mathrm{w}}\left( E\right) \) to ... | Yes |
Theorem 16.18. Let \( S \) be a compact semitopological semigroup. Then the semigroup of left rotations \( \left\{ {{L}_{a} : a \in S}\right\} \) is a weakly compact semigroup of operators on \( \mathrm{C}\left( S\right) \) . | Proof. By Lemma 16.16 we only have to check that for fixed \( f \in \mathbf{C}\left( S\right) \) the orbit \( M \mathrel{\text{:=}} \left\{ {{L}_{a}f : a \in S}\right\} \) is weakly compact in \( \mathrm{C}\left( S\right) \) . By Grothendieck’s Theorem G.5, this is equivalent to \( M \) being compact in \( {\mathrm{C}}... | Yes |
Theorem 16.19. Let \( E \) be a Banach space and \( \mathcal{T} \subseteq {\mathcal{L}}_{\mathrm{w}}\left( E\right) \) a semigroup of operators on \( E \). Then \( \mathcal{T} \) is JdLG-admissible in the following cases:\n\n1) \( \mathcal{T} \) is Abelian and relatively weakly compact.\n\n2) \( E \) is a Hilbert space... | Proof. The case 1) was already mentioned in Theorem 16.5. (Note that by Exercise 14.2 the closure \( \mathcal{S} \) of \( \mathcal{T} \) is Abelian, too.) The case 3) can be reduced to 2) via Corollary 16.14. To prove 2) suppose that \( P \) and \( Q \) are minimal idempotents in \( \mathcal{S} = {\operatorname{cl}}_{\... | Yes |
Theorem 16.20. If \( E \) and its dual space \( {E}^{\prime } \) both are strictly convex, then every relatively weakly compact semigroup of contractions on \( E \) is JdLG-admissible. | Sketch of proof. Recall that a Banach space \( E \) is strictly convex if \( \parallel f\parallel = \parallel g\parallel = 1 \) and \( f \neq g \) implies that \( \parallel f + g\parallel < 2 \) . If \( P \) is a contractive projection on a strictly convex space \( E \), then it has the property in Corollary D.22.a, se... | No |
Proposition 16.23. In the situation above, the minimal idempotent \( Q \) of \( \mathcal{S} \) satisfies\n\n\[ \n{TQ} = {QT}\;\text{ for all }T \in \mathcal{T}.\n\]\n\nIn particular \( {E}_{\mathrm{{rev}}} \) and \( {E}_{\mathrm{{aws}}} \) are invariant under \( \mathcal{S} \) . | Proof. This is an immediate consequence of (v) in Theorem 16.5. | No |
Theorem 16.24. Let \( E = {E}_{\mathrm{{rev}}} \oplus {E}_{\mathrm{{aws}}} \) be the JdLG-decomposition of a Banach space \( E \) with respect to a JdLG-admissible semigroup \( \mathcal{T} \subseteq \mathcal{L}\left( E\right) \) with weak closure \( \mathcal{S} \mathrel{\text{:=}} {\operatorname{cl}}_{\mathrm{w}}\left(... | Proof. a) Recall that \( Q \) acts as the identity on \( {E}_{\text{rev }} = \operatorname{ran}\left( Q\right) \). If \( u \in {E}_{\text{rev }} \), then \( \mathcal{S}u = \mathcal{S}{Qu} = \mathcal{G}u \). Hence if \( v \in \mathcal{S}u \), then there is \( R \in \mathcal{G} \) with \( v = {Ru} \). Since \( \mathcal{G... | Yes |
Example 16.25 (Contraction Semigroups on Hilbert Spaces). Let \( H \) be a Hilbert space and let \( \mathcal{T} \subseteq \mathcal{L}\left( H\right) \) be a semigroup of contractions. Then \( \mathcal{T} \) is JdLG-admissible by Theorem 16.19. The minimal idempotent \( Q \in \mathcal{S} \) is a contraction, hence self-... | \[ H = {H}_{\text{rev }} \oplus {H}_{\text{aws }} \] is an orthogonal decomposition. On \( {H}_{\text{rev }} \) the semigroup \( \mathcal{S} \mathrel{\text{:=}} {\operatorname{cl}}_{\mathrm{w}}\left( \mathcal{T}\right) \) restricts to a compact group of unitary operators. Because \( Q = {Q}^{ * } \), the projection \( ... | Yes |
Proposition 16.27. Let \( E = {E}_{\mathrm{{rev}}} \oplus {E}_{\mathrm{{aws}}} \) be the JdLG-decomposition of a Banach space \( E \) with respect to a JdLG-admissible semigroup \( \mathcal{T} \subseteq \mathcal{L}\left( E\right) \) with weak closure \( \mathcal{S} \mathrel{\text{:=}} {\operatorname{cl}}_{\mathrm{w}}\l... | Proof. Let \( Q \) be the minimal idempotent of \( \mathcal{S} \) . Then \( Q \) restricts to 0 on \( {E}_{\text{aws }} \), and hence a) and b) are evident.\n\nFor c), note first that for each \( u \in E \) the mapping \( T \mapsto {Tu} \) is continuous from \( {\mathcal{L}}_{\mathrm{w}}\left( E\right) \) to \( \left( ... | Yes |
Proposition 16.29. Let \( \mathcal{T} \) be a JdLG-admissible semigroup of bounded linear operators on a Banach space E. Then \[ {E}_{\text{aws }} \cap \{ x \in E : \mathcal{T}x\text{ relatively compact and }\mathop{\inf }\limits_{{T \in \mathcal{T}}}\parallel {Tx}\parallel > 0\} = \varnothing . \] In particular, if \(... | Proof. Let \( Q \) be the minimal idempotent of \( \mathcal{S} \mathrel{\text{:=}} {\operatorname{cl}}_{\mathrm{w}}\mathcal{T} \), i.e., the projection onto \( {E}_{\text{rev }} \) along \( {E}_{\text{aws }} \) . Let \( x \in {E}_{\text{aws }} \) such that \( \mathcal{T}x \) is relatively compact. Then the weak closure... | Yes |
Theorem 16.31 (Jacobs, de Leeuw, Glicksberg). Let \( E = {E}_{\text{rev }} \oplus {E}_{\text{aws }} \) be the JdLG-decomposition of a Banach space \( E \) with respect to a JdLG-admissible semigroup \( \mathcal{T} \subseteq \mathcal{L}\left( E\right) \). Then \[ {E}_{\mathrm{{rev}}} = \overline{\operatorname{lin}}\left... | Proof. We use the previous terminology, i.e., \( \mathcal{S} = {\operatorname{cl}}_{\mathrm{w}}\left( \mathcal{T}\right) \) is the weak operator closure of \( \mathcal{T}, Q \) the minimal idempotent, \( {E}_{\text{rev }} = \operatorname{ran}\left( Q\right) \) and \( \mathcal{G} \mathrel{\text{:=}} \mathcal{S}Q = Q\mat... | Yes |
Theorem 16.33. Let \( T \in \mathcal{L}\left( E\right) \) generate a relatively weakly compact semigroup \( {\mathcal{T}}_{T} \) with associated JdLG-decomposition \( E = {E}_{\mathrm{{rev}}} \oplus {E}_{\mathrm{{aws}}} \) . Then the following assertions hold:\n\na) \( {E}_{\text{rev }} \) is spanned by the eigenvector... | Proof. a) The first assertion follows from Corollary 16.32 since if \( {Tu} = {\lambda u} \), then \( {T}^{n}u = {\lambda }^{n}u \) for all \( n \in {\mathbb{N}}_{0} \) . The second follows from Corollary 16.30. | Yes |
Theorem 16.34. Let \( T \) generate a relatively weakly compact semigroup \( {\mathcal{T}}_{T} \) on a Banach space \( E \), and let \( M \subseteq {E}^{\prime } \) be norm-dense in the dual space \( {E}^{\prime } \) . Then for \( x \in E \) the following assertions are equivalent:\n\n(i) \( \mathop{\operatorname{D-lim... | Proof. We note that \( T \) is necessarily power-bounded. For such operators the pairwise equivalence of (i)-(v) has already been shown in Theorem 9.15 and Proposition 9.17. Note also that the implications (viii) \( \Rightarrow \) (vii) \( \Rightarrow \) (vi) are trivial.\n\nFor the proof of the remaining assertions we... | Yes |
Theorem 16.36. For an operator \( T \in \mathcal{L}\left( E\right) \) on a Banach space \( E \) the following assertions are equivalent:\n\n(i) The operator \( T \) has discrete spectrum.\n\n(ii) The operator \( T \) generates a relatively weakly compact operator semigroup on \( E \), and \( E = {E}_{\mathrm{{rev}}}\le... | Proof. (i) \( \Rightarrow \) (ii): It is clear that the orbit \( \left\{ {{T}^{n}u : n \geq 0}\right\} \) is relatively strongly (hence weakly) compact whenever \( {Tu} = {\lambda u},\lambda \in \mathbb{T} \) . Therefore, by Lemma 16.16, if \( T \) has discrete spectrum, then \( T \) generates a weakly compact operator... | Yes |
Example 17.1 (Fixed Factor). The Kronecker factor associated with the convex semigroup \( \operatorname{conv}\left( \mathcal{T}\right) \) coincides with the fixed factor of \( \mathcal{T} \), i.e., | \[ \operatorname{Kro}\left( {\mathrm{X};\operatorname{conv}\left( \mathcal{T}\right) }\right) = \operatorname{fix}\left( \mathcal{T}\right) = \mathop{\bigcap }\limits_{{T \in \mathcal{T}}}\operatorname{fix}\left( T\right) \] see Example 13.27 and Exercise 16.17. The corresponding Markov projection \( {P}^{2} = P \in \m... | No |
Example 17.2 (Markov Embeddings). If \( \mathcal{T} \) is a semigroup of Markov embeddings on a probability space \( \mathrm{X} \), then\n\n\[ \operatorname{Kro}\left( {\mathrm{X};\mathcal{T}}\right) = \left\{ {f \in {\mathrm{L}}^{1}\left( \mathrm{X}\right) : \mathcal{T}f\text{ is relatively compact }}\right\} .\n\] | This follows from Corollary 16.30. | No |
Lemma 17.3. Let \( \chi : \mathcal{T} \rightarrow \mathrm{U}\left( n\right) \) be fixed, and let \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \in {E}^{n} \) be an associated eigensystem. Then\n\n\[{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {e}_{j}\right| }^{2}\right) }^{1/2} \in \operatorname{fix}\left( \mathcal{... | Proof. For a given eigensystem \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) set \( e \mathrel{\text{:=}} {\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {e}_{j}\right| }^{2}\right) }^{1/2} \) . Let \( f \in \) fix \( \left( \mathcal{T}\right) \cap {\mathrm{L}}^{\infty } \) and \( T \in \mathcal{T} \) . Then \( f{e}... | Yes |
Proposition 17.7. In the situation from above, \( \operatorname{Kro}\left( {\mathrm{X};{\mathcal{T}}_{p}}\right) = \operatorname{Kro}\left( {\mathrm{X};\mathcal{T}}\right) \cap {\mathrm{L}}^{p} \) . | Since \( \operatorname{Kro}\left( {\mathrm{X};\mathcal{T}}\right) \) is a factor, \( \operatorname{Kro}\left( {\mathrm{X};\mathcal{T}}\right) \cap {\mathrm{L}}^{p} \) is dense in \( \operatorname{Kro}\left( {\mathrm{X};\mathcal{T}}\right) \), and hence one can switch freely between the \( {\mathrm{L}}^{1} \) -case and ... | No |
Corollary 17.8. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system with Koopman operator \( {T}_{\varphi } \), and let \( 1 \leq p < \infty \) . Then the following assertions are equivalent:\n\n(i) The system \( \left( {\mathrm{X};\varphi }\right) \) has discrete spectrum.\n\n(ii) The Koopman ... | Proof. We write \( {T}_{p} \) for the restriction of \( {T}_{\varphi } \) to \( {\mathrm{L}}^{p} \) . The equivalences (i) \( \Leftrightarrow \) (ii) for \( p = 1 \) and (i) \( \Leftrightarrow \) (iii) are clear. By Proposition 17.7,(iii) is equivalent to \( \operatorname{Kro}\left( {\mathrm{X};{T}_{p}}\right) = {\math... | Yes |
Let \( G \) be a compact Abelian group and let \( {L}_{a} \) be the Koopman operator induced by the rotation by \( a \in G \). Since every character \( \chi \in {G}^{ * } \) is an eigenfunction of \( {L}_{a} \) corresponding to the eigenvalue \( \chi \left( a\right) \in \mathbb{T} \) and since \( \operatorname{lin}{G}^... | A fortiori, \( {L}_{a} \) has discrete spectrum also on \( {\mathrm{L}}^{p}\left( G\right) \) for every \( 1 \leq p < \infty \). | Yes |
Corollary 17.14. Two ergodic measure-preserving systems with discrete spectrum are (Markov) isomorphic if and only if they are spectrally isomorphic. | Proof. By Corollary 12.12 and by the remark following it, Markov isomorphic systems are spectrally isomorphic.\n\nConversely, if two ergodic measure-preserving systems are spectrally isomorphic, their Koopman operators have the same point spectrum. Under the assumption that both systems have discrete spectrum, they mus... | Yes |
Proposition 17.18. A measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is weakly mixing if and only if it is disjoint from every system with discrete spectrum. | Proof. If \( \left( {\mathrm{X};\varphi }\right) \) is not weakly mixing, then its Kronecker factor is not trivial, and hence the associated Markov embedding is not trivial.\n\nConversely, suppose that \( \left( {\mathrm{Y};\psi }\right) \) has discrete spectrum, \( \left( {\mathrm{X};\varphi }\right) \) is weakly mixi... | Yes |
Theorem 17.19. Let \( \left( {\mathrm{X};\varphi }\right) \) by a measure-preserving system with associated Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) on \( E \mathrel{\text{:=}} {\mathrm{L}}^{p}\left( \mathrm{X}\right), p \in \lbrack 1,\infty ) \) . Then the following assertions are equivalent:\n\n(i)... | Proof. Note that with \( h \mathrel{\text{:=}} f - \langle f,\mathbf{1}\rangle \cdot \mathbf{1} \), assertions (ii)-(iv) can be rewritten equivalently as:\n\n(ii) \( 0 \in {\operatorname{cl}}_{\sigma }\left\{ {{T}^{n}h : n \in {\mathbb{N}}_{0}}\right\} \) .\n\n(iii) \( \mathop{\lim }\limits_{{n \in J}}{T}^{n}h = 0 \) w... | Yes |
Lemma 17.20. Let \( H \) be a Hilbert space, \( \mathcal{T} \subseteq \mathcal{L}\left( H\right) \) a semigroup of operators on \( H \) and \( f \in H \) such that for each \( T \in \mathcal{T} \) there is \( {\lambda }_{T} \in \mathbb{T} \) with \( {Tf} = {\lambda }_{T}f \) . Suppose furthermore that \( R \in \mathcal... | Proof. One has, for every \( T \in \mathcal{T} \) and \( S \in \mathcal{L}\left( H\right) \), \n\n\[ \left( {{T}^{ * }{STf} \mid f}\right) = \left( {{STf} \mid {Tf}}\right) = {\lambda }_{T}\overline{{\lambda }_{T}}\left( {{Sf} \mid f}\right) = \left( {{Sf} \mid f}\right) . \n\]\n\nIt follows from the hypothesis that \(... | Yes |
Theorem 17.22. For \( a, b \in \mathbb{R} \) the following assertions are equivalent:\n\n(i) The Heisenberg system \( \left( {\mathbb{H},\mathrm{m};\left\lbrack {a, b, c}\right\rbrack }\right) \) is ergodic.\n\n(ii) The topological Heisenberg system \( \left( {\mathbb{H};\left\lbrack {a, b, c}\right\rbrack }\right) \) ... | Proof. The equivalence of (i) and (iii) follows from the considerations above in combination with Kronecker's theorem (Theorem 14.18) and the fact that factors of ergodic systems are ergodic. The implication (ii) \( \Rightarrow \) (i) is clear, since in this case \( \mathrm{m} \) is the unique and hence necessarily erg... | No |
Lemma 18.1. In the situation above, \( {\mathrm{{Sp}}}_{A}\left( T\right) = \sigma \left( T\right) \) and the evaluation map\n\n\[ \Gamma \left( A\right) \rightarrow \sigma \left( T\right) ,\;\chi \mapsto \chi \left( T\right) \]\n\nis a homeomorphism. | Proof. Let us abbreviate \( {\rho }_{A}\left( T\right) \mathrel{\text{:=}} \left\{ {\lambda \in \rho \left( T\right) : {\left( \lambda \mathrm{I} - T\right) }^{-1} \in A}\right\} \) . This set is closed in \( \rho \left( T\right) \) . Indeed, let \( {\left( {\lambda }_{n}\right) }_{n \in \mathbb{N}} \) be a sequence in... | Yes |
Theorem 18.2. Let \( T \) be a normal operator on a Hilbert space \( H \), let \( K = \sigma \left( T\right) \) , and let \( {\left( {\mu }_{x, y}\right) }_{x, y \in H} \) be the associated family of measures given by (18.2). Then the following assertions hold:\n\na) The mapping \( H \times H \rightarrow \mathrm{M}\lef... | Proof. a) The sesquilinearity and the symmetry is straightforward from Eq. (18.2), see Exercise 1. For the norm inequality let \( x, y \in H \) . Then, again by (18.2),\n\n\[ \n\left| \left\langle {f,{\mu }_{x, y}}\right\rangle \right| = \left| \left( {f\left( T\right) x \mid y}\right) \right| \leq \parallel f\left( T\... | No |
Corollary 18.3. If \( T \) is a normal operator on a Hilbert space, then \( r\left( T\right) = \parallel T\parallel \) . | Proof. The inequality \( r\left( T\right) \leq \parallel T\parallel \) is a general fact from spectral theory (Appendix C.9). For the converse, in Theorem 18.2.c we put \( f = \mathrm{{id}} \) and obtain\n\n\[ \parallel {Tx}{\parallel }^{2} = {\int }_{K}{\left| z\right| }^{2}\mathrm{\;d}{\mu }_{x}\left( z\right) \leq \... | Yes |
Theorem 18.4 (Spectral Theorem, Multiplier Form). For a bounded, normal operator \( T \) on a Hilbert space \( H \), the pair \( \left( {H, T}\right) \) is unitarily equivalent to \( \left( {{\mathrm{L}}^{2}\left( {\Omega ,\mu }\right), M}\right) \), where \( \mu \) is a positive Baire measure on a locally compact spac... | Proof. We employ the terminology introduced above. For each \( \alpha \) let \( {K}_{\alpha } \mathrel{\text{:=}} K \times \{ \alpha \} \) be a copy of \( K = \sigma \left( T\right) \), so that the sets \( {K}_{\alpha } \) are pairwise disjoint. Let \( \Omega \mathrel{\text{:=}} \mathop{\bigcup }\limits_{\alpha }{K}_{\... | Yes |
Theorem 18.5 (Borel Functional Calculus). In the situation from above, the mapping \( \Psi : \mathrm{{BM}}\left( K\right) \rightarrow \mathcal{L}\left( H\right) \) has the following properties:\n\na) \( \Psi \left( z\right) = T \) .\n\nb) \( \Psi \) is a (contractive) homomorphism of \( {C}^{ * } \) -algebras.\n\nc) \(... | Proof. a) is clear, and c) follows from the identity \( \parallel \Psi \left( f\right) x\parallel = \parallel f\left( T\right) x\parallel = \) \( \parallel f{\parallel }_{{\mathrm{L}}^{2}\left( {\mu }_{x}\right) } \) and from the fact that, by the dominated convergence theorem, bp-convergence implies convergence in \( ... | Yes |
Theorem 18.6 (Bochner-Herglotz). For a scalar sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{Z}} \) the following assertions are equivalent:\n\n(i) There is a Hilbert space \( H \), a unitary operator \( T \) on \( H \) and a vector \( x \in H \) such that\n\n\[ \n{a}_{n} = \left( {x \mid {T}^{n}x}\right) \;\text{... | Proof. (i) \( \Rightarrow \) (ii): This is simply the spectral theorem.\n\n(ii) \( \Rightarrow \) (iii): A short computation shows that\n\n\[ \n\mathop{\sum }\limits_{{n, j}}{a}_{n - j}{\lambda }_{n}\overline{{\lambda }_{j}} = {\int }_{\mathbb{T}}\mathop{\sum }\limits_{{n, j}}{\lambda }_{n}\overline{{\lambda }_{j}}{z}^... | Yes |
Lemma 18.8. If \( T \) is a linear isometry on a Hilbert space \( H \), then for each \( x \in H \) the sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{Z}} \), given by\n\n\[ \n{a}_{n} \mathrel{\text{:=}} \left\{ \begin{array}{ll} \left( {x \mid {T}^{n}x}\right) & \text{ for }n \geq 0, \\ \left( {{T}^{-n}x \mid x}\... | Proof. Fix \( x \in H \) and \( \lambda = {\left( {\lambda }_{j}\right) }_{j \in \mathbb{Z}} \in {\mathrm{c}}_{00}\left( \mathbb{Z}\right) \), and let \( N \geq 0 \) be so large that \( {\lambda }_{j} = 0 \) for all \( \left| j\right| \geq N \) . Then\n\n\[ \n\mathop{\sum }\limits_{{n, j}}{a}_{n - j}{\lambda }_{n}\over... | Yes |
a) One has \( {\mu }_{f\left( T\right) x} = {\left| f\right| }^{2}{\mu }_{x} \) for all \( x \in H \) and \( f \in {\mathrm{L}}^{2}\left( {\mu }_{x}\right) \) . In particular, if \( x \in H \) and \( y \in Z\left( x\right) \), then \( {\mu }_{y} \ll {\mu }_{x} \) . Conversely, if \( x \in H \) and \( \mu \in {\mathrm{M... | a) By Theorem 18.2.b the identity\n\n\[ {\int }_{K}h\mathrm{\;d}{\mu }_{f\left( T\right) x} = {\int }_{K}h{\left| f\right| }^{2}\mathrm{\;d}{\mu }_{x} \]\n\nis true for \( f, h \in \mathrm{C}\left( K\right) \), and by approximation it continues to hold for \( f \in {\mathrm{L}}^{2}\left( {\mu }_{x}\right) \) . This est... | Yes |
Theorem 18.10. For \( I, J \subseteq \mathrm{M}\left( K\right) \) closed ideals the following assertions hold:\n\na) \( H\left( I\right) \) is a closed \( T \) -bi-invariant subspace of \( H \) .\n\nb) If \( I \cap J = \{ 0\} \), then \( I \oplus J \) is a closed ideal and\n\n\[ H\left( {I \oplus J}\right) = H\left( I\... | Proof. a) This follows from (18.6). Note that \( {\mu }_{{Tx}, y} = {\mu }_{x,{T}^{ * }y} \) and \( {\mu }_{{T}^{ * }x, y} = {\mu }_{x,{Ty}} \) for all \( x, y \in H \) .\n\nb) By (18.5) and Lemma B.20, \( \parallel \mu + v\parallel = \left| {\mu + v}\right| \left( K\right) = \left| \mu \right| \left( K\right) + \left|... | No |
Theorem 18.12. Let \( T \in \mathcal{L}\left( H\right) \) be a normal operator on a separable Hilbert space \( H \) with associated family of measures \( {\left( {\mu }_{x, y}\right) }_{x, y \in H} \) on \( K \mathrel{\text{:=}} \sigma \left( T\right) \) . Then there is, up to equivalence, a unique positive measure \( ... | Proof. Uniqueness: Suppose that \( {\mu }_{\max } \) and \( {\mu }_{\max }^{\prime } \) satisfy 1) and 2). By 2), there is \( x \in H \) with \( {\mu }_{x} = {\mu }_{\max } \) and 1) yields that \( {\mu }_{\max } \ll {\mu }_{\max }^{\prime } \) . By symmetry \( {\mu }_{\max } \sim {\mu }_{\max }^{\prime } \) as claimed... | Yes |
Proposition 18.13. Let \( H \) be a separable Hilbert space, and let \( T \) be a normal operator on \( H \) with maximal spectral type \( \left\lbrack {\mu }_{\max }\right\rbrack \) . Then\n\n\[ \n\operatorname{supp}\left( {\mu }_{\max }\right) = \sigma \left( T\right) \n\] | Proof. The inclusion \( \operatorname{supp}\left( {\mu }_{x}\right) \subseteq \sigma \left( T\right) \) is clear. Take \( x \in H \) with \( {\mu }_{x} = {\mu }_{\max } \) and \( \lambda \in \mathbb{C} \smallsetminus \operatorname{supp}\left( {\mu }_{x}\right) \) . Then \( f \mathrel{\text{:=}} {\left( \lambda \mathbf{... | Yes |
Proposition 18.15. In the situation just described, let \( x \in H \) and \( \lambda \in \mathbb{C} \) . a) The following assertions are equivalent: (i) \( {Tx} = {\lambda x} \) . (ii) \( f\left( T\right) x = f\left( \lambda \right) x \) for all \( f \in \mathrm{{BM}}\left( K\right) \) . (iii) \( {\mu }_{x, y} = \left(... | Proof. a) Suppose that \( {Tx} = {\lambda x} \) . Then \( {T}^{n}x = {\lambda }^{n}x \) and hence, by normality, \( {T}^{*n}x = \) \( {\bar{\lambda }}^{n}x \) for each \( n \in \mathbb{N} \) (Lemma D.25). It follows by approximation that \( f\left( T\right) x = f\left( \lambda \right) x \) for all \( f \in \mathrm{C}\l... | Yes |
Proposition 18.17. For a normal operator \( T \in \mathcal{L}\left( H\right) \) we have\n\n\[ \n{H}_{\mathrm{d}} = \overline{\operatorname{lin}}\{ x \in H : x\text{ is an eigenvector of }T\} .\n\] | Proof. If \( x \in H \) is an eigenvector of \( T \) with eigenvalue \( \lambda \in \mathbb{C} \), then \( {\mu }_{x} = \parallel x{\parallel }^{2}{\delta }_{\lambda } \) , and hence \( x \in {H}_{\mathrm{d}} \) . Conversely, suppose that \( x \in {H}_{\mathrm{d}} \), i.e., \( {\mu }_{x} \) is discrete. Then \( {\mu }_... | Yes |
For a unitary operator \( T \) on a Hilbert space \( H \), the JdLG-decomposition and the discrete-continuous decomposition coincide, i.e.\n\n\[ \n{H}_{\mathrm{{rev}}} = {H}_{\mathrm{d}}\;\text{ and }\;{H}_{\mathrm{{aws}}} = {H}_{\mathrm{c}}. \n\] | Proof. Since the JdLG-decomposition is orthogonal (see Example 16.25), the assertion follows from Proposition 18.17 and the description of the subspace \( {H}_{\text{rev }} \) in Theorem 16.33. | No |
Proposition 18.20 (Wiener’s Lemma). Let \( v \in \mathrm{M}\left( \mathbb{T}\right) \) be a complex measure. Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{\left| \widehat{v}\left( n\right) \right| }^{2} = \mathop{\sum }\limits_{{\lambda \in \mathbb{T}}}{\left| ... | Proof. Note that the second equality follows directly from the first one. Abbreviate \( \mu \mathrel{\text{:=}} \left| v\right| \) and let \( T \) be, as above, the operator of multiplication by \( z \) on \( H = {\mathrm{L}}^{2}\left( \mu \right) \) . Then, by Corollary B.23, \( v = {h\mu } \) for some \( h \in \opera... | Yes |
Corollary 18.22. A unitary operator \( T \) on a Hilbert space \( H \) has discrete spectrum if and only if each \( {\mu }_{x}, x \in H \), is a discrete measure. In particular, if \( H \) is separable, then \( T \) has discrete spectrum if and only if \( {\mu }_{\max } \) is a discrete measure. | Proof. Since all eigenvalues of a unitary operator are unimodular, \( T \) has discrete spectrum if and only if \( H = {H}_{\mathrm{d}} \) (cf. Proposition 18.17). | No |
Corollary 18.23. An invertible standard system \( \left( {\mathrm{X};\varphi }\right) \) is ergodic if and only if its maximal spectral type \( \left\lbrack {\mu }_{0}\right\rbrack \) satisfies \( {\mu }_{0}\{ 1\} = 0 \) . | Proof. By Proposition 18.15, \( {\mu }_{0}\{ 1\} = 0 \) if and only if \( \dim \operatorname{fix}\left( {T}_{0}\right) = 0 \), which holds if and only if \( \dim \operatorname{fix}\left( T\right) = 1 \), equivalent to ergodicity by Proposition 7.15. | Yes |
Proposition 18.24. Let \( T \) be a unitary operator on a separable Hilbert space \( H \) with maximal spectral type \( \left\lbrack {\mu }_{\max }\right\rbrack \) . Then \( T \) is almost weakly stable if and only if \( {\mu }_{\max } \) is a continuous measure. | Proof. This follows from Corollary 18.18. Alternatively, avoiding the JdLG-theory, one can argue as follows:\n\nBy definition, \( {\widehat{\mu }}_{x, y}\left( {-n}\right) = \left( {{T}^{n}x \mid y}\right) \) for all \( x, y \in H \) and \( n \in \mathbb{N} \) . Since \( {\mu }_{\max } \) is continuous if and only if \... | Yes |
Lemma 18.26. The set of Rajchman measures\n\n\\[ \n{\\mathrm{M}}_{\\mathrm{R}}\\left( \\mathbb{T}\\right) \\mathrel{\\text{:=}} \\left\\{ {\\mu \\in \\mathrm{M}\\left( \\mathbb{T}\\right) : \\mu \\;\\text{is a Rajchman measure}}\\right\\} \n\\]\n\nis a closed ideal of \\( \\mathrm{M}\\left( \\mathbb{T}\\right) \\) . | Proof. It is straightforward to prove that \\( {\\mathrm{M}}_{\\mathrm{R}}\\left( \\mathbb{T}\\right) \\) is a closed subspace of \\( \\mathrm{M}\\left( \\mathbb{T}\\right) \\) . Fix \\( \\mu \\in {\\mathrm{M}}_{\\mathrm{R}}\\left( \\mathbb{T}\\right) \\) and \\( h \\in \\mathrm{{BM}}\\left( \\mathbb{T}\\right) \\) wit... | Yes |
Proposition 18.27. Let \( T \) be a unitary operator on a separable Hilbert space \( H \) with maximal spectral type \( \left\lbrack {\mu }_{\max }\right\rbrack \) . Then \( {T}^{n} \rightarrow 0 \) in the weak operator topology if and only if \( {\mu }_{\max } \) is a Rajchman measure. | Proof. Since, by Theorems 18.2.d and 18.12, \( {\mu }_{x, y} \ll {\mu }_{\max } \) for every \( x, y \in H \), the assertion follows from Lemma 18.26. | No |
Proposition 18.29. An invertible standard system with Lebesgue spectrum is strongly mixing. | The proof is left as Exercise 19. | No |
Proposition 18.32. For a unitary operator \( T \) on a separable Hilbert space \( H \) the following assertions are equivalent:\n\n(i) T has countable Lebesgue spectrum.\n\n(ii) There is an orthonormal system \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) in \( H \) such that for each \( n \in \mathbb{N} \) the sequ... | Proof. (i) \( \Leftrightarrow \) (ii): For a given \( x \in H \), the sequence \( {\left( {T}^{k}x\right) }_{k \in \mathbb{Z}} \) is an orthonormal basis of \( Z\left( x\right) \) if and only if\n\n\[ {\int }_{\mathbb{T}}{z}^{k - j}\mathrm{\;d}{\mu }_{x}\left( z\right) = \left( {{T}^{k}x \mid {T}^{j}x}\right) = {\delta... | Yes |
Corollary 18.33. A nontrivial two-sided Bernoulli shift \( B\left( {{p}_{0},\ldots ,{p}_{k - 1}}\right) \) has countable Lebesgue spectrum. As a consequence, all such Bernoulli shifts are spectrally isomorphic. | Proof. Consider a Bernoulli shift \( B\left( {{p}_{0},\ldots ,{p}_{k - 1}}\right) = \left( {\mathrm{X};\tau }\right) \), i.e., \( \mathrm{X} = \left( {{\mathcal{W}}_{k},\sum ,\mu }\right) \) with the corresponding Bernoulli measure \( \mu \) on the product \( \sigma \) -algebra \( \sum \) and \( \tau \) the two-sided s... | Yes |
Proposition 18.37 (Halmos, Rokhlin). The measure-preserving system \( \left( {G,\mathrm{\;m};\varphi }\right) \) is ergodic if and only if the group automorphism \( T \mathrel{\text{:=}} {T}_{\varphi } : {G}^{ * } \rightarrow {G}^{ * } \) has no periodic points other than the trivial character \( \mathbf{1} \) . As a c... | Proof. Suppose that the system \( \left( {G,\mathrm{\;m};\varphi }\right) \) is ergodic. Let \( \chi \in {G}^{ * } \) be such that \( {T}^{n}\chi = \) \( \chi \) for some \( n \in \mathbb{N} \) being minimal with this property. Then we have \( \left( {{T}^{j}\chi \mid \chi }\right) = 0 \) for \( j = 1,\ldots, n - 1 \) ... | Yes |
Theorem 18.39. Let \( G \) be a nontrivial compact metrizable Abelian group and let \( \varphi : G \rightarrow G \) be a continuous automorphism of \( G \) . If \( \left( {G,\mathrm{\;m};\varphi }\right) \) is ergodic, then it has countable Lebesgue spectrum. In particular, it is strongly mixing. | Proof. Let \( \chi \in {G}^{ * } \) be a nontrivial character. Then \( {T}^{n}\chi \neq \chi \) by Proposition 18.37 and hence \( \left( {{T}^{n}\chi \mid \chi }\right) = 0 \) for every \( n \in \mathbb{N} \) . This yields that \( {\left( {T}^{n}\chi \right) }_{n \in \mathbb{Z}} \) is an orthonormal basis in the cyclic... | Yes |
Proposition 18.41. a) The Koopman operator \( T \mathrel{\text{:=}} {T}_{{\psi }_{a, m}}\left( \right. \) on \( \left. {{\mathrm{L}}^{2}\left( {\mathbb{T}}^{2}\right) }\right) \) of the system \( \left( {{\mathbb{T}}^{2},{\mathrm{\;m}}_{{\mathbb{T}}^{2}};{\psi }_{a, m}}\right) \) has countable Lebesgue spectrum when re... | The proof is left as Exercise 25. Combining this with the next result yields nonisomorphic but spectrally isomorphic systems. | No |
Theorem 18.42 (Anzai). Let \( a \in \mathbb{T} \) be not a root of unity, and let \( k, m \in \mathbb{N} \) . a) The point isomorphisms between the systems \( \left( {{\mathbb{T}}^{2},{\mathrm{\;m}}_{{\mathbb{T}}^{2}};{\psi }_{a, m}}\right) \) and \( \left( {{\mathbb{T}}^{2},{\mathrm{m}}_{{\mathbb{T}}^{2}};{\psi }_{a, ... | Proof. a) If \( n, b,\xi \) satisfy (18.10), then \( \theta \left( {x, y}\right) \mathrel{\text{:=}} \left( {{bx},\xi \left( x\right) {y}^{n}}\right) \) defines an isomorphism between the two systems (see Exercise 27). For the converse, denote by \( {L}_{a} \) and \( {T}_{a, m} \) the Koopman operators of the left rota... | Yes |
Proposition 19.2. The mapping \[ \iota : S \rightarrow {\beta S},\;s \mapsto {\delta }_{s} \in {\beta S} \] is a homeomorphism onto its range, when \( S \) is endowed with the discrete topology. Furthermore, the range \( \iota \left( S\right) \) is dense. | Proof. The mapping \( \iota \) is trivially injective, and it is continuous since \( \mathrm{S} \) is discrete. Let \( {s}_{0} \in \mathrm{S} \) and \( \varepsilon > 0 \) be given. For \( s \in \mathrm{S} \) define \( x\left( s\right) \mathrel{\text{:=}} \varepsilon \) if \( s \neq {s}_{0} \) and set \( x\left( {s}_{0}... | Yes |
Proposition 19.3. Let \( x : S \rightarrow \mathbb{C} \) be a bounded (continuous) function. Then there is a unique continuous extension \( \widetilde{x} : {\beta S} \rightarrow \mathbb{C} \) . | Proof. For \( p \in {\beta S} \) define\n\n\[ \widetilde{x}\left( p\right) \mathrel{\text{:=}} p\left( x\right) . \]\n\nContinuity of \( \widetilde{x} \) follows since \( {\beta S} \) has the restriction of the weak* topology \( \sigma \left( {{\ell }^{\infty }{\left( S\right) }^{\prime },{\ell }^{\infty }\left( S\righ... | No |
Proposition 19.4. Let \( K \) be a compact space, and let \( x : S \rightarrow K \) be a (continuous) function. Then there is a unique continuous extension \( \widetilde{x} : {\beta S} \rightarrow K \), i.e., a unique continuous function \( \widetilde{x} \) such that the diagram below commutes. | Proof. Let \( f \in \mathrm{C}\left( K\right) \) be arbitrary. By Proposition 19.3 we can uniquely extend \( f \circ x \) to a continuous function \( \left( {f \circ x}\right) : {\beta S} \rightarrow \mathbb{C} \) . The mapping\n\n\[ \mathrm{C}\left( K\right) \rightarrow \mathrm{C}\left( {\beta S}\right) ,\;f \mapsto \... | Yes |
Proposition 19.5. Let \( K, L \) be compact spaces, let \( x : S \rightarrow K \) be a function, and let \( f : K \rightarrow L \) be continuous. Then for every \( p \in {\beta S} \) we have\n\n\[ f\left( {\mathop{\lim }\limits_{{s \rightarrow p}}x\left( s\right) }\right) = \mathop{\lim }\limits_{{s \rightarrow p}}f\le... | Proof. Let \( \widetilde{x} : {\beta S} \rightarrow \mathbb{C} \) be the unique extension of \( x \) yielded by Proposition 19.4. For \( s \in S \) we have \( f \circ \widetilde{x}\left( s\right) = \left( {f \circ x}\right) \widetilde{}\left( s\right) \) . By continuity of \( f \circ \widetilde{x} \) and by uniqueness ... | Yes |
Lemma 19.6. a) Let \( p \in {\beta S} \) and \( A \subseteq S \) . Then \( p\left( {\mathbf{1}}_{A}\right) \in \{ 0,1\} \) . | Proof. a) We have \( {\mathbf{1}}_{A} = {\mathbf{1}}_{A}{\mathbf{1}}_{A} \), so by multiplicativity of \( p \) we obtain\n\n\[ p{\left( {\mathbf{1}}_{A}\right) }^{2} = p\left( {\mathbf{1}}_{A}\right) p\left( {\mathbf{1}}_{A}\right) = p\left( {{\mathbf{1}}_{A}{\mathbf{1}}_{A}}\right) = p\left( {\mathbf{1}}_{A}\right) ,\... | Yes |
Proposition 19.7. a) If \( x \in {\ell }^{\infty }\left( \mathbb{N}\right) \) is a convergent sequence, then for every \( p \in \beta \mathbb{N} \smallsetminus \mathbb{N} \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = p\left( x\right) \] | Proof. a) Denote by \( \alpha \) the limit of \( x = {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) and let \( \varepsilon > 0 \) be fixed. Take \( {n}_{0} \in \mathbb{N} \) such that\n\n\[ \left| {{x}_{n} - \alpha }\right| \leq \varepsilon \;\text{ for all }n \in \mathbb{N}\text{ with }n > {n}_{0}, \]\n\nand let \( A ... | Yes |
Proposition 19.8. a) For \( s, t \in S \) we have\n\n\[{\delta }_{s} * {\delta }_{t} = {\delta }_{st}\] | Proof: a) Let \( x \in {\ell }^{\infty }\left( S\right) \) and let \( s, t \in S \) . Then\n\n\[{\delta }_{s} * {\delta }_{t}\left( x\right) = {\delta }_{s}\left( {r \mapsto {\delta }_{t}\left( {{L}_{r}x}\right) }\right) = {\delta }_{s}\left( {r \mapsto x\left( {rt}\right) }\right) = x\left( {st}\right) = {\delta }_{st... | Yes |
Proposition 19.9. a) For each \( p \in {\beta S} \) the right multiplication by \( p \)\n\n\[ q \mapsto q * p \]\n\nis continuous on \( {\beta S} \) . | a) Let \( \varepsilon > 0 \), let \( x \in {\ell }^{\infty }\left( S\right) \), and let \( q \in {\beta S} \) . For \( {q}^{\prime } \in {\beta S} \) we have\n\n\[ \left| {q * p\left( x\right) - {q}^{\prime } * p\left( x\right) }\right| = \left| {\left( {q - {q}^{\prime }}\right) \left( {r \mapsto p\left( {{L}_{r}x}\ri... | No |
Proposition 19.10. Let \( K \) be a compact space, let \( x : S \rightarrow K \) and let \( p, q \in {\beta S} \) . Then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow p * q}}x\left( r\right) = \mathop{\lim }\limits_{{s \rightarrow p}}\left( {\mathop{\lim }\limits_{{t \rightarrow q}}x\left( {st}\right) }\right) \] | Proof. The statement for scalar-valued sequences is just the definition of the convolution. For the general case one can employ the standard argument using Urysohn's lemma, already familiar from the proof of Proposition 19.7. | No |
Let \( \left( {K;\varphi }\right) \) be a topological system. For \( x \in K \) and \( p \in \beta \mathbb{N} \) we define\n\n\[ \n{\varphi }^{p}\left( x\right) \mathrel{\text{:=}} \mathop{\lim }\limits_{{n \rightarrow p}}{\varphi }^{n}\left( x\right) \n\]\n\nThen\n\n\[ \n\mathcal{E} \mathrel{\text{:=}} \left\{ {{\varp... | For \( p, q \in \beta \mathbb{N} \) we have by Propositions 19.10 and 19.5 that\n\n\[ \n{\varphi }^{p + q}\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow p}}\mathop{\lim }\limits_{{m \rightarrow q}}{\varphi }^{n + m}\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow p}}{\varphi }^{n}\left( {\mathop{\lim }\... | Yes |
For a topological system \( \left( {K;\varphi }\right) \) define\n\n\[ \mathcal{T} \mathrel{\text{:=}} \overline{\left\{ {\varphi }^{n} : n \in \mathbb{N}\right\} } \subseteq {K}^{K} \]\n\nwhere the closure is understood in the topology of pointwise convergence of the compact space \( {K}^{K} \) (see Tychonoff’s Theore... | Proof. For \( \psi \in \mathcal{T} \) there is a net \( {\left( {n}_{\alpha }\right) }_{\alpha } \) with \( {\varphi }^{{n}_{\alpha }} \rightarrow \psi \) in \( {K}^{K} \) . Since \( \beta \mathbb{N} \) is compact, \( {\left( {n}_{\alpha }\right) }_{\alpha } \) has a convergent subnet \( {\left( {n}_{{\alpha }^{\prime ... | Yes |
Proposition 19.13. Let \( \left( {K;\varphi }\right) \) be a topological system.\n\na) For \( x \in K \)\n\n\[ \n{\overline{\operatorname{orb}}}_{ > 0}\left( x\right) = \left\{ {{\varphi }^{p}\left( x\right) : p \in \beta \mathbb{N}}\right\} .\n\]\n\nb) If \( \left( {L;\varphi }\right) \) is a subsystem, then \( L \) i... | Proof. a) Let \( \mathcal{T} \) be as in Proposition 19.12. By compactness of \( \mathcal{T} \) and by the continuity of point evaluations on \( {K}^{K} \) the equality \( \mathcal{T}x = \{ \psi \left( x\right) : \psi \in \mathcal{T}\} = \) \( {\operatorname{orb}}_{ > 0}\left( x\right) \) holds. Since \( \mathcal{E} = ... | No |
Proposition 19.14. For a topological system \( \left( {K;\varphi }\right), K \) metrizable with metric \( d \) , and for \( x, y \in K \) the following statements are equivalent:\n\n(i) \( x, y \in K \) are proximal.\n\n(ii) For every \( \varepsilon > 0 \) there is \( n \in \mathbb{N} \) with \( d\left( {{\varphi }^{n}... | We leave the proof as Exercise 3. | No |
Proposition 19.15 (Idempotents and Proximal Points). Let \( \left( {K;\varphi }\right) \) be a topological system. For \( x, y \in K \) the following assertions (i)-(iii) are equivalent:\n\n(i) The points \( x \) , \( y \) are proximal.\n\n(ii) There is \( p \in \beta \mathbb{N} \) with\n\n\[{\varphi }^{p}\left( x\righ... | Proof. (i) \( \Rightarrow \) (ii): Suppose \( x, y \in K \) are proximal points, and let \( z \in K \) be point as in the definition of proximality. Then for every open neighborhood \( U \) of \( z \) we can take the corresponding minimal \( {n}_{U} \in \mathbb{N} \) with \( {\varphi }^{{n}_{U}}\left( x\right) ,{\varph... | Yes |
Theorem 19.16 (Auslander, Ellis). Let \( \\left( {K;\\varphi }\\right) \) be a topological system. For \( x \\in K \) let \( L \\mathrel{\\text{:=}} {\\overline{\\operatorname{orb}}}_{ > 0}\\left( x\\right) \). a) Then for any subsystem \( \\left( {M;\\varphi }\\right) \) of \( \\left( {L;\\varphi }\\right) \) there is... | Proof. a) Define \[ S \\mathrel{\\text{:=}} \\left\\{ {p \\in \\beta \\mathbb{N} : {\\varphi }^{p}\\left( x\\right) \\in M}\\right\\} \] which is nonempty by Proposition 19.13 and a closed subset of \( \\beta \\mathbb{N} \) by the definition of the topology of pointwise convergence and by the continuity of \( p \\mapst... | Yes |
Proposition 19.18 (Ellis). A topological system \( \left( {K;\varphi }\right) \) is distal if and only if its enveloping semigroup \( \mathcal{E} \) is a group. | Proof. Suppose that the enveloping semigroup is a group. Then the only idempotent element in \( \mathcal{E} \) is id : \( K \rightarrow K \), i.e., \( {\varphi }^{p} = \) id for every idempotent \( p \in \beta \mathbb{N} \) . By Proposition 19.15 a point can be only proximal to itself.\n\nSuppose \( \left( {K;\varphi }... | Yes |
Proposition 19.19. A distal topological system is a disjoint union of its minimal subsystems. | Proof. By Theorem 19.16 every point \( x \) is proximal to a uniformly recurrent point \( z \) in its orbit closure. By distality we must have \( x = z \), hence every point is uniformly recurrent. This proves the statement by Theorem 3.11 and Remark 3.2.4. | Yes |
Proposition 19.20. For given \( r \in \mathbb{N} \) consider the shift system \( \left( {{\mathcal{W}}_{r}^{ + };\tau }\right) \) . Two points \( x, y \in {\mathcal{W}}_{r}^{ + } \) are proximal if and only if the set\n\n\[ \left\{ {n \in {\mathbb{N}}_{0} : x\left( n\right) = y\left( n\right) }\right\} \]\n\nis thick, ... | Proof. By the definition of the topology on \( {\mathcal{W}}_{r}^{ + } \), for each compatible metric \( d \) on \( {\mathcal{W}}_{r}^{ + } \) and every \( N \in \mathbb{N} \) there is \( \varepsilon > 0 \) such that \( d\left( {x, y}\right) < \varepsilon \) if or only if the first \( N \) letters of \( x \) and \( y \... | No |
Corollary 19.24. a) Let \( I \subseteq \mathbb{N} \) be an IP set and let \( I = {A}_{1} \cup {A}_{2} \cup \cdots \cup {A}_{r} \) be a partition. Then there is \( {j}_{0} \in \{ 1,\ldots, r\} \) such that \( {A}_{{j}_{0}} \) is an IP set. | Proof. a) Let \( {\left( {n}_{k}\right) }_{k \in \mathbb{N}} \) be a sequence in \( \mathbb{N} \) with \( \operatorname{FS}\left( {n}_{k}\right) \subseteq I \) . For \( j = 1,\ldots, r \) we define\n\n\[ \n{\mathcal{A}}_{j} \mathrel{\text{:=}} \left\{ {\alpha : \mathop{\sum }\limits_{{k \in \alpha }}{n}_{k} \in {A}_{j}... | Yes |
Theorem 19.25 (Van der Waerden). Let\n\n\[ \mathbb{N} = {A}_{1} \cup {A}_{2} \cup \ldots \cup {A}_{r}\;\text{ for some }r \in \mathbb{N}. \]\n\nThen there is \( {j}_{0} \in \{ 1,\ldots, r\} \) such that \( {A}_{{j}_{0}} \) contains arithmetic progressions of arbitrary length, i.e., for all \( k \in \mathbb{N} \) there ... | We prepare the proof by defining the sets\n\n\[ {I}_{k} \mathrel{\text{:=}} \{ \left( {a, a + n, a + {2n},\ldots, a + \left( {k - 1}\right) n}\right) : a \in \mathbb{N}, n \in \mathbb{N}\} \subseteq \mathbb{N} \times \mathbb{N} \times \cdots \times \mathbb{N} \]\n\nfor all \( k \in \mathbb{N} \) . A set \( A \subseteq ... | Yes |
Proposition 19.26. Let \( S \) be subsemigroup in a compact right-topological semigroup \( H \) such that the left multiplications by elements from \( S \) are continuous, and let \( I \) be a two-sided ideal in \( S \). Then \( \bar{S} \) is a compact right-topological subsemigroup of \( H \), and \( \bar{I} \) is a t... | Proof. Compactness of \( \bar{S} \) and the continuity of right multiplications is clear. To show that \( \bar{S} \) is a semigroup take \( x, y \in \bar{S} \) and \( W \) an open neighborhood of \( {xy} \). By the continuity of the right multiplication by \( y \) there is an open neighborhood \( V \) of \( x \) with \... | Yes |
Theorem 19.27. Let \( p \in \beta \mathbb{N} \) be contained in a minimal right ideal of \( \beta \mathbb{N} \) . Then\n\n\[ \left( {p, p,\ldots, p}\right) \in \overline{{I}_{k}}\text{.} \] | Proof. Notice first that \( \left( {p, p,\ldots, p}\right) \in \overline{{S}_{k}} \) . Since by Proposition 19.26 \( \overline{{I}_{k}} \) is a two-sided ideal, by Lemma 16.4 it is enough to show that \( \left( {p, p,\ldots, p}\right) \) is contained in a minimal right ideal. Let \( R \) be a minimal right ideal contai... | Yes |
Theorem 19.28 (Furstenberg, Weiss). Let \( \left( {K;\varphi }\right) \) be a topological system.\n\na) Let \( {U}_{1},{U}_{2},\ldots ,{U}_{r} \subseteq K \) be open subsets covering \( K \) . Then there is \( {j}_{0} \in \) \( \{ 1,\ldots, r\} \) such that for all \( k \in \mathbb{N} \) there is \( n \in \mathbb{N} \)... | Proof. a) Let \( x \in K \) be arbitrary, and for \( j \in \{ 1,\ldots, r\} \) let\n\n\[ \n{A}_{j} \mathrel{\text{:=}} \left\{ {n \in \mathbb{N} : {\varphi }^{n}\left( x\right) \in {U}_{j}}\right\}\n\]\n\nNow, by van der Waerden’s Theorem 19.25, there is \( {j}_{0} \in \{ 1,\ldots, r\} \) such that \( {A}_{{j}_{0}} \) ... | Yes |
Theorem 19.29 (Multiple Recurrence). Let \( \left( {K;\varphi }\right) \) be a topological system with \( \left( {K, d}\right) \) a metric space, and let \( k \in \mathbb{N} \) . Then there is a simultaneously \( k \) -recurrent point \( x \in K \) . If the system is minimal, then the set of \( k \) -recurrent points i... | Proof. By passing to a subsystem we may assume that \( \left( {K;\varphi }\right) \) is minimal, see Theorem 3.5, so it suffices to show the second assertion only. Consider the set\n\n\[ \n{G}_{\varepsilon } \mathrel{\text{:=}} \left\{ {x \in K : d\left( {{\varphi }^{ni}\left( x\right), x}\right) < \varepsilon \text{ f... | Yes |
Theorem 19.31. Let \( K \) be a compact space, let \( {\varphi }_{1},\ldots ,{\varphi }_{k} : K \rightarrow K \) be commuting homeomorphisms, and let \( I \subseteq \mathbb{N} \) be an IP set. Take \( {U}_{1},{U}_{2},\ldots ,{U}_{r} \subseteq K \) open subsets covering \( K \) . Then there is \( {j}_{0} \in \{ 1,\ldots... | For the proof we need some preparations. Let \( \mathcal{S} \) be a semigroup of continuous self-mappings of the compact space \( K \) . We say that \( \mathcal{S} \) acts minimally on \( K \) if whenever a closed set \( F \subseteq K \) is invariant under every \( \varphi \in \mathcal{S} \), then either \( F = \varnot... | No |
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