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Theorem 8.35. A bounded amenable (e.g., Abelian) semigroup \( \mathcal{T} \) on a Banach space \( E \) is mean ergodic if \( \mathcal{T} \) is relatively weakly compact. | Recall again that a bounded semigroup on a reflexive Banach space is always relatively weakly compact, so bounded amenable semigroups on reflexive Banach spaces are mean ergodic. This is a direct generalization of Theorem 8.22. | No |
It was shown in Proposition 6.20 that a Bernoulli shift \( B\left( {{p}_{0},\ldots ,{p}_{k - 1}}\right) \) is strongly mixing. | The same is true for general Bernoulli shifts, see Exercise 16, and Example 4 in Section 18.4 below. | No |
Theorem 9.4. For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) ,\mathrm{X} = \left( {X,\sum ,\mu }\right) \), its associated Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \), and \( p \in \lbrack 1,\infty ) \) the following assertions are equivalent:\n\n(i) \( \left( {\mathrm{X};\varphi ... | Proof. The implications (i) \( \Rightarrow \) (ii) and (iii) \( \Rightarrow \) (i) are trivial, so suppose that (ii) holds. Fix \( A \in \sum \) and \( k, l \in {\mathbb{N}}_{0} \), and let \( f \mathrel{\text{:=}} {T}^{k}{\mathbf{1}}_{A}, g \mathrel{\text{:=}} {T}^{l}{\mathbf{1}}_{A} \) . Then for \( n \geq l - k \) w... | Yes |
Theorem 9.6. For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) with Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) on \( E = {\mathrm{L}}^{p}\left( \mathrm{X}\right) ,1 \leq p < \infty \), the following assertions are equivalent:\n\n(i) \( \left( {\mathrm{X};\varphi }\right) \) is s... | Proof. (i) \( \Leftrightarrow \) (iv) and (i) \( \Rightarrow \) (v) are clear from (iii) of Theorem 9.4.\n\n\( \left( \mathrm{v}\right) \Rightarrow \left( \mathrm{{vi}}\right) \) : By mean ergodicity of \( T \), we have \( E = \mathrm{{fix}}\left( T\right) \oplus \overline{\mathrm{{ran}}}\left( {\mathrm{I} - T}\right) ... | Yes |
Proposition 9.7. For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) and \( k \in \mathbb{N} \) the following assertions are equivalent:\n\n(i) The measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is strongly mixing.\n\n(vii) Its \( {k}^{\text{th }} \) iterate \( \left( {\mathrm... | Proof. By (iv) of Theorem 9.6 it suffices to show that \( {E}_{\mathrm{{ws}}}\left( T\right) = {E}_{\mathrm{{ws}}}\left( {T}^{k}\right) \) . The inclusion \ | No |
Proposition 9.8. For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) the following assertions are equivalent:\n\n(i) \( \left( {\mathrm{X};\varphi }\right) \) is strongly mixing.\n\n(ii) \( \left( {\mathrm{X} \otimes \mathrm{Y};\varphi \times \psi }\right) \) is strongly mixing for any strongly mi... | Proof. For the implication (i) \( \Rightarrow \) (ii) suppose that \( \left( {\mathrm{X};\varphi }\right) \) and \( \left( {\mathrm{Y};\psi }\right) \) are mixing and let as above \( T \) and \( S \) be the respective Koopman operators. Then for \( f, u \in {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) and \( g, v \in {\... | Yes |
Theorem 9.11 (Blum-Hanson). Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a measure-preserving system with Koopman operator \( T = {T}_{\\varphi } \), and let \( 1 \\leq p < \\infty \) . Then \( \\left( {\\mathrm{X};\\varphi }\\right) \) is strongly mixing if and only if for every subsequence \( {\\left( {n}_{k}... | The equivalence stated in this theorem is rather elementary if strong convergence is replaced by weak convergence. Namely, it follows from the fact that a sequence \( {\\left( {a}_{n}\\right) }_{n \\in \\mathbb{N}} \) is convergent if and only if each of its subsequences is Cesàro convergent (cf. Exercise 1 and the pro... | No |
Proposition 9.17 (Jones-Lin). One can add the assertion\n\n\[\n\\text{(v)}\\mathop{\\sup }\\limits_{{{x}^{\\prime } \\in {E}^{\\prime },\\begin{Vmatrix}{x}^{\\prime }\\end{Vmatrix} \\leq 1}}\\frac{1}{n}\\mathop{\\sum }\\limits_{{j = 0}}^{{n - 1}}{\\left| \\left\\langle {T}^{j}x,{x}^{\\prime }\\right\\rangle \\right| }^... | Proof. First, we suppose that \( T \) is a contraction. The dual unit ball \( {\\mathrm{B}}^{\\prime } \\mathrel{\\text{:=}} \\left\\{ {{x}^{\\prime } \\in }\\right. \) \( \\left. {{E}^{\\prime } : \\begin{Vmatrix}{x}^{\\prime }\\end{Vmatrix} \\leq 1}\\right\\} \) is compact with respect to the weak*-topology (Banach-A... | Yes |
Corollary 9.18. Let \( T \) be a power-bounded operator on the Banach space \( E \) . Then the set \( {E}_{\mathrm{{aws}}}\left( T\right) \) of weakly almost stable vectors is a closed \( T \) -invariant subspace of \( E \) contained in \( \overline{\operatorname{ran}}\left( {\mathrm{I} - T}\right) \) . Moreover, \( {E... | Proof. It is trivial from (ii) in Theorem 9.15 that \( {E}_{\text{aws }} \) is \( T \) -invariant. That it is a closed subspace follows from characterization (iv) (see Exercise 8). If \( x \in {E}_{\text{aws }} \) then by (iv) \( {\mathrm{A}}_{n}\left\lbrack T\right\rbrack x \rightarrow 0 \) weakly, hence \( x \in \ove... | No |
Theorem 9.19. Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be 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 equ... | Proof. Note that (ii) simply says that \( f - \\langle f,\\mathbf{1}\\rangle \\mathbf{1} \\in {E}_{\\text{aws }}\\left( T\\right) \) for all \( f \\in E \) . Hence, the equivalence (ii) \( \\Leftrightarrow \) (iii) follows from Theorems 9.15 and 9.17. The equivalences (ii) \( \\Leftrightarrow \) (iv) and (ii) \( \\Left... | Yes |
For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) and \( k \in \mathbb{N} \) the following assertions are equivalent:\n\n(i) The measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is weakly mixing.\n\n(vii) Its \( {k}^{\text{th }} \) iterate \( \left( {\mathrm{X};{\varphi }^{k}}... | Proof. By (iv) from Theorem 9.19 it suffices to have \( {E}_{\mathrm{{aws}}}\left( T\right) = {E}_{\mathrm{{aws}}}\left( {T}^{k}\right) \), which has already been established in Corollary 9.18. | Yes |
Consider any rotation \( \left( {\mathbb{T};a}\right) \) with Koopman operator \( T \) and \( f\left( z\right) \mathrel{\text{:=}} z \) . Then \( \langle f,\mathbf{1}\rangle = 0 \) . | But \( \left\langle {{T}^{n}f,\bar{f}}\right\rangle = {a}^{n} \) does not converge to \( 0 = \langle f,\mathbf{1}\rangle \) along any subsequence \( J \subseteq \mathbb{N} \), so \( \left( {\mathbb{T};a}\right) \) is not weakly mixing. | No |
Corollary 9.24. A nontrivial ergodic group rotation system \( \left( {G,\mathrm{\;m};a}\right) \) is not weakly mixing. | Proof. Let \( 0 \leq f \in \mathrm{C}\left( G\right) \) and consider the function \( k\left( {x, y}\right) \mathrel{\text{:=}} f\left( {{x}^{-1}y}\right) \) for \( x, y \in G \) . Then \( k \) is invariant under left rotations by \( \left( {a, a}\right) \) and \( {\int }_{G \times G}k = {\int }_{G}f \) . By Urysohn’s l... | Yes |
Proposition 9.27. a) If a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is weakly mixing of order \( m \in \mathbb{N} \), then it is weakly mixing of order \( k \) for all \( k \leq m \) . | Proof. a) To check weak mixing of order \( k \) specialize \( {A}_{k} = \cdots = {A}_{m - 1} = X \) . | No |
Theorem 9.31. Every weakly mixing measure-preserving system \( \left( {\mathrm{X};\varphi }\right) ,\mathrm{X} = \) \( \left( {X,\sum ,\mu }\right) \), is weakly mixing of all orders, i.e., | Proof. Since \( \left( {\mathrm{X} \otimes \mathrm{X};\varphi \times \varphi }\right) \) is again weakly mixing by Theorem 9.23, we can apply Corollary 9.30 to \( \varphi \times \varphi \) and the sets \( {A}_{0} \times {A}_{0},\ldots ,{A}_{k - 1} \times {A}_{k - 1} \) to obtain\n\n\[ \mathop{\lim }\limits_{{N \rightar... | Yes |
Theorem 10.1 (Markov-Kakutani). Let \( C \) be a nonempty compact, convex subset of a Hausdorff topological vector space, and let \( \Gamma \) be a set of pairwise commuting, affine, and continuous mappings \( T : C \rightarrow C \) . Then these mappings have a common fixed point in \( C \) . | Proof. First we suppose that \( \Gamma = \{ T\} \) is a singleton. Let \( f \in C \) . Then, by compactness of \( C \), the sequence \( {\mathrm{A}}_{n}\left\lbrack T\right\rbrack f \) has a cluster point \( g \in C \) . Since \( C \) is compact, \( \frac{1}{n}\left( {f - {T}^{n}f}\right) \rightarrow 0 \) . Hence, by L... | Yes |
Theorem 10.2 (Krylov-Bogoljubov). For every topological system \( \left( {K;\varphi }\right) \) there exists at least one \( \varphi \) -invariant probability measure on \( K \) . More precisely, for every \( 0 \neq f \in \operatorname{fix}\left( {T}_{\varphi }\right) \) in \( \mathrm{C}\left( K\right) \) there exists ... | Proof. We apply the Markov-Kakutani theorem to \( \Gamma = \left\{ {T}_{\varphi }^{\prime }\right\} \) and to the weakly* compact, convex set\n\n\[ C \mathrel{\text{:=}} {\mathrm{M}}^{1}\left( K\right) \cap \left\{ {\mu \in \mathrm{M}\left( K\right) : \langle f,\mu \rangle = \parallel f{\parallel }_{\infty }}\right\} .... | Yes |
Proposition 10.4. Let \( \left( {K;\varphi }\right) \) be a topological system. Then \( \mu \in {\mathrm{M}}_{\varphi }^{1}\left( K\right) \) is an extreme point of \( {\mathrm{M}}_{\varphi }^{1}\left( K\right) \) if and only if the measure-preserving system \( \left( {K,\mu ;\varphi }\right) \) is ergodic. | Proof. Assume that \( \varphi \) is not ergodic. Then there exists a set \( A \) with \( 0 < \mu \left( A\right) < 1 \) such that \( {\varphi }^{ * }A = A \) . The probability measure \( {\mu }_{A} \), defined by\n\n\[{\mu }_{A}\left( B\right) \mathrel{\text{:=}} \frac{\mu \left( {A \cap B}\right) }{\mu \left( A\right)... | Yes |
Proposition 10.5. Let \( \left( {K;\varphi }\right) \) be a topological system. Then \( {\mathrm{M}}_{\varphi }\left( K\right) \) is a lattice, i.e., if \( v \in {\mathrm{M}}_{\varphi }\left( K\right) \), then also \( \left| v\right| \in {\mathrm{M}}_{\varphi }\left( K\right) \) . Consequently, \( {\mathrm{M}}_{\varphi... | Proof. Let \( T \) be the associated Koopman operator on \( \mathrm{C}\left( K\right) \) and take \( v \in \operatorname{fix}\left( {T}^{\prime }\right) = \) \( {\mathrm{M}}_{\varphi }\left( K\right) \) . By Exercise 9 we obtain \( \left| v\right| = \left| {{T}^{\prime }v}\right| \leq {T}^{\prime }\left| v\right| \) an... | No |
Theorem 10.6. Let \( \left( {K;\varphi }\right) \) be a topological system with Koopman operator \( T \) on \( \mathrm{C}\left( K\right) \) . The following assertions are equivalent:\n\n(i) \( \left( {K;\varphi }\right) \) is uniquely ergodic, i.e., \( {\mathrm{M}}_{\varphi }^{1}\left( K\right) \) is a singleton.\n\n(i... | Proof. The equivalence (i) \( \Leftrightarrow \) (ii) is clear by Proposition 10.5. The equivalence (i) \( \Leftrightarrow \) (iii) is a consequence of Proposition 10.4 and has been discussed in the paragraph after that proposition.\n\nIf (ii) holds, then \( \operatorname{fix}\left( T\right) \) is one-dimensional since... | Yes |
Lemma 10.7. Let \( \left( {K;\varphi }\right) \) be a topological system and \( \mu \in {\mathrm{M}}^{1}\left( K\right) \) a \( \varphi \) -invariant measure. Then the support \( \operatorname{supp}\left( \mu \right) \) of \( \mu \) is \( \varphi \) -stable, i.e., it satisfies \( \varphi \left( {\operatorname{supp}\lef... | Proof. Let \( M \mathrel{\text{:=}} \operatorname{supp}\left( \mu \right) \) . Since \( T \) is a lattice homomorphism,\n\n\[{\int }_{K}\left| {Tf}\right| \mathrm{d}\mu = {\int }_{K}T\left| f\right| \mathrm{d}\mu = {\int }_{K}\left| f\right| \mathrm{d}\mu\]\n\nwhence by (10.2) \( f \in {I}_{M} \) if and only if \( {Tf}... | No |
Proposition 10.8. Let \( \\left( {K;\\varphi }\\right) \) be a topological system, and let \( M \\subseteq K \) be closed and \( \\varphi \) -invariant.\n\n a) There exists an ergodic probability measure \( \\mu \\in {\\mathrm{M}}_{\\varphi }^{1}\\left( K\\right) \) with \( \\operatorname{supp}\\left( \\mu \\right) \\s... | Proof. a) By the Krylov-Bogoljubov theorem and the arguments following Proposition 10.4, there is an ergodic \( \\mu \\in {\\mathbf{M}}_{\\varphi }^{1}\\left( M\\right) \) . Its natural extension to \( K \) is also ergodic since \( {\\mathrm{L}}^{1}\\left( {K,\\mu }\\right) = {\\mathrm{L}}^{1}\\left( {M,\\mu }\\right) ... | Yes |
Corollary 10.9. Let \( \left( {K;\varphi }\right) \) be a topological system with Koopman operator \( T \) on \( \mathrm{C}\left( K\right) \) . The following assertions are equivalent:\n\n(i) The topological system \( \left( {K;\varphi }\right) \) is strictly ergodic.\n\n(ii) The topological system \( \left( {K;\varphi... | Proof. (i) \( \Rightarrow \) (ii): Suppose that \( \left( {K;\varphi }\right) \) is strictly ergodic, and let \( \mu \) be the unique invariant probability measure on \( K \) . By Theorem 10.6, \( T \) is mean ergodic. Now let \( \varnothing \neq M \subseteq K \) be \( \varphi \) -invariant. Then by Theorem 10.2 there ... | No |
Proposition 10.10. The Koopman operator \( {L}_{a} \) associated with a rotation system \( \left( {\mathbb{T};a}\right) \) is mean ergodic on \( \mathrm{C}\left( \mathbb{T}\right) \) . | Proof. We write \( T = {L}_{a} \) for the Koopman operator and \( {\mathrm{A}}_{n} = {\mathrm{A}}_{n}\left\lbrack T\right\rbrack \) for its Cesàro averages. The linear hull of the functions \( {\chi }_{n} : z \mapsto {z}^{n}, n \in \mathbb{Z} \), is a dense subalgebra of \( \mathrm{C}\left( \mathbb{T}\right) \), by the... | Yes |
Proposition 10.11. Let \( G \) be a compact group and let \( f \in \mathrm{C}\left( G\right) \) . Then the mappings\n\n\[ G \rightarrow \mathrm{C}\left( G\right) ,\;a \mapsto {L}_{a}f\;\text{ and } \]\n\n\[ G \rightarrow \mathrm{C}\left( G\right) ,\;a \mapsto {R}_{a}f \] \n\nare continuous. | Proof. This is a direct application of Theorem 4.17. | Yes |
Corollary 10.12. The Koopman operator associated with a compact group rotation system \( \left( {G;a}\right) \) is mean ergodic on \( \mathrm{C}\left( G\right) \) . | Proof. Let \( f \in \mathrm{C}\left( G\right) \) . Then by Proposition 10.11 the set \( \left\{ {{L}_{g}f : g \in G}\right\} \subseteq \mathrm{C}\left( G\right) \) is compact. A fortiori, the set \( \left\{ {{L}_{a}^{n}f : n \in {\mathbb{N}}_{0}}\right\} \) is relatively compact. Consequently, its closed convex hull is... | Yes |
Theorem 10.13 (Minimal Group Rotations). Let \( G \) be a compact group with Haar probability measure \( \mathrm{m} \), and consider a group rotation \( \left( {G;a}\right) \) with associated Koopman operator \( {L}_{a} \) on \( \mathrm{C}\left( G\right) \) . Then the following assertions are equivalent:\n\n(i) \( \lef... | Proof. The equivalence of (i)-(iv) has been shown in Theorems 2.36 and 3.4. The implication (ii) \( \Rightarrow \) (v) follows from Lemma 4.20. Combining \( \dim \operatorname{fix}\left( {L}_{a}\right) = 1 \) with the mean ergodicity of \( {L}_{a} \) (Corollary 10.12) we obtain the implication (v) \( \Rightarrow \) (vi... | Yes |
Theorem 10.15. Let \( v \) be a \( \psi \) -invariant probability measure on \( K \times G \), and let \( \mu \mathrel{\text{:=}} {\pi }_{ * }v \) . If \( \mu \otimes \mathrm{m} \) is ergodic, then \( v = \mu \otimes \mathrm{m} \) . | Proof: The product measure \( \mu \otimes \mathrm{m} \) is not only \( \psi \) -invariant, but also invariant under \( {\rho }_{a} \) for all \( a \in G \) . Fix \( f \in \mathrm{C}\left( K\right) \) and \( g \in \mathrm{C}\left( G\right) \) . Since the measure-preserving system \( \left( {K \times G,\mu \otimes \mathr... | Yes |
Corollary 10.16 (Furstenberg). If \( \left( {K;\varphi }\right) \) is uniquely ergodic with invariant probability measure \( \mu \), and if \( \mu \otimes \mathrm{m} \) is ergodic, then \( \left( {K \times G;\psi }\right) \) is uniquely ergodic. | The usual proof of Furstenberg's result relies on the pointwise ergodic theorem (which is the subject of the next chapter) and on generic points (see Exercise 11.4). A completely different, more conceptual, approach is given in Chapter 15, see in particular Theorem 15.31. | No |
Proposition 10.17. Let \( \alpha \in \lbrack 0,1) \) be an irrational number. Then the skew shift \( \left( {\lbrack 0,1{)}^{2},{\lambda }^{2};{\psi }_{\alpha }}\right) \) is ergodic. | Proof. To prove the assertion we use Proposition 7.15 and show that the fixed space of the Koopman operator \( T \) of \( {\psi }_{\alpha } \) on \( {\mathrm{L}}^{2}\left( {\left\lbrack 0,1\right) }^{2}\right) = {\mathrm{L}}^{2}\left( {\left\lbrack 0,1\right\rbrack }^{2}\right) \) consists of the constant functions onl... | Yes |
For an irrational number \( \alpha \in \mathbb{R} \) the skew shift \( \left( {K;{\psi }_{\alpha }}\right) \), where\n\n\[ K = \lbrack 0,1{)}^{2}\;\text{ and }\;{\psi }_{\alpha }\left( {x, y}\right) = \left( {x + \alpha \;\left( {\;\operatorname{mod}\;1}\right), x + y\;\left( {\;\operatorname{mod}\;1}\right) }\right) ,... | is strictly ergodic and its associated Koopman operator is mean ergodic on \( \mathrm{C}\left( K\right) \) . | No |
Theorem 10.20 (Weyl). For \( \alpha \in \lbrack 0,1) \smallsetminus \mathbb{Q} \) the sequence \( {\left( n\alpha \left( \;\operatorname{mod}1\right) \right) }_{n \in {\mathbb{N}}_{0}} \) is equidistributed in \( \lbrack 0,1) \) . | We consider the translation system \( \left( {\lbrack 0,1);\alpha }\right) \) as in Example 2.7. Recall from Example 2.15 that this topological system is isomorphic to the rotation system \( \left( {\mathbb{T};a}\right) \), where \( a = {\mathrm{e}}^{{2\pi }\mathrm{i}\alpha } \) . As we saw in the previous section, the... | Yes |
Proposition 10.21. Let \( \alpha \in \lbrack 0,1) \smallsetminus \mathbb{Q} \). Then the Koopman operator associated with the translation system \( \left( {\lbrack 0,1}\right) ;\alpha ) \) is mean ergodic on \( \mathrm{R}\left\lbrack {0,1}\right\rbrack \), and the mean ergodic projection is given by \[ {Pf} = {\int }_{... | Proof. We denote as usual the Koopman operator by \( T \) and its Cesàro averages by \( {\mathrm{A}}_{n}, n \in \mathbb{N} \). We shall identify functions defined on \( \lbrack 0,1) \) with their 1-periodic extensions to \( \mathbb{R} \). Let \( \chi \) be any characteristic function of an interval (open, closed, or ha... | Yes |
Theorem 10.23 (Weyl’s Equidistribution Theorem for Polynomials). Let \( p\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{d}{a}_{j}{x}^{j} \in \mathbb{R}\left\lbrack x\right\rbrack \) be a polynomial such that for some \( j \neq 0 \) the coefficient \( {a}_{j} \) is irrational. Then the sequence \( {\left( p\left( n\... | Proof. Let \( d = \deg p\left( {d \geq 1\text{by assumption}}\right) \) and suppose the leading coefficient \( {a}_{d} \) of \( p \) is rational, i.e., \( {a}_{d} = \frac{q}{r} \) with \( q, r \in \mathbb{Z} \) . For \( i = 0,\ldots, r - 1 \) consider the polynomials \( {q}_{i}\left( x\right) \mathrel{\text{:=}} p\left... | Yes |
Theorem 11.6 (Akcoglu's Ergodic Theorem). Let \( \mathrm{X} \) be a measure space and let \( T \) be a positive contraction on \( {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) for some \( 1 < p < \infty \) . Then \( T \) is pointwise ergodic. | For \( p = 2 \) and \( T \) self-adjoint this is due to Stein and has an elementary proof, see Stein (1961b). In the general case the proof is quite involved and beyond the scope of this book, see Krengel (1985, Sec. 5.2 ) or Kern et al. (1977) and Nagel and Palm (1982). | No |
Lemma 11.7. In the situation described above, the sequence \( {\left( {\mathrm{A}}_{n}f\right) }_{n \in \mathbb{N}} \) is \( \parallel \cdot {\parallel }_{\infty } \) - convergent for every \( f \in F \) . | Proof. Write \( f = g + \left( {\mathrm{I} - T}\right) h \) with \( g \in \mathrm{{fix}}\left( T\right) \) and \( h \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) . Then as already seen before, \( {\mathrm{A}}_{n}f = g + \frac{1}{n}\left( {h - {T}^{n}h}\right) \), and since \( \mathop{\sup }\limits_{n}{\begin{Vm... | Yes |
Proposition 11.9 (Banach’s Principle). Let \( \\mathrm{X} = \\left( {X,\\sum ,\\mu }\\right) \) be a measure space, \( 1 \\leq p < \\infty \), and \( {\\left( {T}_{n}\\right) }_{n \\in \\mathbb{N}} \) a sequence of bounded linear operators on \( E = {\\mathrm{L}}^{p}\\left( \\mathrm{X}\\right) \). If the associated max... | Proof. Since the operators \( {T}_{n} \) are linear, \( F \) is a subspace of \( E \). To see that it is closed, let \( f \\in E \) and \( g \\in F \). For any natural numbers \( k, l \) we have \[ \\left| {{T}_{k}f - {T}_{l}f}\\right| \\leq \\left| {{T}_{k}\\left( {f - g}\\right) }\\right| + \\left| {{T}_{k}g - {T}_{l... | Yes |
Lemma 11.10. Let \( 1 \leq p < \infty \) and \( 0 \leq f \in {\mathrm{L}}^{p}\left( {\mathrm{X};\mathbb{R}}\right) \) and \( \lambda > 0 \) . Then the following assertions hold:\n\na) \( \mu \left\lbrack {f > \lambda }\right\rbrack \leq {\lambda }^{-p}\parallel f{\parallel }_{p}^{p} < \infty \) .\n\nb) \( {\left( f - \... | Proof. a) Let \( A \mathrel{\text{:=}} \left\lbrack {f > \lambda }\right\rbrack \) . Then \( {\lambda }^{p}{\mathbf{1}}_{A} \leq {f}^{p}{\mathbf{1}}_{A} \), and integrating proves the claim. For b) use the same set \( A \) to write\n\n\[{\left( f - \lambda \right) }^{ + } = \left( {f - \lambda }\right) {\mathbf{1}}_{A}... | Yes |
Theorem 11.11 (Maximal Ergodic Theorem). Let \( T \) be a positive Dunford-Schwartz operator on \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) ,\mathrm{X} \) some measure space, let \( p \in \lbrack 1,\infty ) \), and let \( 0 \leq f \in {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) . Then for each \( \lambda > 0 \) and \( ... | Proof. Take \( k \in \{ 2,\ldots, n\} \) . Then, by Lemma 11.10.c,\n\n\[ {S}_{k}f - {k\lambda } = f - \lambda + T{S}_{k - 1}f - \left( {k - 1}\right) \lambda \leq f - \lambda + T{\left( {S}_{k - 1}f - \left( k - 1\right) \lambda \right) }^{ + }\n\n\[ \leq f - \lambda + T{\left( {M}_{n}^{\lambda }f\right) }^{ + }\n\nBy ... | Yes |
Corollary 11.12 (Maximal Inequality). Let \( T \) be a positive Dunford-Schwartz operator on \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) ,\mathrm{X} \) some measure space, let \( p \in \lbrack 1,\infty ) \), and let \( 0 \leq f \in {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) . Then\n\n\[ \mu \left\lbrack {{\mathrm{A}}^... | Proof. By the Maximal Ergodic Theorem 11.11 and by Hölder's inequality,\n\n\[ \mu \left\lbrack {{\mathrm{A}}_{n}^{ * }\left| f\right| > \lambda }\right\rbrack \leq \frac{1}{\lambda }{\int }_{\left\lbrack {\mathrm{A}}_{n}^{ * }\left| f\right| > \lambda \right\rbrack }\left| f\right| \mathrm{d}\mu \leq \frac{1}{\lambda }... | Yes |
Corollary 11.14. If \( \alpha \in \left\lbrack {0,1}\right\rbrack \smallsetminus \mathbb{Q} \), then for every Borel set \( B \subseteq \left\lbrack {0,1}\right\rbrack \) we have\n\n\[ \frac{1}{n}\operatorname{card}\left\{ {j \in \lbrack 0, n) \cap {\mathbb{N}}_{0} : x + {j\alpha }\left( {\;\operatorname{mod}\;1}\right... | Proof. This is Exercise 3. | No |
Theorem 11.16 (Kolmogorov). Let \( \\left( {\\Omega ,\\mathcal{F},\\mathrm{P}}\\right) \) be a probability space, and let \( {\\left( {X}_{n}\\right) }_{n \\in \\mathbb{N}} \\subseteq {\\mathrm{L}}^{1}\\left( {\\Omega ,\\mathcal{F},\\mathrm{P}}\\right) \) be a sequence of independent and identically distributed real ra... | Proof. Since the \( {X}_{j} \) are identically distributed, \( v \\mathrel{\\text{:=}} {\\mathrm{P}}_{{X}_{j}} \) (the distribution of \( {X}_{j} \) ) is a Borel probability measure on \( \\mathbb{R} \), independent of \( j \), and\n\n\[ \n\\mathrm{E}\\left( {X}_{1}\\right) = {\\int }_{\\mathbb{R}}t\\mathrm{\\;d}v\\lef... | No |
Consider a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) with Koopman operator \( T = {T}_{\varphi } \). Fix \( M \in {\sum }_{\mathrm{X}} \) and define \( \theta : X \rightarrow {\mathcal{W}}_{2}^{ + } = \{ 0,1{\} }^{{\mathbb{N}}_{0}} \) by\n\n\[ \theta \left( x\right) \mathrel{\text{:=}} {\left(... | Indeed, \( \varphi \left( x\right) \in \left\lbrack {{\varphi }^{n} \in M}\right\rbrack \) if and only if \( x \in \left\lbrack {{\varphi }^{n + 1} \in M}\right\rbrack \), thus\n\n\[ \theta \left( {\varphi \left( x\right) }\right) = \tau \left( {\theta \left( x\right) }\right) . \]\n\nLet \( v = {\theta }_{ * }\mu \) b... | Yes |
Theorem 12.10. Every Markov embedding \( S : {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \rightarrow {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) induces a map \( \Theta : \sum \left( \mathrm{Y}\right) \rightarrow \sum \left( \mathrm{X}\right) \) via\n\n\[ S{\mathbf{1}}_{A} = {\mathbf{1}}_{\Theta \left( A\right) }\;\text{ ... | Proof. Since \( S \) is a positive operator, it maps real-valued functions to real-valued functions. Now, a moment’s thought reveals that a real-valued function \( f \in {\mathrm{L}}^{1} \) is a characteristic function if and only if \( f \land \left( {\mathbf{1} - f}\right) = 0 \) . Hence, \( S \) maps characteristic ... | No |
Theorem 12.14 (Von Neumann). Two measure-preserving systems on standard probability spaces are isomorphic if and only if they are point isomorphic. | Proof. One implication is trivial. The converse is a straightforward consequence of Theorem 7.20. Indeed, let \( \left( {\mathrm{X};\varphi }\right) \) and \( \left( {\mathrm{Y};\psi }\right) \) be measure-preserving systems with Koopman operators \( {T}_{\varphi } \) and \( {T}_{\psi } \), respectively, and let \( \Ph... | Yes |
Theorem 12.15. For measure-preserving systems \( \left( {\mathrm{X};\varphi }\right) \) and \( \left( {\mathrm{Y};\psi }\right) \) with associated Koopman operators \( {T}_{\varphi } \) and \( {T}_{\psi } \), respectively, consider the following assertions:\n\n(i) The systems \( \left( {\mathrm{X};\varphi }\right) \) a... | Proof. (i) \( \Rightarrow \) (ii) is simply Remark 12.8. The converse, in the case when the underlying spaces are standard probability spaces, is von Neumann's Theorem 12.14.\n\n(ii) \( \Rightarrow \) (iii): By Corollary 12.12 we find a Markov isomorphism \( S : {\mathrm{L}}^{1}\left( \mathrm{X}\right) \rightarrow {\ma... | Yes |
Each measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) gives rise to an abstract system \( \left( {\mathrm{X};T}\right) \) where \( T \mathrel{\text{:=}} {T}_{\varphi } \) is the Koopman operator. | According to Proposition 7.12, the system \( \left( {\mathrm{X};\varphi }\right) \) is invertible if and only if its abstract counterpart \( \left( {\mathrm{X};{T}_{\varphi }}\right) \) is invertible. Moreover, by Corollary 12.12 above, two measure-preserving systems \( \left( {\mathrm{X};\varphi }\right) \) and \( \le... | Yes |
Theorem 12.20. Let \( \left( {\mathrm{X};T}\right) \) be an abstract measure-preserving system. Then \( A \subseteq \) \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) is a \( T \) -invariant \( {C}^{ * } \) -subalgebra if and only if there exists a faithful topological measure-preserving system \( \left( {K,\mu ;\... | Let us call a subalgebra \( A \) of \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) full if \( {\mathrm{{cl}}}_{{\mathrm{L}}^{1}}A = {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) . If \( A \) is full, then the Markov embedding \( \Phi \) in Theorem 12.20 is surjective, hence\n\n\[\n\Phi : \left( {K,\mu ;{T}_{\psi }}... | No |
Theorem 12.22 (Metric Models). An abstract measure-preserving system (X; T) has a metric model if and only if \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) is a separable Banach space. | Proof. Let \( \left( {K,\mu ;\psi }\right) \) be a metric model for \( \left( {\mathrm{X};T}\right) \) . By Theorem 4.7, \( \mathrm{C}\left( K\right) \) is a separable Banach space, and as any dense subset of \( \mathrm{C}\left( K\right) \) is also dense in \( {\mathrm{L}}^{1}\left( {K,\mu }\right) \) , the latter spac... | Yes |
Proposition 12.23. Let \( K \) be the Stone representation space obtained above. Then \( K \) is extremally disconnected, i.e., the closure of every open set is open. | Proof. We know that \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) and \( \mathrm{C}\left( K\right) \) are isomorphic Banach lattices. Since \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) is order complete (Remark 7.11), so is \( \mathrm{C}\left( K\right) \) . Let \( G \subseteq K \) be an open set. Conside... | Yes |
Proposition 12.25. The canonical map \( \mathrm{C}\left( K\right) \rightarrow {\mathrm{L}}^{\infty }\left( {K,\mu }\right) \) is bijective. In other words, every equivalence class \( f \in {\mathrm{L}}^{\infty }\left( {K,\mu }\right) \) contains a (unique) continuous function. | Proof. Since \( \mu \) is strictly positive, the canonical map \( J : \mathrm{C}\left( K\right) \rightarrow {\mathrm{L}}^{\infty }\left( {K,\mu }\right) \) (mapping each continuous function onto its equivalence class modulo equality \( \mu \) - almost everywhere) is an isometry (Exercise 5.11). Note that the lattice is... | No |
Proposition 12.26. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system, let \( A \subseteq {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) be an invariant \( {C}^{ * } \) -subalgebra, let \( \left( {K,\mu ;\psi }\right) \) be a faithful topological measure-preserving system, and let\n\n\[ \Ph... | Proof. By construction, \( {T}_{\psi } \) is mean ergodic on \( \mathrm{C}\left( K\right) \) if and only if \( {T}_{\varphi } \) is mean ergodic on \( A \) . The fixed spaces are related by\n\n\[ \operatorname{fix}\left( {{T}_{\psi } \cap \mathrm{C}\left( K\right) }\right) = {\Phi }^{-1}\left( {\operatorname{fix}\left(... | No |
Proposition 12.28. For an ergodic measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) the following assertions are equivalent:\n\n(i) The Koopman operator \( {T}_{\varphi } \) is mean ergodic on \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \).\n\n(ii) \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\righ... | Proof. Since a finite-dimensional Banach space is reflexive, by Theorem 8.22 every power-bounded operator thereon is mean ergodic, whence the implication (ii) \( \Rightarrow \) (i) follows.\n\nFor the converse we use the Stone representation \( \left( {K,\mu ;\psi }\right) \) with its (unique) invariant measure \( \mu ... | Yes |
Theorem 13.2. a) The set \( \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) of all Markov operators is a convex subset of \( \mathcal{L}\left( {{\mathrm{L}}^{1}\left( \mathrm{X}\right) ;{\mathrm{L}}^{1}\left( \mathrm{Y}\right) }\right) \) . Composition of Markov operators yields a Markov operator, so in particular \... | Proof. The first assertion a) is straightforward from the definition. | No |
Lemma 13.3. Let \( 1 \leq p < \infty \), and let \( f, g \in {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) . Then \( f \cdot g \in {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) if and only if there are sequences \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}},{\left( {g}_{n}\right) }_{n \in \mathbb{N}} \subseteq {\mathrm{L}... | Proof. For one implication, pass to subsequences that converge almost everywhere and then apply Fatou’s lemma to conclude that the product belongs to \( {\mathrm{L}}^{p} \) . For the converse, use the approximation\n\n\[{f}_{n} \mathrel{\text{:=}} f \cdot {\mathbf{1}}_{\left\lbrack \left| f\right| \leq n\right\rbrack }... | Yes |
Proposition 13.4. Let \( S \in \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) and \( f \in {\mathrm{L}}^{1}\left( \mathrm{X}\right), g \in {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \) . If \( g \cdot {Sf} \in {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \) , then \( f \cdot {S}^{\prime }g \in {\mathrm{L}}^{1}\left( \mat... | Proof. Take \( {g}_{n} \in {\mathrm{L}}^{\infty }\left( \mathrm{Y}\right) \) with \( \left| {g}_{n}\right| \leq \left| g\right| \) such that \( {g}_{n} \rightarrow g \) almost everywhere and in \( {\mathrm{L}}^{1} \) . Then \( \left| {{g}_{n}{Sf}}\right| \leq \left| g\right| \cdot \left| {Sf}\right| = \left| {gSf}\righ... | Yes |
Proposition 13.6. Let \( \mathrm{X} \), Y be probability spaces and let \( 1 \leq p \leq \infty \) . Then the restriction mapping\n\n\[ \n{\Phi }_{p} : \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \rightarrow {\mathrm{M}}_{p}\left( {\mathrm{X};\mathrm{Y}}\right) ,\;{\Phi }_{p}\left( S\right) \mathrel{\text{:=}} {\le... | Proof. Each \( S \in {\mathrm{M}}_{p}\left( {\mathrm{X};\mathrm{Y}}\right) \) satisfies\n\n\[ \n\parallel {Sf}{\parallel }_{1} = {\int }_{\mathrm{Y}}\left| {Sf}\right| \leq {\int }_{\mathrm{Y}}S\left| f\right| = {\int }_{\mathrm{X}}\left| f\right| = \parallel f{\parallel }_{1} \n\] \n\nfor all \( f \in {\mathrm{L}}^{p}... | No |
Theorem 13.8. The set of Markov operators \( \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) is compact with respect to the weak operator topology. If both \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) and \( {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \) are separable, then both weak and strong operator topologies on... | Proof. The set \( \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) of Markov operators is weakly closed in the set of all contractions \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \rightarrow {\mathrm{L}}^{2}\left( \mathrm{Y}\right) \) . Hence, the first assertion follows from Theorem D.7.\n\nThe separability of \( {\... | No |
Theorem 13.9. For a Markov operator \( S \in \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) the following assertions are equivalent:\n\n(i) \( S\left( {f \cdot g}\right) = {Sf} \cdot {Sg} \) for all \( f, g \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \).\n\n(ii) \( {S}^{\prime }S = \mathrm{I} \).\n\n(iii) Th... | Proof. If (i) holds, then \( \langle {Sf},{Sg}{\rangle }_{\mathrm{Y}} = \int \left( {Sf}\right) \cdot \left( {Sg}\right) = \int S\left( {f \cdot g}\right) = \int f \cdot g = \langle f, g{\rangle }_{\mathrm{X}} \) for all \( f, g \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \). This means that \( {S}^{\prime }{Sf}... | Yes |
Theorem 13.11. On the set of Markov embeddings \( \operatorname{Emb}\left( {\mathrm{X};\mathrm{Y}}\right) \) the weak and the strong operator topologies coincide. | Proof. Let \( \Phi : \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \rightarrow {\mathrm{M}}_{2}\left( {\mathrm{X};\mathrm{Y}}\right) \) be the restriction mapping \( \Phi \left( S\right) \mathrel{\text{:=}} {\left. S\right| }_{{\mathrm{L}}^{2}} \) . By Theorem 13.9(iv) one has \( \Phi \left( {\operatorname{Emb}\left(... | Yes |
Theorem 13.12. For a Markov operator \( P \in \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) the following assertions are equivalent:\n\n(i) \( P \) is a factor map, i.e., \( {P}^{\prime } \) is an embedding.\n\n(ii) \( P{P}^{\prime } = \mathrm{I} \).\n\n(iii) There is \( T \in \mathrm{M}\left( {\mathrm{Y};\mathrm{... | Proof. If (i) holds, then \( {P}^{\prime } \) is an embedding, whence \( \mathrm{I} = {\left( {P}^{\prime }\right) }^{\prime }{P}^{\prime } = P{P}^{\prime } \), i.e.,(ii follows. The implication (ii) \( \Rightarrow \) (iii) is trivial. If \( {PT} = \mathrm{I} \), then taking adjoints we obtain \( {T}^{\prime }{P}^{\pri... | Yes |
Corollary 13.14. For a Markov operator \( S \in \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) the following assertions are equivalent:\n\n(i) There are \( {T}_{1},{T}_{2} \in \mathrm{M}\left( {\mathrm{Y};\mathrm{X}}\right) \) such that \( {T}_{1}S = \mathrm{I} \) and \( S{T}_{2} = \mathrm{I} \) .\n\n(ii) \( S \) i... | Proof. By Theorems 13.9 and 13.12 it is clear that (i)-(iii) are equivalent, and any one of them implies (iv) and (v). Suppose that (iv) holds. Then \( {S}^{\prime }S = \mathrm{I} \) and \( S \) is surjective. Hence, it is bijective, and \( {S}^{-1} = {S}^{\prime } \) is a Markov operator, whence (i) follows. The proof... | Yes |
Proposition 13.15. For a probability space \( \mathrm{X} \) the set \( \operatorname{Aut}\left( \mathrm{X}\right) \) of Markov automorphisms is a topological group with respect to the strong \( \left( { = \text{weak}}\right) \) operator topology. If \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) is separable, then \( \... | Proof. Clearly, \( \operatorname{Aut}\left( \mathrm{X}\right) \) is a group with the neutral element \( \mathrm{{Id}} \) . The operator multiplication is jointly continuous for the strong operator topology (since all Markov operators are contractions). Inversion coincides with taking the adjoint and is therefore contin... | Yes |
Every Markov projection \( Q \in \mathrm{M}\left( \mathrm{X}\right) \) is self-adjoint, i.e., satisfies \( {Q}^{\prime } = Q \) and restricts to an orthogonal projection on the Hilbert space \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) . Conversely, if \( Q \) is an orthogonal projection on \( {\mathrm{L}}^{2}\left( ... | Proof. If \( Q \) is a Markov projection, it restricts to a contractive projection on the Hilbert space \( H \mathrel{\text{:=}} {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) . Hence, by Theorem D.21, \( Q \) is an orthogonal projection and in particular self-adjoint, see also Remark 13.7.\n\nIf, conversely, \( Q \) is a... | Yes |
Every Markov projection \( Q \in \mathrm{M}\left( \mathrm{X}\right) \) is uniquely determined by its range \( \operatorname{ran}\left( Q\right) \) . | Let \( P, Q \) be Markov projections with \( \operatorname{ran}\left( P\right) = \operatorname{ran}\left( Q\right) \) . Then\n\n\[ \operatorname{ran}\left( {\left. P\right| }_{{\mathrm{L}}^{2}}\right) = \operatorname{ran}\left( P\right) \cap {\mathrm{L}}^{2} = \operatorname{ran}\left( Q\right) \cap {\mathrm{L}}^{2} = \... | Yes |
Corollary 13.18. Let \( \mathrm{X},\mathrm{Y} \) be probability spaces, let \( Q \in \mathrm{M}\left( \mathrm{Y}\right) \) be a Markov projection, and let \( S \in \mathrm{M}\left( {\mathrm{X};\mathrm{Y}}\right) \) be such that \( {QS} \) is an embedding. Then \( {QS} = S \) . | Proof. Let \( f \in {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) . Since \( {QS} \) is an embedding, it is isometric on \( {\mathrm{L}}^{2} \), and hence\n\n\[ \parallel f{\parallel }_{2} = \parallel {QSf}{\parallel }_{2} \leq \parallel {Sf}{\parallel }_{2} \leq \parallel f{\parallel }_{2}. \]\n\nHence, \( {QSf} = {Sf} ... | Yes |
Proposition 13.19. Let \( \mathrm{X} = \left( {X,\sum ,\mu }\right) \) be a probability space. Then the following assertions hold:\n\na) The range \( F \mathrel{\text{:=}} \operatorname{ran}\left( Q\right) \) of a Markov projection \( Q \in \mathbf{M}\left( \mathrm{X}\right) \) is a unital Banach sublattice of \( {\mat... | Proof. a) Obviously \( \mathbf{1} = Q\mathbf{1} \in F \) . For \( f = {Qf} \in F \) we have\n\n\[ \left| f\right| = \left| {Qf}\right| \leq Q\left| f\right| ,\text{ i.e.,}Q\left| f\right| - \left| f\right| \geq 0.\]\n\nTherefore, since \( {Q}^{2} = Q \) and \( Q \) is a Markov operator,\n\n\[ 0 \leq {\int }_{\mathrm{X}... | Yes |
Theorem 13.20. Let \( \mathrm{X} = \left( {X,\sum ,\mu }\right) \) be a probability space. The assignments\n\n\[ \nQ \mapsto P = {\left. Q\right| }_{{\mathrm{L}}^{2}},\;Q \mapsto \operatorname{ran}\left( Q\right) ,\;Q \mapsto {\sum }^{\prime } = \left\{ {A \in \sum : Q{\mathbf{1}}_{A} = {\mathbf{1}}_{A}}\right\} \n\]\n... | Proof. Only the last assertion remains to be proved. If \( F \) is a unital Banach sublattice of \( {\mathrm{L}}^{1} \), then clearly \( E \mathrel{\text{:=}} F \cap {\mathrm{L}}^{p} \) is a unital Banach sublattice of \( {\mathrm{L}}^{p} \) . If \( 0 \leq f \in F \) , then \( {f}_{n} \mathrel{\text{:=}} f \land n\math... | Yes |
Theorem 13.22. For a Markov operator \( Q \in \mathrm{M}\left( \mathrm{X}\right) \) the following assertions are equivalent:\n\n(i) \( {Q}^{2} = Q \), i.e., \( Q \) is a Markov projection.\n\n(ii) \( Q = S{S}^{\prime } \) for some Markov embedding \( S \) .\n\n(iii) \( Q = {P}^{\prime }P \) for some Markov factor map \... | Proof. By the duality of embeddings and factor maps, (ii) and (iii) are equivalent. The equivalence of (i) and (ii) follows from Proposition 13.19 and the remarks following it. The implication (iii) \( \Rightarrow \) (iv) is obtained from\n\n\[ Q\left( {{Qf} \cdot g}\right) = {P}^{\prime }P\left( {{P}^{\prime }{Pf} \cd... | Yes |
Example 13.24 (Mean Ergodic Projections I). Every Markov operator \( T \in \mathrm{M}\left( \mathrm{X}\right) \) , \( \mathrm{X} = \left( {X,\sum ,\mu }\right) \), is a Dunford-Schwartz operator. Hence, it is mean ergodic by Theorem 8.24, i.e., the limit\n\n\[ \n{P}_{T} \mathrel{\text{:=}} \mathop{\lim }\limits_{{n \ri... | Since \( T \) is a Markov operator, so is \( {P}_{T} \), i.e., \( {P}_{T} \) is a Markov projection. By Proposition 13.19, \( \operatorname{fix}\left( T\right) \) is a Banach sublattice. Note that by Corollary 8.7 we have\n\n\[ \n\operatorname{fix}\left( T\right) = \operatorname{fix}\left( {T}^{\prime }\right) \;\text{... | Yes |
Example 13.25 (Mean Ergodic Projections II). More generally, let \( \mathcal{T} \subseteq \mathrm{M}\left( \mathrm{X}\right) \) be a semigroup of Markov operators. Then the fixed space \( \operatorname{fix}\left( \mathcal{T}\right) = \mathop{\bigcap }\limits_{{T \in \mathcal{T}}}\operatorname{fix}\left( T\right) \) is ... | \[ {Pf} \in \overline{\operatorname{conv}}\{ {Tf} : T \in \mathcal{T}\} \;\text{ for all }f \in {\mathrm{L}}^{1}\left( \mathrm{X}\right) ,\] because this is true for \( f \in {\mathrm{L}}^{2}\left( \mathrm{X}\right) \), see Theorem 8.32. It follows that, in the terminology of Definition 8.31, the semigroup \( \mathcal{... | No |
Lemma 13.28. Let \( \mathrm{X},\mathrm{Y} \) be probability spaces, and let \( P \in \mathrm{M}\left( \mathrm{X}\right) \) and \( Q \in \mathrm{M}\left( \mathrm{Y}\right) \) be Markov projections with ranges \( F = \operatorname{ran}\left( P\right) \) and \( G = \operatorname{ran}\left( Q\right) \) . Then for a Markov ... | Proof. The equivalence in a) is trivial. For the equivalences in b) see Proposition D.23. | No |
Lemma 13.30. Let \( S : {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \rightarrow {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) be a Markov embedding, \( F \mathrel{\text{:=}} \operatorname{ran}\left( S\right) \) its range, and \( \pi = {\pi }_{S} \) the corresponding restriction map as in (13.2). Then for \( R, T \in \mathrm... | Proof. The proof of a)-d) is simple and is left as Exercise 4. For e) suppose first that \( \pi \left( T\right) = {S}^{\prime }{TS} \) is a Markov embedding. Then \( {S\pi }\left( T\right) = S{S}^{\prime }{TS} = {QTS} \) is an embedding, where \( Q \) is the Markov projection with range \( F \) . By Corollary 13.18 it ... | No |
Theorem 13.35. Let \( \\left( {\\left( {{\\mathrm{X}}_{i};T}\\right) ,{\\left( {J}_{ij}\\right) }_{i \\leq j}}\\right) \) be an inductive system and let \( \\left( {\\mathrm{X};T}\\right) \) be another system together with a family of homomorphisms \( {\\left( {J}_{i} : \\left( {\\mathrm{X}}_{i};T\\right) \\rightarrow ... | Proof. For \( i \\leq j \) one has \( {J}_{i} = {J}_{j}{J}_{ji} \) and hence \( \\operatorname{ran}\\left( {\\mathrm{E}}_{i}\\right) = \\operatorname{ran}\\left( {J}_{i}\\right) \\subseteq \\operatorname{ran}\\left( {J}_{j}\\right) = \\operatorname{ran}\\left( {\\mathrm{E}}_{j}\\right) \) . This yields (13.4).\n\nNote ... | Yes |
Proposition 13.36. Let \( \left( {\left( {\mathrm{X};T}\right) ,{\left( {J}_{i}\right) }_{i \in I}}\right) \) be an inductive limit of an inductive system \( \left( {\left( {{\mathrm{X}}_{i};T}\right) ,{\left( {J}_{ij}\right) }_{i \leq j}}\right) \) . Further, let \( \left( {\mathrm{Y};T}\right) \) be another system, \... | Proof. a) The hypothesis yields \( {TS}{J}_{i} = T{S}_{i} = {S}_{i}T = S{J}_{i}T = {ST}{J}_{i} \) and hence \( {TS}{\mathrm{E}}_{i} = {ST}{\mathrm{E}}_{i} \) for all \( i \in I \) . It follows that \( {ST} = {TS} \) . | Yes |
Corollary 13.37. The inductive limit of an inductive system of ergodic/weakly mixing systems is again ergodic/weakly mixing. | Proof. Let \( \left( {\left( {\mathrm{X};T}\right) ,{\left( {J}_{i}\right) }_{i \in I}}\right) \) be an inductive limit of an inductive system \( \left( {\left( {{\mathrm{X}}_{i};T}\right) ,{\left( {J}_{ij}\right) }_{i \leq j}}\right) \) . For each \( i \in I \) let \( {P}_{i} : {\mathrm{L}}^{1}\left( {\mathrm{X}}_{i}\... | Yes |
Theorem 13.38. Each inductive system \( \left( {\left( {{\mathrm{X}}_{i};T}\right) ,{\left( {J}_{ij}\right) }_{i \leq j}}\right) \) of abstract dynamical systems has an inductive limit. | Proof. Pick for each \( i \in I \) a faithful topological model \( \left( {{K}_{i},{\mu }_{i};{\varphi }_{i}}\right) \) of \( \left( {{\mathrm{X}}_{i};T}\right) \) . Without loss of generality we may identify \( {\mathrm{L}}^{1}\left( {\mathrm{X}}_{i}\right) \) with \( {\mathrm{L}}^{1}\left( {{K}_{i},{\mu }_{i}}\right)... | Yes |
Lemma 13.39. The inductive limit system \( \left( {\mathrm{Y};T}\right) \) is invertible. | Proof. Since \( T \) is an isometry, it is sufficient to show that \( \operatorname{ran}\left( T\right) \) is dense, and for this it is sufficient to show that \( \operatorname{ran}\left( T\right) \) contains \( \mathop{\bigcup }\limits_{{k \in {\mathbb{N}}_{0}}}\operatorname{ran}\left( {J}_{k}\right) \) . Now note tha... | No |
Theorem 13.40. Let \( \left( {\mathrm{X};T}\right) \) be an abstract system with minimal invertible extension \( J : \left( {\mathrm{X};T}\right) \rightarrow \left( {\mathrm{Y};T}\right) \) . If \( \widetilde{J} : \left( {\mathrm{X};T}\right) \rightarrow \left( {\mathrm{Z};T}\right) \) is another invertible extension, ... | Proof. Note that if \( S : \left( {\mathrm{Y};T}\right) \rightarrow \left( {\mathrm{Z};T}\right) \) is a homomorphism with \( {SJ} = \widetilde{J} \), then \( \widetilde{J} = {SJ} = S{J}_{i}{T}^{i} = {T}^{i}S{J}_{i} \) and hence \( S{J}_{j} = {T}^{-i}\widetilde{J} \), since \( T \) is invertible on \( {\mathrm{L}}^{1}\... | Yes |
Theorem 13.42 (Minimal Invertible Extension). Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a standard measure-preserving system. Then there exists an invertible standard system \( \\left( {\\mathrm{Y};\\psi }\\right) \) and a point factor map \( \\pi : \\left( {\\mathrm{Y};\\psi }\\right) \\rightarrow \\left( {... | Proof. Since \( \\mathrm{X} \) is a standard probability space, \( {\\mathrm{L}}^{1}\\left( \\mathrm{X}\\right) \) is separable. Let \( J : \\left( {\\mathrm{X};{T}_{\\varphi }}\\right) \\rightarrow \\left( {\\mathrm{Y};T}\\right) \) be an invertible extension of the abstract system \( \\left( {\\mathrm{X};{T}_{\\varph... | Yes |
Lemma 14.1. For a topological group \( G \) the following statements hold:\n\na) If \( V \) is an open neighborhood of 1, then there is an open, symmetric set \( W \) with \( 1 \in W \) and \( {WW} \subseteq V \) . | Proof. a) Let \( V \) be an open neighborhood of 1 . Since the multiplication is continuous we find \( U \) open with \( 1 \in U \) and \( {UU} \subseteq V \) . Then \( W \mathrel{\text{:=}} U \cap {U}^{-1} \) has the desired property. | Yes |
Theorem 14.2 (Haar Measure). On a compact group \( G \) there is a unique left invariant Baire probability measure. This measure is also right invariant, inversion invariant, and strictly positive. | Proof. It is easy to see that if \( \mu \in {\mathrm{M}}^{1}\left( G\right) \) is left (right) invariant, then \( \widetilde{\mu } \in {\mathrm{M}}^{1}\left( G\right) \) , defined by\n\n\[ \langle f,\widetilde{\mu }\rangle = {\int }_{G}f\left( {x}^{-1}\right) \mathrm{d}\mu \left( x\right) \;\left( {f \in \mathrm{C}\lef... | Yes |
Theorem 14.4. Let \( G \) be a locally compact Abelian group. Then its dual group \( {G}^{ * } \) separates the points of \( G \), i.e., for every \( 1 \neq a \in G \) there is a character \( \chi \in {G}^{ * } \) such that \( \chi \left( a\right) \neq 1 \) . | We shall not prove this theorem in its full generality, but only in the two cases most interesting to us, namely when \( G \) is discrete or when \( G \) is compact. For discrete groups, the proof reduces to pure algebra, see Proposition 14.28 in the supplement to this chapter. The other case is treated in the next cha... | No |
Proposition 14.6 (Orthogonality). Let \( G \) be a compact Abelian group. Then \( {G}^{ * } \) is an orthonormal set in \( {\mathrm{L}}^{2}\left( G\right) \) . | Proof. Let \( {\chi }_{1},{\chi }_{2} \in {G}^{ * } \) be two characters and consider \( \chi \mathrel{\text{:=}} {\chi }_{1}\overline{{\chi }_{2}} \) . Then\n\n\[ \alpha \mathrel{\text{:=}} \left( {{\chi }_{1} \mid {\chi }_{2}}\right) = {\int }_{G}{\chi }_{1}\overline{{\chi }_{2}}\mathrm{\;{dm}} = {\int }_{G}\chi \lef... | Yes |
Proposition 14.7. Let \( G \) be a compact Abelian group and let \( X \subseteq {G}^{ * } \) be a subset separating the points of \( G \) . Then the subgroup \( \langle X\rangle \) generated by \( X \) is equal to \( {G}^{ * } \) . Moreover, \( \operatorname{lin}\langle X\rangle \) is dense in the Banach space \( \math... | Proof. Consider \( A = \operatorname{lin}\langle X\rangle \) which is a conjugation invariant subalgebra of \( \mathrm{C}\left( G\right) \) separating the points of \( G \) . So by the Stone-Weierstraß Theorem 4.4 it is dense in \( \mathrm{C}\left( G\right) \) . If there is \( \chi \in {G}^{ * } \smallsetminus \langle ... | Yes |
Corollary 14.8. For a compact Abelian group \( G \) we have the following:\na) The set \( \operatorname{lin}\left( {G}^{ * }\right) \) of trigonometric polynomials is dense in \( \mathrm{C}\left( G\right) \) .\nb) The dual group \( {G}^{ * } \) forms an orthonormal basis of \( {\mathrm{L}}^{2}\left( G\right) \) . | Proof. a) follows from Proposition 14.7 with \( X = {G}^{ * } \) since \( {G}^{ * } \) separates the points of \( G \) by Theorem 14.4. A fortiori, \( \operatorname{lin}\left( {G}^{ * }\right) \) is dense in \( {\mathrm{L}}^{2}\left( G\right) \), and hence b) follows from Proposition 14.6. | Yes |
Let \( G \) be a compact Abelian group with Haar measure \( \mathrm{m} \), let \( a \in G \) and consider the measure-preserving rotation system \( \left( {G,\mathrm{\;m};a}\right) \) with Koopman operator \( {L}_{a} \) on \( {\mathrm{L}}^{2}\left( G\right) \). Then\n\n\[ \n{L}_{a}f = \mathop{\sum }\limits_{{\chi \in {... | Proof. By Corollary 14.8 we can write \( f = \mathop{\sum }\limits_{{\chi \in {G}^{ * }}}\left( {f \mid \chi }\right) \chi \), then apply \( {L}_{a} \) to obtain\n\n\[ \n\mathop{\sum }\limits_{{\chi \in {G}^{ * }}}\lambda \left( {f \mid \chi }\right) \chi = {\lambda f} = {L}_{a}f = \mathop{\sum }\limits_{{\chi \in {G}^... | Yes |
Proposition 14.11. Let \( G \) be a locally compact Abelian group. If \( G \) is compact, then \( {G}^{ * } \) is discrete; and if \( G \) is discrete, then \( {G}^{ * } \) is compact. | Proof. Suppose that \( G \) is compact. Take \( \chi \in {G}^{ * } \), then \( \chi \left( G\right) \subseteq \mathbb{T} \) is a compact subgroup of \( \mathbb{T} \) . Therefore, if \( \chi \) satisfies\n\n\[ \parallel \mathbf{1} - \chi {\parallel }_{\infty } = \mathop{\sup }\limits_{{g \in G}}\left| {1 - \chi \left( g... | Yes |
Lemma 14.13. The mapping\n\n\[ \Phi : G \rightarrow {G}^{* * } \mathrel{\text{:=}} {\left( {G}^{ * }\right) }^{ * } \]\n\nis continuous. | Proof. If \( G \) is discrete, then \( \Phi \) is trivially a continuous mapping. If \( G \) is compact, then \( {G}^{ * } \) is discrete. So the compact sets in \( {G}^{ * } \) are just the finite ones, and the topology on \( {G}^{* * } \) is the topology of pointwise convergence. So \( \Phi \) is continuous. | Yes |
Theorem 14.14 (Pontryagin Duality Theorem). For a locally compact Abelian group \( G \) the mapping \( \Phi : G \rightarrow {G}^{* * } \) defined in (14.2) is a topological isomorphism. | Proof. After the preceding discussion it remains to prove that \( \Phi \) is surjective and its inverse in continuous.\n\nFirst, suppose that \( G \) is discrete. Then, by Proposition 14.11, \( {G}^{ * } \) is compact and the subgroup \( \operatorname{ran}\left( \Phi \right) \subseteq {G}^{* * } \) clearly separates th... | Yes |
Proposition 14.17. Let \( G \) be a locally compact Abelian group and \( \mathrm{b}G \) its Bohr compactification. Then the following assertions hold:\n\na) If \( G \) is compact, then\n\n\[ \mathrm{b}G = {\left( {G}^{ * }\right) }^{ * } \simeq G.\]\n\nb) The dual group of \( \mathrm{b}G \) is topologically isomorphic ... | Proof. a) Since \( \Phi \) is continuous, \( \Phi \left( G\right) \) is compact, and hence closed in \( {\mathbb{T}}^{{G}^{ * }} \) . So \( \mathrm{b}G = \Phi \left( G\right) = {\left( {G}^{ * }\right) }^{ * }.\n\nb) For \( \chi \in {G}^{ * } \) the projection \( {\pi }_{\chi } : {\mathbb{T}}^{{G}^{ * }} \rightarrow \m... | Yes |
Theorem 14.18 (Kronecker). For \( a = \left( {{a}_{1},\ldots ,{a}_{d}}\right) \in {\mathbb{T}}^{d} \) the rotation system \( \left( {{\mathbb{T}}^{d};a}\right) \) is topologically transitive \( \left( { = \text{minimal}}\right) \) if and only if \( {a}_{1},{a}_{2},\ldots ,{a}_{d} \) are linearly independent in the \( \... | Proof. If \( {a}_{1}^{{k}_{1}}{a}_{2}^{{k}_{2}}\cdots {a}_{n}^{{k}_{d}} = 1 \) for \( 0 \neq \left( {{k}_{1},{k}_{2},\ldots ,{k}_{d}}\right) \in {\mathbb{Z}}^{d} \), then the set\n\n\[ A \mathrel{\text{:=}} \left\{ {x \in {\mathbb{T}}^{d} : {x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{d}^{{k}_{d}} = 1}\right\} \]\n\ni... | Yes |
Proposition 14.20 (Weil's Lemma). Let \( G \) be a locally compact monothetic group with generating element \( a \in G \). Then \( G \) is either topologically isomorphic to \( \mathbb{Z} \) or compact, and the latter happens if and only if already \( \left\{ {{a}^{n} : n \in \mathbb{N}}\right\} \) is dense in \( G \). | Proof. We define \( A \mathrel{\text{:=}} \left\{ {{a}^{n} : n \geq 1}\right\} \) and suppose that \( U \cap A = \varnothing \) for some nonempty open subset \( U \subseteq G \). Since \( \langle a\rangle \) is dense, we can find \( k \geq 0 \) such that \( {a}^{-k} \in U \), and then pick a symmetric open neighborhood... | Yes |
Proposition 14.21. For a compact group \( G \) with Haar measure \( \mathrm{m} \) and an element \( a \in G \) the following statements are equivalent:\n\n(i) The topological system \( \left( {G;a}\right) \) is minimal.\n\n(ii) The set \( \left\{ {{a}^{n} : n \in \mathbb{N}}\right\} \) is dense in \( G \) .\n\n(iii) Th... | Proof. The implications (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) are trivial, while (iii) \( \Rightarrow \) (ii) follows from Weil’s Lemma 14.20. Implication (ii) \( \Rightarrow \) (i) is proved in Theorem 3.4.\n\nSuppose that (iii) holds. Then \( G \) is Abelian, and by continuity, two characters are equal i... | Yes |
Proposition 14.22. A compact Abelian group \( G \) is monothetic if and only if its dual group \( {G}^{ * } \) is algebraically isomorphic to a subgroup of \( \mathbb{T} \) . In this case, the isomorphism is given by\n\n\[ \chi \mapsto \chi \left( a\right) \in {G}^{ * }\left( a\right) \mathrel{\text{:=}} \left\{ {\chi ... | Proof. Suppose that \( G \) is monothetic with generating element \( a \) . Then the set \( {G}^{ * }\left( a\right) \) is a subgroup of \( \mathbb{T} \) and the evaluation mapping \( {G}^{ * } \ni \chi \mapsto \chi \left( a\right) \) is a surjective group homomorphism. Furthermore, it is even injective by (iii) of Pro... | Yes |
Proposition 14.24. Let \( G \) be a compact Abelian group with Haar measure \( \mathrm{m} \), and \( {L}_{a} \) be the Koopman operator of the rotation by some element \( a \in G \), considered either as an operator on \( \mathrm{C}\left( G\right) \) or on \( {\mathrm{L}}^{p}\left( G\right), p \in \lbrack 1,\infty ) \)... | Proof. If \( {L}_{a} \) is regarded on \( {\mathrm{L}}^{2}\left( G\right) \), the statement was proved in Proposition 14.9. Since characters are continuous and are eigenvectors of \( {L}_{a} \), the statement follows also for \( \mathrm{C}\left( G\right) \) . So the inclusion \( {G}^{ * }\left( a\right) \subseteq {\sig... | Yes |
Proposition 14.25. Let \( G, H \) be compact monothetic groups with Haar measures \( {\mathrm{m}}_{G} \) and \( {\mathrm{m}}_{H} \), respectively, and with generating elements \( a \in G, b \in H \) . Then the following statements are equivalent:\n\n(i) The topological systems \( \left( {G;a}\right) \) and \( \left( {H... | Proof. (iii) \( \Rightarrow \) (i),(ii): If \( \Phi \) is a topological group isomorphism with \( \Phi \left( a\right) = b \), then \( \Phi \) is also an isomorphism of the dynamical systems.\n\n(i) or (ii) \( \Rightarrow \) (iv): If the dynamical systems are isomorphic, the Koopman operators on the corresponding space... | Yes |
Theorem 14.26. Consider the compact group \( G \mathrel{\text{:=}} {\mathbb{T}}^{I}, I \) a nonempty set. Then \( G \) is monothetic if and only if \( \operatorname{card}\left( I\right) \leq \operatorname{card}\left( \mathbb{T}\right) \) . | Proof. If \( G \) is monothetic, then by Proposition 14.22 we have that \( \operatorname{card}\left( {G}^{ * }\right) \leq \) \( \operatorname{card}\left( \mathbb{T}\right) \) . Since the projections \( {\pi }_{i} : {\mathbb{T}}^{I} \rightarrow \mathbb{T}, i \in I \), are all different characters of \( G \) , we obtain... | Yes |
Proposition 14.27. Let \( G \) be an Abelian group, \( H \subseteq G \) a subgroup, and let \( \psi : H \rightarrow \mathbb{T} \) be a homomorphism. Then there is a homomorphism \( \chi : G \rightarrow \mathbb{T} \) extending \( \psi \) . | Proof. Consider the following collection of pairs\n\n\[ \mathcal{M} \mathrel{\text{:=}} \left\{ {\left( {K,\varphi }\right) : H \leq K\text{ subgroup of }G,\varphi : K \rightarrow \mathbb{T}\text{ homomorphism,}{\left. \varphi \right| }_{H} = \psi }\right\} . \]\n\nThis set is partially ordered by the relation \( \left... | Yes |
Proposition 14.28. If \( G \) is a (discrete) Abelian group, then the characters separate the points of \( G \) . | Proof. Let \( x \in G, x \neq 1 \) . If there is \( n \geq 2 \) with \( {x}^{n} = 1 \), then take \( 1 \neq \alpha \in \mathbb{T} \) an \( {n}^{\text{th }} \) root of unity. If the order of \( x \) is infinite, then take any \( 1 \neq \alpha \in \mathbb{T} \) . For \( k \in \mathbb{Z} \) set \( \psi \left( {x}^{k}\righ... | Yes |
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