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Example 2.37 (Kronecker’s Theorem). The rotation \( \left( {\mathbb{T};a}\right) \) is topologically transitive if and only if \( a \in \mathbb{T} \) is not a root of unity. | Proof. If \( {a}^{{n}_{0}} = 1 \) for some \( {n}_{0} \in \mathbb{N} \), then \( \left\{ {z \in \mathbb{T} : {z}^{{n}_{0}} = 1}\right\} \) is closed and \( \varphi \) -invariant, so by Theorem 2.36, \( \left( {\mathbb{T};a}\right) \) is not transitive.\n\nFor the converse, suppose that \( a \) is not a root of unity an... | Yes |
The product of two topologically transitive systems need not be topologically transitive. | Consider \( {a}_{1} = {\mathrm{e}}^{\mathrm{i}},{a}_{2} = {\mathrm{e}}^{2\mathrm{i}} \), and the product system \( \left( {{\mathbb{T}}^{2};\left( {{a}_{1},{a}_{2}}\right) }\right) \) . Then \( M = \left\{ {\left( {x, y}\right) \in {\mathbb{T}}^{2} : {x}^{2} = y}\right\} \) is a nontrivial, closed invariant set which c... | Yes |
Lemma 2.40. For \( n \in \mathbb{N} \) one has \( {\left\lbrack {A}^{n}\right\rbrack }_{ij} > 0 \) if and only if there is \( x \in F \) with \( {x}_{0} = i \) and \( {x}_{n} = j \) . | Proof. We argue by induction on \( n \in \mathbb{N} \), the case \( n = 1 \) being clear by the definition of the transition matrix \( A \) . Suppose that the claimed equivalence is proved for \( n \geq 1 \) . The inequality \( {\left\lbrack {A}^{n + 1}\right\rbrack }_{ij} > 0 \) holds if and only if there is \( m \) s... | Yes |
Proposition 2.41. Let \( \left( {F;\tau }\right) \) be a subshift of order 2 and suppose that every letter occurs in some word in \( F \) . Consider the next assertions.\n\n(i) The transition matrix \( A \) of \( \left( {F;\tau }\right) \) is irreducible.\n\n(ii) \( \left( {F;\tau }\right) \) is forward transitive.\n\n... | Proof. (i) \( \Rightarrow \) (ii): Since \( F \) is metrizable, we can apply Proposition 2.35. It suffices to consider open sets \( U \) and \( V \) intersecting \( F \) that are of the form\n\n\[ U = \left\{ {x : {x}_{0} = {u}_{0},\ldots ,{x}_{n} = {u}_{n}}\right\} \text{ and }V = \left\{ {x : {x}_{0} = {v}_{0},\ldots... | Yes |
Proposition 3.3. For a topological system \( \left( {K;\varphi }\right) \) the following assertions are equivalent:\n\n(i) \( \left( {K;\varphi }\right) \) is minimal.\n\n(ii) \( {\operatorname{orb}}_{ + }\left( x\right) \) is dense in \( K \) for each \( x \in K \) .\n\n(iii) \( K = \mathop{\bigcup }\limits_{{n \in {\... | Proof. This is Exercise 4. | No |
Theorem 3.5. Every topological system \( \left( {K;\varphi }\right) \) has at least one minimal subsystem. | Proof. Let \( \mathcal{M} \) be the family of all nonempty closed \( \varphi \) -invariant subsets of \( K \) . Then, of course, \( K \in \mathcal{M} \), so \( \mathcal{M} \) is nonempty. Further, \( \mathcal{M} \) is ordered by set inclusion. Given a chain \( \mathcal{C} \subseteq \mathcal{M} \) the set \( C \mathrel{... | Yes |
Proposition 3.6. An isometric topological system \( \left( {K;\varphi }\right) \) is minimal if and only if it is topologically transitive. | Proof. Suppose that \( {x}_{0} \in K \) has dense forward orbit and pick \( y \in K \) . By Proposition 3.3 it suffices to prove that \( {x}_{0} \in {\overline{\operatorname{orb}}}_{ + }\left( y\right) \) . Let \( \varepsilon > 0 \) be arbitrary. Then there is \( m \in {\mathbb{N}}_{0} \) such that \( d\left( {{\varphi... | Yes |
Corollary 3.7 (“Structure Theorem” for Isometric Systems). An isometric system is a (possibly infinite) disjoint union of minimal subsystems. | Proof. By Remark 3.2.4, different minimal subsystems must be disjoint. Hence, the statement is equivalent to saying that every point in \( K \) is contained in a minimal system. But for any \( x \in K \) the system \( \left( {{\overline{\operatorname{orb}}}_{ + }\left( x\right) ;\varphi }\right) \) is topologically tra... | Yes |
c) By a) and b) a recurrent, but not uniformly recurrent point is easy to find: Let \( k = 2 \), and enumerate the words formed from the alphabet \( \{ 0,1\} \) according to the lexicographical ordering. Now write these words into one infinite word \( x \in {\mathcal{W}}_{2}^{ + } \) : | ## \(0\left| 1\right| {00}\left| {01}\right| {10}\left| {11}\right| {000}\left| {001}\right| {010}\left| {011}\right| {100}\left| {101}\right| {110}\left| {111}\right| {0000}\cdots .\) All blocks of \( x \) occur as a sub-block of later finite blocks, hence they are repeated infinitely often, and hence \( x \) is recur... | Yes |
Theorem 3.11. Let \( \left( {K;\varphi }\right) \) be a topological system and \( x \in K \) . Then the following assertions are equivalent:\n\n(i) \( x \) is uniformly recurrent.\n\n(ii) \( \left( {{\overline{\operatorname{orb}}}_{ + }\left( x\right) ;\varphi }\right) \) is minimal.\n\n(iii) \( x \) is contained in a ... | Proof. The equivalence of (ii) and (iii) is evident. Suppose that (iii) holds. Then we may suppose without loss of generality that \( \left( {K;\varphi }\right) \) is minimal. Let \( U \subseteq K \) be a nonempty open neighborhood of \( x \) . By Proposition 3.3, \( \mathop{\bigcup }\limits_{{n \in {\mathbb{N}}_{0}}}{... | Yes |
Proposition 3.12. a) If a topological system contains a forward transitive uniformly recurrent point, then it is minimal. | Proof. a) and b) are immediate from Theorem 3.11, and c) follows from Theorem 3.11 together with Corollary 3.7. | No |
Let \( \alpha \in \lbrack 0,1) \) and \( I \subseteq \lbrack 0,1) \) be an interval containing \( \alpha \) in its interior. Then the set\n\n\[
\{ n \in \mathbb{N} : {n\alpha } - \lfloor {n\alpha }\rfloor \in I\}
\]\n\nhas bounded gaps. | Proof. Consider \( \left( {\lbrack 0,1}\right) ;\alpha ) \), the translation mod 1 by \( \alpha \) (Example 2.7). It is isomorphic to the group rotation \( \left( {\mathbb{T};{\mathrm{e}}^{{2\pi }\mathrm{i}\alpha }}\right) \), so the claim follows from Proposition 3.12. | No |
Proposition 3.15. Let \( \left( {K;\varphi }\right) \) be a topological system, \( G \) a compact group, and \( \left( {H;\psi }\right) \) the group extension along some \( \Phi : K \rightarrow G \) . If \( {x}_{0} \in K \) is a recurrent point, then \( \left( {{x}_{0}, g}\right) \in H \) is recurrent in \( H \) for al... | Proof. It suffices to prove the assertions for \( g = 1 \in G \) . Indeed, for every \( g \in G \) the map \( {\rho }_{g} : H \rightarrow H,{\rho }_{g}\left( {x, h}\right) = \left( {x,{hg}}\right) \), is an automorphism of \( \left( {H;\psi }\right) \), and hence maps recurrent points to recurrent points. For every \( ... | Yes |
Proposition 3.16. Let \( \left( {K;\varphi }\right) \) be a topological system, \( G \) a compact group, and \( \left( {H;\psi }\right) \) the group extension along \( \Phi : K \rightarrow G \) . If \( {x}_{0} \in K \) is a uniformly recurrent point, then \( \left( {{x}_{0}, g}\right) \in H \) is uniformly recurrent in... | Proof. As before, it suffices to prove that \( \left( {{x}_{0}, h}\right) \) is uniformly recurrent for one \( h \in G \) . The set \( {\overline{\operatorname{orb}}}_{ + }\left( {x}_{0}\right) \) is minimal by Theorem 3.11, so by passing to a subsystem we can assume that \( \left( {K;\varphi }\right) \) is minimal. No... | Yes |
Corollary 3.17. Let \( \alpha \in \mathbb{R} \) and \( \varepsilon > 0 \) be given. Then there exists \( n \in \mathbb{N}, m \in \mathbb{Z} \) such that\n\n\[ \left| {{n}^{2}\alpha - m}\right| \leq \varepsilon \] | Proof. Consider the topological system \( \left( {\lbrack 0,1}\right) ;\alpha ) \) from Example 2.7, and recall that, endowed with the appropriate metric and with addition modulo 1 , it is a compact group isomorphic as a topological group to \( \mathbb{T} \) . We consider a group extension similar to Example 2.22. Let\... | Yes |
Proposition 3.18. Let \( p \in \mathbb{R}\left\lbrack x\right\rbrack \) be a polynomial of degree \( k \in \mathbb{N} \) with \( p\left( 0\right) = 0 \) . Then for every \( \varepsilon > 0 \) there is \( n \in \mathbb{N} \) and \( m \in \mathbb{Z} \) with\n\n\[ \left| {p\left( n\right) - m}\right| < \varepsilon \] | Proof. Start from a polynomial \( p\left( x\right) \) of degree \( k \) and define\n\n\[ {p}_{k}\left( x\right) \mathrel{\text{:=}} p\left( x\right) ,\;{p}_{k - i}\left( x\right) \mathrel{\text{:=}} {p}_{k - i + 1}\left( {x + 1}\right) - {p}_{k - i + 1}\left( x\right) \;\left( {i = 1,\ldots, k}\right) . \]\n\nThen each... | Yes |
Lemma 4.1. Let \( A, B \) be disjoint closed subsets of a compact space \( K \) . Then there are disjoint open sets \( U, V \subseteq K \) with \( A \subseteq U \) and \( B \subseteq V \) ; or, equivalently, \( A \subseteq U \) and \( \bar{U} \cap B = \varnothing \) . | Proof. Let \( x \in A \) be fixed. For every \( y \in B \) there are disjoint open neighborhoods \( U\left( {x, y}\right) \) of \( x \) and \( V\left( {x, y}\right) \) of \( y \) . Finitely many of the \( V\left( {x, y}\right) \) cover \( B \) by compactness, i.e., \( B \subseteq V\left( {x,{y}_{1}}\right) \cup \cdots ... | Yes |
Theorem 4.3 (Tietze). Let \( K \) be a compact space, let \( A \subseteq K \) be closed, and let \( f \in \mathrm{C}\left( A\right) \) . Then there is \( g \in \mathrm{C}\left( K\right) \) such that \( {\left. g\right| }_{A} = f \) . | Proof. The real and imaginary parts of a continuous function are continuous, hence it suffices to consider the case that \( f \) is real-valued. Then, since \( A \) is compact, \( f\left( A\right) \) is a compact subset of \( \mathbb{R} \), and by scaling and shifting we may suppose that \( f\left( A\right) \subseteq \... | Yes |
Theorem 4.4 (Stone-Weierstraß). Let \( A \) be a complex conjugation invariant subalgebra of \( \mathrm{C}\left( K\right) \) containing the constant functions and separating the points of \( K \) . Then \( A \) is dense in \( \mathrm{C}\left( K\right) \) . | For the proof we note that the closure \( \bar{A} \) of \( A \) also satisfies the hypotheses of the theorem, hence we may suppose without loss of generality that \( A \) is closed. The next result is the key to the proof. | Yes |
Proposition 4.5. Let \( A \) be a closed conjugation invariant subalgebra of \( \mathrm{C}\left( K\right) \) containing the constant function 1 . Then any positive function \( f \in A \) has a unique real square root \( g \in A \), i.e., \( f = {g}^{2} = \bar{g} \cdot g \) . | Proof. We prove that the square root of \( f \), defined pointwise, belongs to \( A \) . By normalizing first we can assume \( \parallel f{\parallel }_{\infty } \leq 1 \) . Recall that the binomial series\n\n\[ \n{\left( 1 + x\right) }^{\frac{1}{2}} = \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( \begin{array}{l} \fr... | No |
Lemma 4.6. Each compact metric space is separable. | Proof. For fixed \( m \in \mathbb{N} \) the balls \( \mathrm{B}\left( {x,\frac{1}{m}}\right), x \in K \), cover \( K \), so there is a finite set \( {F}_{m} \subseteq K \) such that\n\n\[ K \subseteq \mathop{\bigcup }\limits_{{x \in {F}_{m}}}B\left( {x,\frac{1}{m}}\right) \]\n\nThen the set \( F \mathrel{\text{:=}} \ma... | Yes |
Theorem 4.7. A compact topological space \( K \) is metrizable if and only if \( \mathrm{C}\left( K\right) \) is separable. | Proof. Suppose that \( \mathrm{C}\left( K\right) \) is separable, and let \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence in \( \mathrm{C}\left( K\right) \) such that \( \left\{ {{f}_{n} : n \in \mathbb{N}}\right\} \) is dense in \( \mathrm{C}\left( K\right) \) . Define\n\n\[ \Phi : K \rightarrow \Omega... | Yes |
Theorem 4.8. Let \( I \subseteq \mathrm{C}\left( K\right) \) be a closed algebra ideal. Then there is a closed subset \( F \subseteq K \) such that \( I = {I}_{F} \) . | Proof. Define\n\n\[ F \mathrel{\text{:=}} \{ x \in K : f\left( x\right) = 0\text{ for all }f \in I\} = \mathop{\bigcap }\limits_{{f \in I}}\left\lbrack {f = 0}\right\rbrack .\n\]\n\nObviously, \( F \) is closed and \( I \subseteq {I}_{F} \) . Fix \( f \in {I}_{F},\varepsilon > 0 \) and define \( {F}_{\varepsilon } \mat... | Yes |
Lemma 4.9. An ideal \( I \) of \( \mathrm{C}\left( K\right) \) is maximal if and only if \( I = {I}_{\{ x\} } \) for some \( x \in K \) . | Proof. It is straightforward to see that each \( {I}_{\{ x\} }, x \in K \), is a maximal ideal. Suppose conversely that \( I \) is a maximal ideal. By Theorem 4.8 it suffices to show that \( I \) is closed. Since \( \bar{I} \) is again an ideal and \( I \) is maximal, \( \bar{I} = I \) or \( \bar{I} = \mathrm{C}\left( ... | Yes |
Lemma 4.10. A nonzero linear functional \( \psi : \mathrm{C}\left( K\right) \rightarrow \mathbb{C} \) is multiplicative if and only if \( \psi = {\delta }_{x} \) for some \( x \in K \) . | Proof. Let \( \gamma : \mathrm{C}\left( K\right) \rightarrow \mathbb{C} \) be nonzero and multiplicative. Then there is \( f \in \mathrm{C}\left( K\right) \) such that \( \gamma \left( f\right) = 1 \) . Hence\n\n\[ 1 = \gamma \left( f\right) = \gamma \left( {\mathbf{1}f}\right) = \gamma \left( \mathbf{1}\right) \gamma ... | Yes |
Lemma 4.12. Let \( K \) be a compact space, \( \Omega \) a topological space, and let \( \varphi : \Omega \rightarrow \) \( K \) be a mapping. Then \( \varphi \) is continuous if and only if \( f \circ \varphi \) is continuous for all \( f \in \mathrm{C}\left( K\right) \) . | Proof. Clearly, if \( \varphi \) is continuous, then also \( f \circ \varphi \) is continuous for every \( f \in \mathrm{C}\left( K\right) \) . Conversely, if this condition holds, then \( {\varphi }^{-1}\left\lbrack {\left| f\right| > 0}\right\rbrack = \left\lbrack {\left| {f \circ \varphi }\right| > 0}\right\rbrack \... | Yes |
Theorem 4.13. Let \( K \), L be (nonempty) compact spaces and let \( T : \mathrm{C}\left( K\right) \rightarrow \mathrm{C}\left( L\right) \) be linear. Then the following assertions are equivalent:\n\n(i) \( T \) is an algebra homomorphism.\n\n(ii) There is a continuous mapping \( \varphi : L \rightarrow K \) such that ... | Proof. Urysohn’s lemma yields that \( \varphi \) as in (ii) is uniquely determined, and it is clear from (ii) that \( \parallel T\parallel = 1 \) . For the proof of the implication (i) \( \Rightarrow \) (ii) take \( y \in L \) . Then\n\n\[ {T}^{\prime }{\delta }_{y} \mathrel{\text{:=}} {\delta }_{y} \circ T : \mathrm{C... | Yes |
Lemma 4.14. Let \( K, L \) be compact spaces, and let \( \varphi : L \rightarrow K \) be continuous, with Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } : \mathrm{C}\left( K\right) \rightarrow \mathrm{C}\left( L\right) \) . Then the following hold:\n\na) \( \varphi \) is surjective if and only if \( T \) is i... | Proof. This is Exercise 1. | No |
Lemma 4.16. Let \( \Omega, K \) be topological spaces, \( K \) compact. For a mapping \( \Phi \) : \( \Omega \times K \rightarrow \mathbb{C} \) the following assertions are equivalent:\n\n(i) \( \Phi \) is continuous.\n\n(ii) For each \( x \in \Omega \) the mapping \( \widetilde{\Phi }\left( x\right) \mathrel{\text{:=}... | Proof. (i) \( \Rightarrow \) (ii): Fix \( x \in \Omega \) and \( \varepsilon > 0 \) . For each \( y \in K \) there are open sets \( {U}_{y} \subseteq \Omega \) and \( {V}_{y} \subseteq K \) with \( x \in {U}_{y}, y \in {V}_{y} \) such that\n\n\[ \left| {\Phi \left( {x, y}\right) - \Phi \left( {{x}^{\prime },{y}^{\prime... | Yes |
Theorem 4.17. Let \( \Omega, K, L \) be topological spaces, and suppose that \( K \) and \( L \) are compact. Let \( \Phi : \Omega \times K \rightarrow L \) be a mapping such that for every \( x \in \Omega \) the mapping \( \Phi \left( {x, \cdot }\right) : K \rightarrow L \) is continuous. Let \( {T}_{x} : \mathrm{C}\l... | Proof. Note that \( \Phi \) is continuous if and only if for every \( f \in \mathrm{C}\left( L\right) \) the mapping \( f \circ \Phi \) is continuous. This follows from Lemma 4.12. Hence, the assertion is a direct consequence of Lemma 4.16 above. | No |
Lemma 4.18. Let \( \\left( {K;\\varphi }\\right) \) be a topological system with Koopman operator \( T = {T}_{\\varphi } \) and let \( A \\subseteq K \) be a closed subset. Then \( A \) is \( \\varphi \) -invariant if and only if the ideal \( {I}_{A} \) is \( T \) -invariant. | Proof. Suppose that \( A \) is \( \\varphi \) -invariant and \( f \\in {I}_{A} \) . If \( x \\in A \), then \( \\varphi \\left( x\\right) \\in A \) and hence \( \\left( {Tf}\\right) \\left( x\\right) = f\\left( {\\varphi \\left( x\\right) }\\right) = 0 \) since \( f \) vanishes on \( A \) . Thus \( {Tf} \) vanishes on ... | Yes |
Lemma 4.20. Let \( \\left( {K;\\varphi }\\right) \) be a topological system with Koopman operator \( T = {T}_{\\varphi } \) on \( \\mathrm{C}\\left( K\\right) \). If \( \\left( {K;\\varphi }\\right) \) is topologically transitive, then \( \\mathrm{{fix}}\\left( T\\right) \) is one-dimensional. | Proof. As already remarked, if \( x \\in K \) and \( f \\in \\operatorname{fix}\\left( T\\right) \), then \( f\\left( {{\\varphi }^{n}\\left( x\\right) }\\right) = \\left( {{T}^{n}f}\\right) \\left( x\\right) = \) \( f\\left( x\\right) \) is independent of \( n \\geq 0 \), and hence \( f \) is constant on \( {\\overlin... | Yes |
Theorem 4.21. Let \( \left( {K;\varphi }\right) \) be a topological system with Koopman operator \( T = {T}_{\varphi } \). Then the peripheral point spectrum of \( T \) is a union of subgroups of \( \mathbb{T} \). If \( \operatorname{fix}\left( T\right) \) is one-dimensional, then the peripheral point spectrum is a gro... | Proof. Let \( \lambda \in \mathbb{T} \) be an eigenvalue of \( T \), and let \( 0 \neq f \in \mathrm{C}\left( K\right) \) be a corresponding eigenvector with \( \parallel f{\parallel }_{\infty } = 1 \). Then for each \( n \in \mathbb{N} \) we have that \( {\lambda }^{n} \) is an eigenvalue with eigenvector \( {f}^{n} \... | Yes |
In order to determine this subgroup, take \( \lambda \in \mathbb{T} \) and \( \chi \in \mathrm{C}\left( G\right) \) such that \( \left| \chi \right| = \mathbf{1} \) and \( {L}_{a}\chi = {\lambda \chi } \), i.e., \( \chi \left( {ax}\right) = {\lambda \chi }\left( x\right) \) for all \( x \in G \) . Without loss of gener... | \[ \chi \left( {{a}^{n}{a}^{m}}\right) = \chi \left( {a}^{n + m}\right) = {\lambda }^{n + m} = {\lambda }^{n}{\lambda }^{m} = \chi \left( {a}^{n}\right) \chi \left( {a}^{m}\right) \] for all \( n, m \in {\mathbb{N}}_{0} \) . By continuity of \( \chi \) and since the powers of \( a \) are dense in \( G \), this implies ... | Yes |
Proposition 4.26. Let \( A \) be a commutative complex Banach algebra. If \( \psi \in \Gamma \left( A\right) \) , then \( \psi \) is continuous with \( \parallel \psi \parallel \leq 1 \) . | Proof. Suppose by contradiction that there is \( a \in A \) with \( \parallel a\parallel < 1 \) and \( \psi \left( a\right) = 1 \) . The series\n\n\[ b \mathrel{\text{:=}} \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}^{n} \]\n\nis absolutely convergent in \( A \) with \( {ab} + a = b \) . Since \( \psi \) is a homomorph... | Yes |
Lemma 4.27. a) For every \( a \in A \) we have \( r\left( a\right) \leq \parallel a\parallel \). | Proof. a) This follows from the submultiplicativity of the norm. | No |
Proposition 4.29. Let \( A \) be a complex Banach algebra and let \( a \in A \) . Then the following are true:\n\na) The spectrum \( \operatorname{Sp}\left( a\right) \) is a compact subset of \( \mathbb{C} \) and is contained in the closed ball \( \overline{\mathrm{B}}\left( {0, r\left( a\right) }\right) \). | Proof. a) If \( \lambda \in \mathbb{C} \) is such that \( \left| \lambda \right| > r\left( a\right) \), then Lemma 4.28 implies that \( \lambda \mathrm{e} - a \) is invertible, hence \( \operatorname{Sp}\left( a\right) \subseteq \overline{\mathrm{B}}\left( {0, r\left( a\right) }\right) \) . We now show that the complem... | Yes |
Theorem 4.30 (Gelfand-Mazur). Let \( A \neq \{ 0\} \) be a complex Banach algebra such that every nonzero element in \( A \) is invertible. Then \( A \) is isomorphic to \( \mathbb{C} \) . | Proof. Let \( a \in A \) . Then by Proposition 4.29 there is \( {\lambda }_{a} \in \operatorname{Sp}\left( a\right) \) with \( \left| {\lambda }_{a}\right| = r\left( a\right) \) . By assumption, \( {\lambda }_{a}\mathrm{e} - a = 0 \), so \( a = {\lambda }_{a}\mathrm{e} \), hence \( A \) is one-dimensional. This proves ... | Yes |
Proposition 4.31. Let \( A \) be a commutative complex Banach algebra. If \( \psi \in \Gamma \left( A\right) \) , then \( \ker \left( \psi \right) \) is a maximal ideal. Conversely, if \( I \subseteq A \) is a maximal ideal, then \( I \) is closed and there is a unique \( \psi \in \Gamma \left( A\right) \) such that \(... | Proof. Clearly, \( \ker \left( \psi \right) \) is a closed ideal. Since \( \ker \left( \psi \right) \) is of codimension one, it must be maximal.\n\nFor the second assertion let \( I \) be a maximal ideal. Consider its closure \( \bar{I} \), still an ideal. Since \( \mathrm{B}\left( {\mathrm{e},1}\right) \) consists of... | Yes |
Theorem 4.32. Let \( A \) be a commutative unital Banach algebra and let \( a \in A \) . Then\n\n\[ \operatorname{Sp}\left( a\right) = \{ \psi \left( a\right) : \psi \in \Gamma \left( A\right) \} = \widehat{a}\left( {\Gamma \left( A\right) }\right) . \] | Proof. Since \( \psi \left( \mathrm{e}\right) = 1 \) for \( \psi \in \Gamma \left( A\right) \), one has \( \psi \left( {\psi \left( a\right) \mathrm{e} - a}\right) = 0 \), so \( \psi \left( a\right) \mathrm{e} - a \) cannot be invertible, i.e., \( \psi \left( a\right) \in \operatorname{Sp}\left( a\right) \) .\n\nOn the... | Yes |
Lemma 4.34. Let \( A \) be a commutative \( {C}^{ * } \) -algebra. For \( a \in A \) with \( a = {a}^{ * } \) the following assertions are true:\n\na) \( r\left( a\right) = \parallel a\parallel \) .\n\nb) \( \operatorname{Sp}\left( a\right) \subseteq \mathbb{R} \) ; equivalently, \( \psi \left( a\right) \in \mathbb{R} ... | Proof. a) We have \( a{a}^{ * } = {a}^{2} \), so \( \begin{Vmatrix}{a}^{2}\end{Vmatrix} = \parallel a{\parallel }^{2} \) holds. By induction one can prove \( \begin{Vmatrix}{a}^{{2}^{n}}\end{Vmatrix} = \parallel a{\parallel }^{{2}^{n}} \) for all \( n \in {\mathbb{N}}_{0} \) . From this we obtain\n\n\[ r\left( a\right)... | Yes |
Lemma 4.35. Let \( A \) be a commutative \( {C}^{ * } \) -algebra and let \( \psi \in \Gamma \left( A\right) \) . Then\n\n\[ \psi \left( {a}^{ * }\right) = \overline{\psi \left( a\right) }\;\text{ for all }a \in A, \]\n\ni.e., \( \psi \) is a \( * \) -homomorphism. | Proof. For \( a \in A \) fixed define\n\n\[ x \mathrel{\text{:=}} \frac{a + {a}^{ * }}{2}\;\text{ and }\;y \mathrel{\text{:=}} \frac{b - {b}^{ * }}{2\mathrm{i}}. \]\n\nThen we have \( x = {x}^{ * } \) and \( y = {y}^{ * } \) and \( a = x + \mathrm{i}y \) . By Lemma 4.34.b, \( \psi \left( x\right) ,\psi \left( y\right) ... | Yes |
Lemma 5.3. Every finite positive Baire measure is regular in the sense that for every \( B \in \operatorname{Ba}\left( K\right) \) one has\n\n\[ \mu \left( B\right) = \sup \{ \mu \left( A\right) : A \in \mathrm{{Ba}}\left( K\right), A\text{ compact,}A \subseteq B\} \]\n\n\[ = \inf \{ \mu \left( O\right) : O \in \operat... | Proof. This is a standard argument involving Dynkin systems (Theorem B.1); for a different proof, see Bogachev (2007, II, 7.1.8). | No |
Proposition 5.4. If \( \mu, v \) are regular finite positive Borel measures on \( K \), then to each Borel set \( A \in \operatorname{Bo}\left( K\right) \) there is a Baire set \( B \in \operatorname{Ba}\left( K\right) \) such that\n\n\[ \mu \left( {A\bigtriangleup B}\right) = 0 = v\left( {A\bigtriangleup B}\right) . \... | Proof. Let \( A \subseteq K \) be a Borel set. By regularity, there are open sets \( {O}_{n},{O}_{n}^{\prime } \), closed sets \( {L}_{n},{L}_{n}^{\prime } \) such that \( {L}_{n},{L}_{n}^{\prime } \subseteq A \subseteq {O}_{n},{O}_{n}^{\prime } \) and \( \mu \left( {{O}_{n} \smallsetminus {L}_{n}}\right), v\left( {{O}... | Yes |
Lemma 5.5. If \( \mu, v \in \mathrm{M}\left( K\right) \) with \( {\int }_{K}f\mathrm{\;d}\mu = {\int }_{K}f\mathrm{\;d}v \) for all \( f \in \mathrm{C}\left( K\right) \), then \( \mu = v \) . | Proof. By passing to \( \mu - v \) we may suppose that \( v = 0 \) . By Exercise 8.a and standard measure theory it suffices to prove that \( \mu \left( A\right) = 0 \) for each compact \( {G}_{\delta } \) -subset \( A \) of \( K \) . Given such a set, one can find open subsets \( {O}_{n} \) of \( K \) such that \( {O}... | Yes |
Theorem 5.7 (Riesz’ Representation Theorem). Let \( K \) be a compact space. Then the mapping\n\n\[ \n\mathrm{M}\left( K\right) \rightarrow \mathrm{C}{\left( K\right) }^{\prime },\;\mu \mapsto \langle \cdot ,\mu \rangle \n\]\n\nis an isometric isomorphism. | For the convenience of the reader, we have included a proof in Appendix E; see also Rudin (1987, 2.14) or Lang (1993, IX.2). | No |
Proposition 5.9. Let \( 0 \leq \mu \in \mathrm{M}\left( K\right) \) . Then\n\n\[ \operatorname{supp}\left( \mu \right) = \{ x \in K : \mu \left( U\right) > 0\text{ for each open neighborhood }U\text{ of }x\} . \] | Proof. Let \( M \mathrel{\text{:=}} \operatorname{supp}\left( \mu \right) \) and let \( L \) denote the right-hand side of (5.3). Let \( x \in M \) and \( U \) be an open neighborhood of \( x \) . By Urysohn’s lemma there is \( f \in \mathrm{C}\left( K\right) \) such that \( x \in \left\lbrack {f \neq 0}\right\rbrack \... | Yes |
Theorem 5.10 (Krylov-Bogoljubov). Let \( \left( {K;\varphi }\right) \) be a topological system. Then there is at least one \( \varphi \) -invariant Baire probability measure on \( K \) . | Proof. We postpone the proof of this theorem to Chapter 10, see Theorem 10.2. Note however that under the identification \( \mathrm{M}\left( K\right) = \mathrm{C}{\left( K\right) }^{\prime } \) from above, the \( \varphi \) -invariance of \( \mu \) just means that \( {T}_{\varphi }^{\prime }\left( \mu \right) = \mu \),... | No |
Consider the rotation topological system \( \left( {\mathbb{T};a}\right) \) for some \( a \in \mathbb{T} \). Obviously, the normalized arc-length measure is invariant. If \( a \) is an \( {n}^{\text{th }} \) root of unity, then the convex combination of point measures | \[ \mu \mathrel{\text{:=}} \frac{1}{n}\mathop{\sum }\limits_{{j = 1}}^{n}{\delta }_{{a}^{j}} \] is another invariant probability measure, see also Exercise 5. | No |
We claim that for every Borel set \( M \subseteq G/\Gamma \) \n\n\[ \n\lambda \left( {A \cap {q}^{-1}\left( M\right) }\right) = \lambda \left( {B \cap {q}^{-1}\left( M\right) }\right) .\n\] | Indeed, by the invariance of \( \lambda \) and since \( {q}^{-1}\left( M\right) h = {q}^{-1}\left( M\right) \) for all \( h \in \Gamma \) \n\n\[ \n\lambda \left( {A \cap {q}^{-1}\left( M\right) }\right) = \lambda \left( {A \cap {q}^{-1}\left( M\right) \cap {\mathbb{R}}^{3}}\right) = \lambda \left( {A \cap {q}^{-1}\left... | Yes |
Theorem 5.18. Let \( G \) be a locally compact group, and let \( \Gamma \) be an unimodular, closed and cocompact subgroup of \( G \) . Then \( G \) is unimodular and there is a unique Baire probability measure \( \mathrm{m} \) on \( G/\Gamma \) that is invariant under all rotations by elements of \( G \) . | For the proof we need some auxiliary results. Integration against the measures \( {\mathrm{m}}_{G} \) and \( {\mathrm{m}}_{\Gamma } \) on \( G \) and \( \Gamma \) is denoted by \( \mathrm{d}x \) and \( \mathrm{d}y \), respectively. The modular function on \( G \) is abbreviated by \( \Delta \) . We identify continuous ... | No |
Lemma 5.19. In the situation described above, the following assertions hold:\n\na) \( \Phi : {\mathrm{C}}_{\mathrm{c}}\left( G\right) \rightarrow \mathrm{C}\left( {G/\Gamma }\right) \) is linear and surjective. More precisely, there is a positive linear operator \( \Psi : \mathrm{C}\left( {G/\Gamma }\right) \rightarrow... | Proof. a) Linearity is clear. By Exercise 2.16 there is a compact subset \( K \subseteq G \) with \( {K\Gamma } = G \) . By Exercise 14.b we can find a function \( 0 \leq h \in {\mathrm{C}}_{\mathrm{c}}\left( G\right) \) with \( K \subseteq \left\lbrack {h = 1}\right\rbrack \) . Then \( \left( {\Phi h}\right) \left( x\... | No |
Lemma 6.1 (Approximation). Let \( \left( {X,\sum ,\mu }\right) \) be a finite measure space and let \( \mathcal{E} \subseteq \) \( \sum \) be an algebra of subsets such that \( \sigma \left( \mathcal{E}\right) = \sum \) . Then \( \mathcal{E} \) is dense in the measure algebra, i.e., for every \( A \in \sum \) and \( \v... | Proof. This is just Lemma B. 17 from Appendix B. Its proof is standard measure theory using Dynkin systems. | No |
Example 6.7. Let \( X \mathrel{\text{:=}} \{ 0,1\} \) with the trivial \( \sigma \) -algebra \( \sum \mathrel{\text{:=}} \{ \varnothing, X\} \) and the unique probability measure thereon. Consider the measure-preserving mappings \( \varphi \) and \( \psi \) on \( X \) defined by\n\n\[ \varphi \left( x\right) \mathrel{\... | We also note that the map \( \varphi \) in Example 6.7 does not have an essential inverse although its induced map \( {\varphi }^{ * } \) is invertible. Hence, the converse of Corollary 6.5 does not hold in general. | No |
Proposition 6.10. Let \( \varphi ,\psi : \mathrm{X} \rightarrow \mathrm{Y} \) be measure-preserving maps between probability spaces \( \mathrm{X} \) and \( \mathrm{Y} \) such that \( {\varphi }^{ * } = {\psi }^{ * } \) . If \( \mathrm{Y} \) is a standard probability space, then \( \varphi = \psi \) almost everywhere. | Proof. Note that if \( \mathrm{Y} \) is a Borel probability space then the hypotheses of Lemma 6.9 are satisfied since there is countable collection of open balls separating the points. In the general case we can find a Borel probability space \( {\mathrm{Y}}^{\prime } \) and an essentially invertible measure-preservin... | Yes |
Lemma 6.12. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system. Then the following statements are equivalent:\n\n(i) Every \( A \in {\sum }_{\mathrm{X}} \) is recurrent.\n\n(ii) Every \( A \in {\sum }_{\mathrm{X}} \) is infinitely recurrent.\n\n(iii) For every \( \varnothing \neq A \in \sum \l... | Proof. The implication (ii) \( \Rightarrow \) (i) is evident. For the converse take \( A \in {\sum }_{\mathrm{X}} \) and apply \( {\varphi }^{ * } \) to (6.1) to obtain \( {\varphi }^{ * }A \subseteq \mathop{\bigcup }\limits_{{n \geq 2}}{\varphi }^{*n}A \) . Inserting this back into (6.1) yields\n\n\[ A \subseteq \math... | Yes |
Theorem 6.13 (Poincaré). Every measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is (infinitely) recurrent, i.e., every set \( A \in {\sum }_{\mathrm{X}} \) is infinitely recurrent. | Proof. Let \( A \in \sum \left( \mathrm{X}\right) \) be such that \( A \cap {\varphi }^{*n}A = \varnothing \) for all \( n \geq 1 \) . Thus for \( n > m \geq 0 \) we have\n\n\n\nFig. 6.1 What happens after removing t... | Yes |
Lemma 6.15. Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a measure-preserving system, \( \\mathrm{X} = \\left( {X,\\sum ,\\mu }\\right) \), and let \( A, B \\in \\sum \) . Then, for \( n \\geq 1 \) , | Proof. Using the \( \\varphi \) -invariance of \( \\mu \) we write\n\n\[ \n\\mu \\left( B\\right) = \\mu \\left( {{\\varphi }^{ * }B}\\right) = \\mu \\left( {A \\cap {\\varphi }^{ * }B}\\right) + \\mu \\left( {{A}^{\\mathrm{c}} \\cap {\\varphi }^{ * }B}\\right) ,\n\]\n\nand this is (6.4) when \( n = 1 \) and with \( X ... | Yes |
Theorem 6.16. Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a measure-preserving system, \( \\mathrm{X} = \\left( {X,\\sum ,\\mu }\\right) \), and let \( A \\in \\sum \) with \( \\mu \\left( A\\right) > 0 \) . Then\n\n\[ \n{\\int }_{A}{n}_{A}\\mathrm{\\;d}{\\mu }_{A} = \\frac{\\mu \\left( {\\mathop{\\bigcup }\\l... | Proof. We specialize \( B = X \) in Lemma 6.15. Note that\n\n\[ \nA \\cap \\mathop{\\bigcap }\\limits_{{j = 1}}^{{k - 1}}{\\varphi }^{*j}{A}^{\\mathrm{c}} = \\mathop{\\bigcup }\\limits_{{j = k}}^{\\infty }{A}_{j}\\;\\left( {k \\geq 1}\\right)\n\]\n\nsince \( A \) is recurrent by Poincaré’s theorem. Hence, by (6.4) in L... | Yes |
Lemma 6.19. For a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) the following statements are equivalent:\n\n(i) The measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is ergodic.\n\n(ii) For every \( \varnothing \neq A \in \sum \left( \mathrm{X}\right) \) one has\n\n\[ \mathop{\bi... | Proof. (i) \( \Rightarrow \) (ii): For a set \( A \in {\sum }_{\mathrm{X}} \) and \( n \geq 0 \) the set\n\n\[ {A}^{\left( n\right) } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{k \geq n}}{\varphi }^{*k}A \]\n\nsatisfies \( {\varphi }^{ * }{A}^{\left( n\right) } \subseteq {A}^{\left( n\right) } \) and hence is an in... | Yes |
Proposition 6.20. Let \( \left( {{\mathcal{W}}_{k}^{ + },\sum ,\mu ;\tau }\right) = B\left( {{p}_{0},\ldots ,{p}_{k - 1}}\right) \) be a Bernoulli shift. Then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mu \left( {{\tau }^{*n}A \cap B}\right) = \mu \left( A\right) \mu \left( B\right) \]\n\nfor all \( A, B \in... | Proof. We use the notation of Example 5.1.5. Let \( \mathcal{E} \) denote the algebra of cylinder sets on \( {\mathcal{W}}_{k}^{ + } = {L}^{{\mathbb{N}}_{0}} \) . If \( B \in \mathcal{E} \), then \( B = {B}_{0} \times \mathop{\prod }\limits_{{k \geq {n}_{0}}}L \) for some \( {n}_{0} \in \mathbb{N} \) and \( {B}_{0} \su... | No |
Corollary 6.22 (Kac). Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be an ergodic measure-preserving system, \( X = \\left( {X,\\sum ,\\mu }\\right) \), and \( A \\in \\sum \) with \( \\mu \\left( A\\right) > 0 \) . Then for the expected return time to A one has\n\n\[ \n{\\int }_{A}{n}_{A}\\mathrm{\\;d}{\\mu }_{A} ... | Proof. Since the system is ergodic, the implication (i) \( \\Rightarrow \) (iii) of Lemma 6.19 shows that \( X = \\mathop{\\bigcup }\\limits_{{n \\geq 0}}{\\varphi }^{*n}A \) . Hence, the claim follows from Theorem 6.16. | No |
Theorem 6.23. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system, \( \mathrm{X} = \left( {X,\sum ,\mu }\right) \), and let \( A \in \sum \) be a set of positive measure. Then the induced transformation \( {\varphi }_{A} \) is measurable with respect to \( {\sum }_{A} \) and preserves the induc... | Proof. Take \( B \in \sum, B \subseteq A \) . Then\n\n\[ \left\lbrack {{\varphi }_{A} \in B}\right\rbrack = \mathop{\bigcup }\limits_{{n \geq 1}}{A}_{n} \cap \left\lbrack {{\varphi }^{n} \in B}\right\rbrack \]\n\nshowing that \( {\varphi }_{A} \) is indeed \( {\sum }_{A} \) -measurable. To see that \( {\varphi }_{A} \)... | Yes |
Corollary 6.25. Let \( \left( {\mathrm{X};\varphi }\right) \) be an ergodic measure-preserving system such that there are sets with arbitrarily small positive measure. Then for each \( \varepsilon > 0 \) and \( n \in \mathbb{N} \) there is a Rokhlin tower \( B,{\varphi }^{ * }B,{\varphi }^{*2}B,\ldots ,{\varphi }^{*\le... | Proof of Theorem 6.24. Define\n\n\[ {A}_{0} \mathrel{\text{:=}} A,\;{A}_{k + 1} \mathrel{\text{:=}} {\varphi }^{ * }{A}_{k} \cap {A}^{\mathrm{c}}\;\left( {k \geq 0}\right) . \]\n\nThen \( {A}_{n} = {\varphi }^{*n}A \cap \mathop{\bigcap }\limits_{{j = 0}}^{{n - 1}}{\varphi }^{*j}{A}^{\mathrm{c}} \) is the set of points ... | Yes |
Lemma 7.5. Let \( E, F \) be Banach lattices and let \( S : E \rightarrow F \) be a positive operator. Then the following assertions hold:\n\na) \( f \leq g\; \Rightarrow \;{Sf} \leq {Sg} \) for all \( f, g \in {E}_{\mathbb{R}} \) . | Proof. a) follows from linearity of \( S \) . | No |
Theorem 7.6. Let \( \mathrm{X} \) be a measure space and let \( 1 \leq p < \infty \) . Let \( \mathcal{F} \subseteq {\mathrm{L}}_{ + }^{p}\left( \mathrm{X}\right) \) be a \( \vee \) -stable set such that\n\n\[ s \mathrel{\text{:=}} \sup \left\{ {\parallel f{\parallel }_{p} : f \in \mathcal{F}}\right\} < \infty . \]\n\n... | Proof. Take a sequence \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) in \( \mathcal{F} \) with \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{p} \rightarrow s \) . By passing to the sequence \( {\left( {f}_{1} \vee {f}_{2} \vee \cdots \vee {f}_{n}\right) }_{n \in \mathbb{N}} \) we may suppose that \( {\left( {f}_{n}\rig... | Yes |
Corollary 7.8. Let \( \mathrm{X} \) be a measure space and let \( 1 \leq p < \infty \) . Then the Banach lattice \( {\mathrm{L}}^{p}\left( {\mathrm{X};\mathbb{R}}\right) \) is order complete. | Proof. Let \( {\mathcal{F}}^{\prime } \subseteq {\mathrm{L}}^{p}\left( {\mathrm{X};\mathbb{R}}\right) \) and suppose that there exists \( F \in {\mathrm{L}}^{p}\left( {\mathrm{X};\mathbb{R}}\right) \) with \( f \leq F \) for all \( f \in {\mathcal{F}}^{\prime } \) . We may suppose without loss of generality that \( {\m... | Yes |
Theorem 7.10. Let \( \mathrm{X} \) be a finite measure space and \( 1 \leq p < \infty \) . Then each closed lattice ideal \( I \subseteq {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) has the form \( {I}_{A} \) for some \( A \in {\sum }_{\mathrm{X}} \) . | Proof. Let \( I \subseteq {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) be a closed lattice ideal. The set\n\n\[ J \mathrel{\text{:=}} \{ f \in I : 0 \leq f \leq 1\} \]\n\nis nonempty, closed, \( \vee \) -stable and has upper bound \( \mathbf{1} \in {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) (since \( {\mu }_{\mathrm{X}... | Yes |
Proposition 7.12. A measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) is invertible if and only if its Koopman operator \( {T}_{\varphi } \) is invertible on \( {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) for one/each \( 1 \leq p \leq \infty \) . | Proof. Fix \( 1 \leq p \leq \infty \) and abbreviate \( T \mathrel{\text{:=}} {T}_{\varphi } \) . Let \( \left( {\mathrm{X};\varphi }\right) \) be invertible, i.e., \( {\varphi }^{ * } \) is surjective (Definition 6.2). Since \( T{\mathbf{1}}_{A} = {\mathbf{1}}_{{\varphi }^{ * }A} \) for any \( A \in \sum \left( \mathr... | Yes |
Consider the space \( {\mathrm{L}}^{1}\left( {\{ 0,\ldots, n - 1\} }\right) = {\mathbb{R}}^{n} \) and a positive operator \( T \) on it, identified with its \( n \times n \) -matrix. Then the irreducibility of \( T \) according to Definition 7.13 coincides with that notion introduced in Section 2.4 on page 27. Namely, ... | After a permutation of the points we may suppose that \( A = \{ k,\ldots, n - 1\} \) for \( 0 < k < n \), and this means that the representing matrix (with respect to the canonical basis) has the form:  | Yes |
Proposition 7.15. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system with Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) on \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) . Then for \( 1 \leq p \leq \infty \) the space \( \operatorname{fix}\left( T\right) \cap {\mathrm{L}}^{p} \) ... | Proof. That \( \operatorname{fix}\left( T\right) \cap {\mathrm{L}}^{p} \) is a Banach sublattice of \( {\mathrm{L}}^{p} \) is clear from the identities\n\n\[ T\left( \bar{f}\right) = \overline{Tf} = \bar{f}\;\text{ and }\;T\left| f\right| = \left| {Tf}\right| = \left| f\right| \]\n\nfor \( f \in \operatorname{fix}\left... | No |
Proposition 7.16. For \( a \in \mathbb{T} \) the rotation measure-preserving system \( \left( {\mathbb{T},\mathrm{m};a}\right) \) is ergodic if and only if \( a \) is not a root of unity. | Proof. Let \( T \mathrel{\text{:=}} {L}_{a} \) be the Koopman operator on \( {\mathrm{L}}^{2}\left( \mathbb{T}\right) \) (cf. Example 4.22) and suppose that \( f \in \operatorname{fix}\left( T\right) \) . The functions \( {\chi }_{n} : x \mapsto {x}^{n}, n \in \mathbb{Z} \), form a complete orthonormal system in \( {\m... | Yes |
Proposition 7.18. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system with Koopman operator \( {T}_{q} \mathrel{\text{:=}} {T}_{\varphi } \) on \( {\mathrm{L}}^{q}\left( \mathrm{X}\right) \left( {1 \leq q \leq \infty }\right) \). Then the following assertions hold:\n\na) \( \ker \left( {\lambda... | Proof. a) Fix \( \lambda \in \mathbb{T} \) and \( f \in {\mathrm{L}}^{q} \) with \( {Tf} = {\lambda f} \). Then \( \left| f\right| = \left| {\lambda f}\right| = \left| {Tf}\right| = T\left| f\right| \), whence \( \left| f\right| \in \operatorname{fix}\left( T\right) \). Hence, for any \( n \geq 0,{g}_{n} \mathrel{\text... | Yes |
Proposition 7.19. Let \( \varphi ,\psi : \mathrm{X} \rightarrow \mathrm{Y} \) be measure-preserving mappings between standard probability spaces \( \mathrm{X},\mathrm{Y} \), and let \( {T}_{\varphi },{T}_{\psi } : {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \rightarrow {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) be the in... | Proof. Only one implication is not trivial. If \( {T}_{\varphi } = {T}_{\psi } \), then \( {\mathbf{1}}_{{\varphi }^{ * }A} = {T}_{\varphi }{\mathbf{1}}_{A} = {T}_{\psi }{\mathbf{1}}_{A} = \) \( {\mathbf{1}}_{{\psi }^{ * }A} \) almost everywhere for every \( A \in {\sum }_{\mathrm{Y}} \), i.e., \( {\varphi }^{ * } = {\... | No |
Corollary 7.21. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system over a standard probability space \( \mathrm{X} \) . Then the system \( \left( {\mathrm{X};\varphi }\right) \) is invertible if and only if \( \varphi \) is essentially invertible. | Proof. One implication is Corollary 6.5. For the converse, suppose that \( \left( {\mathrm{X};\varphi }\right) \) is invertible. Then its Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) is invertible, by Proposition 7.12. The inverse \( {T}^{-1} \) satisfies properties 1) and 2) from Theorem 7.20, hence by ... | Yes |
Theorem 7.23. Let \( E = \mathrm{C}\left( K\right) \) or \( E = {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \), and let \( A \subseteq E \) be a closed, conjugation invariant linear subspace with \( \mathbf{1} \in A \) . Then the following assertions are equivalent:\n\n(i) \( A \) is a subalgebra of \( E \) .\n\n(ii... | Proof. The proof of the first part is left as Exercise 16. | No |
Theorem 7.24 (Hölder’s Inequality for Positive Operators). Let \( E = \mathrm{C}\left( K\right) \) or \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \), and let \( A \subseteq E \) be a \( {C}^{ * } \) -subalgebra of \( E \) . Furthermore, let \( \mathrm{Y} \) be any measure space, and let \( T : A \rightarrow {\mat... | Proof. We start with the representation\n\n\[ {x}^{1/p} = \mathop{\inf }\limits_{{t > 0}}\frac{1}{p}{t}^{-1/q}x + \frac{1}{q}{t}^{1/p} \]\n\nwhich is (7.5) with \( p \) replaced by \( \frac{1}{p} \) . For fixed \( y > 0 \), multiply this identity with \( {y}^{1/q} \) and arrive at\n\n\[ {x}^{1/p}{y}^{1/q} = \mathop{\in... | Yes |
Lemma 8.2. Let \( E \) be a Banach space and let \( T : E \rightarrow E \) be a bounded linear operator on \( E \) . Then, with \( {\mathrm{A}}_{n} \mathrel{\text{:=}} {\mathrm{A}}_{n}\left\lbrack T\right\rbrack \), the following assertions hold:\n\na) If \( f \in \mathrm{{fix}}\left( T\right) \), then \( {\mathrm{A}}_... | Proof. a) is trivial, and the formulae (8.2)-(8.4) are established by simple algebraic manipulations. The remaining statements then follow from these formulae. | No |
Lemma 8.3. Let \( T \) be a bounded linear operator on a Banach space \( E \) . Then\n\n\[ F \mathrel{\text{:=}} \left\{ {f \in E : {P}_{T}f \mathrel{\text{:=}} \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mathrm{A}}_{n}f\text{ exists }}\right\} \]\n\nis a \( T \) -invariant subspace of \( E \) containing \( \opera... | Proof. It is clear that \( F \) is a subspace of \( E \) and \( {P}_{T} : F \rightarrow E \) is linear. By Lemma 8.2.b, \( F \) is \( T \) -invariant, \( \operatorname{ran}\left( {P}_{T}\right) \subseteq \operatorname{fix}\left( T\right) \) and \( {P}_{T}{Tf} = T{P}_{T}f = {P}_{T}f \) for all \( f \in F \) . Finally it... | Yes |
Theorem 8.5. Let \( T \in \mathcal{L}\left( E\right) \) , \( E \) a Banach space. Suppose that \( \mathop{\sup }\limits_{{n \in \mathbb{N}}}\begin{Vmatrix}{\mathrm{A}}_{n}\end{Vmatrix} < \infty \) and that \( \frac{1}{n}{T}^{n}f \rightarrow 0 \) for all \( f \in E \) . Then the subspace\n\n\[ F \mathrel{\text{:=}} \lef... | Proof. By Lemma 8.3, all that remains to show is that \( F \) is closed, \( {P}_{T} \) is bounded, and \( \ker \left( {P}_{T}\right) = \overline{\operatorname{ran}}\left( {\mathrm{I} - T}\right) \) . The closedness of \( F \) and the boundedness of \( {P}_{T} \) are solely due to the uniform boundedness of the operator... | No |
Theorem 8.6 (Mean Ergodic Theorem on Hilbert Spaces). Let \( H \) be a Hilbert space and let \( T \in \mathcal{L}\left( H\right) \) be a contraction, i.e., \( \parallel T\parallel \leq 1 \) . Then\n\n\[ \n{P}_{T}f \mathrel{\text{:=}} \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{j = ... | Proof. If \( T \) is a contraction, then the powers \( {T}^{n} \) and hence the Cesàro averages \( {\mathrm{A}}_{n}\left\lbrack T\right\rbrack \) are contractions, too, and \( \frac{1}{n}{T}^{n} \rightarrow 0 \) . Therefore, Theorem 8.5 can be applied and so the subspace \( F \) is closed and \( {P}_{T} : F \rightarrow... | Yes |
Corollary 8.7. Let \( T \) be a contraction on a Hilbert space \( H \) . Then \( \operatorname{fix}\left( T\right) = \operatorname{fix}\left( {T}^{ * }\right) \) and \( {P}_{T} = {P}_{{T}^{ * }} \) . | Proof. Note that \( f \in \operatorname{fix}\left( {T}^{ * }\right) \) implies that \( \left( {{Tf} \mid f}\right) = \left( {f \mid {T}^{ * }f}\right) = \parallel f{\parallel }^{2} \) and hence \( {Tf} = f \) as in the proof of Theorem 8.6. By symmetry, \( \operatorname{fix}\left( T\right) = \operatorname{fix}\left( {T... | Yes |
Theorem 8.8. Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a measure-preserving system, and consider the Koopman operator \( T = {T}_{\\varphi } \) on the space \( {\\mathrm{L}}^{1} = {\\mathrm{L}}^{1}\\left( \\mathrm{X}\\right) \) . Then\n\n\[ \n{P}_{T}f \\mathrel{\\text{:=}} \\mathop{\\lim }\\limits_{{n \\righ... | Proof. Let \( f \\in {\\mathrm{L}}^{\\infty } \) . Then, by von Neumann’s theorem, the limit \( {P}_{T}f = \) \( \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}{\\mathrm{A}}_{n}f \) exists in \( {\\mathrm{L}}^{2} \), thus a fortiori in \( {\\mathrm{L}}^{1} \) . Moreover, since \( \\left| {{T}^{n}f}\\right| \\leq \... | Yes |
Theorem 8.10. Let \( \\left( {\\mathrm{X};\\varphi }\\right) \) be a measure-preserving system, \( \\mathrm{X} = \\left( {X,\\sum ,\\mu }\\right) \), with associated Koopman operator \( T \\mathrel{\\text{:=}} {T}_{\\varphi } \) on \( {\\mathrm{L}}^{1}\\left( \\mathrm{X}\\right) \), and let \( 1 \\leq p < \\infty \) . ... | Proof. The equivalence of (i)-(iv) has been proved in Proposition 7.15.\n\n(ii) \( \\Rightarrow \) (v): Let \( f \\in {\\mathrm{L}}^{1} \) . Then \( {P}_{T}f \\in \\operatorname{fix}\\left( T\\right) \), so \( {P}_{T}f = c \\cdot \\mathbf{1} \) by (ii). Integrating yields \( c = {\\int }_{\\mathrm{X}}f \)\n\n\( \\left(... | Yes |
Lemma 8.13. For a row-stochastic \( k \times k \) -matrix \( S \) the following assertions are equivalent:\n\n(i) \( S \) is irreducible.\n\n(ii) There is \( m \in \mathbb{N} \) such that \( {\left( \mathrm{I} + S\right) }^{m} \) is strictly positive.\n\n(iii) \( Q = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac... | Proof. (i) \( \Rightarrow \) (ii): Simply expand \( {\left( \mathrm{I} + S\right) }^{m} = \mathop{\sum }\limits_{{j = 0}}^{m}\left( \begin{matrix} m \\ j \end{matrix}\right) {S}^{j} \) . If \( S \) is irreducible, then for large enough \( m \in \mathbb{N} \), the resulting matrix must have each entry strictly positive.... | Yes |
Theorem 8.14. Let \( S \) be a row-stochastic \( k \times k \) -matrix with fixed probability vector p. Then \( p \) is strictly positive and the Markov shift \( \left( {{\mathcal{W}}_{k}^{ + },\sum ,\mu \left( {S, p}\right) ;\tau }\right) \) is ergodic, if and only if \( S \) is irreducible. | Proof. Let \( {i}_{0},\ldots ,{i}_{l} \in L \) and \( {j}_{0},\ldots ,{j}_{r} \in L \) . Then for \( n \in \mathbb{N} \) we have\n\n\[ \mu \left( {\left\{ {i}_{0}\right\} \times \cdots \times \left\{ {i}_{l}\right\} \times {L}^{n - 1} \times \left\{ {j}_{0}\right\} \times \cdots \times \left\{ {j}_{r}\right\} \times \p... | Yes |
Lemma 8.16. If \( E \) is a Banach space and \( T \in \mathcal{L}\left( E\right) \) is mean ergodic, then \( T \) is Cesàro bounded and \( \frac{1}{n}{T}^{n}f \rightarrow 0 \) for every \( f \in E \) . | Proof. As \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mathrm{A}}_{n}f \) exists for every \( f \in E \) and \( E \) is a Banach space, it follows from the uniform boundedness principle (Theorem C.1) that \( \mathop{\sup }\limits_{{n \in \mathbb{N}}}\begin{Vmatrix}{\mathrm{A}}_{n}\end{Vmatrix} < \infty \) . From... | Yes |
Lemma 8.17. Let \( C \) be a convex subset of a Hausdorff topological vector space, \( T : C \rightarrow C \) be a continuous affine mapping and \( g \in C \) . Then \( {Tg} = g \) if and only if there is \( f \in C \) and a subsequence \( {\left( {n}_{j}\right) }_{j \in \mathbb{N}} \) such that\n\n\[ \text{(1)}\frac{1... | Proof. If \( {Tg} = g \), then (1) and (2) hold for \( f = g \) and every subsequence \( {\left( {n}_{j}\right) }_{j \in \mathbb{N}} \) . Conversely, suppose that (1) and (2) hold for some \( f \in E \) and a subsequence \( {\left( {n}_{j}\right) }_{j \in \mathbb{N}} \) . Then\n\n\[ g - {Tg} = \left( {\mathrm{I} - T}\r... | Yes |
Proposition 8.18. Let \( T \) be a Cesàro bounded operator on some Banach space \( E \) such that \( \frac{1}{n}{T}^{n}h \rightarrow 0 \) for each \( h \in E \) . Then for \( f, g \in E \) the following statements are equivalent:\n\n(i) \( {\mathrm{A}}_{n}f \rightarrow g \) in the norm of \( E \) as \( n \rightarrow \i... | Proof. The implication \( \left( \mathrm{v}\right) \Rightarrow \left( \mathrm{i}\right) \) follows from Theorem 8.5 while the implications (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) are trivial. If (iii) holds, then \( g \in \operatorname{fix}\left( T\right) \) by Lemma 8.17. Moreover,\n\n\[ g \in {\operatornam... | Yes |
Theorem 8.20 (Mean Ergodic Operators). Let \( T \) be a Cesàro bounded operator on some Banach space \( E \) such that \( \frac{1}{n}{T}^{n}f \rightarrow 0 \) weakly for each \( f \in E \) . Further, let \( D \subseteq E \) be a dense subset of \( E \) . Then the following assertions are equivalent:\n\n(i) \( T \) is m... | Proof. The equivalence of (i)-(iv) and (vi) is immediate from Proposition 8.18 and Theorem 8.5. Suppose that (vi) holds and \( 0 \neq {f}^{\prime } \in \operatorname{fix}\left( {T}^{\prime }\right) \) . Then there is \( f \in E \) such that \( \left\langle {f,{f}^{\prime }}\right\rangle \neq 0 \) . By (vi) we can write... | No |
Theorem 8.22. Every power-bounded linear operator on a reflexive Banach space is mean ergodic. | Proof. Let \( f \in E \) . Then, by power-boundedness of \( T \) the set \( \left\{ {{\mathrm{A}}_{n}f : n \in \mathbb{N}}\right\} \) is norm-bounded. Since \( E \) is reflexive, it is even relatively weakly compact, hence the sequence \( {\left( {\mathrm{A}}_{n}f\right) }_{n \in \mathbb{N}} \) has a weak cluster point... | Yes |
Theorem 8.23. Let \( T : {\mathrm{L}}^{1}\left( \mathrm{X}\right) \rightarrow {\mathrm{L}}^{1}\left( \mathrm{Y}\right) \) be a Dunford-Schwartz operator. Then\n\n\[ \parallel {Tf}{\parallel }_{p} \leq \parallel f{\parallel }_{p}\;\text{ for all }f \in {\mathrm{L}}^{p} \cap {\mathrm{L}}^{1},1 \leq p \leq \infty . \] | Proof. The claim is a direct consequence of the Riesz-Thorin interpolation theorem (Folland 1999, Thm. 6.27). If \( T \) is positive, there is a more elementary proof, which we give for convenience. For \( p = 1,\infty \) there is nothing to show, so let \( 1 < p < \infty \) . Take \( f \in {\mathrm{L}}^{p} \cap {\math... | Yes |
Theorem 8.24. A Dunford-Schwartz operator \( T \) on \( {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) over a finite measure space \( \mathrm{X} \) is mean ergodic. | Proof. By Theorem 8.23, \( T \) restricts to a contraction on \( {\mathrm{L}}^{2} \), which is a Hilbert space. By Theorem 8.6, \( T \) is mean ergodic on \( {\mathrm{L}}^{2} \), which means that for \( f \in {\mathrm{L}}^{2} \) the limit \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mathrm{A}}_{n}\left\lbrack T\... | Yes |
Theorem 8.26. Let \( E \) be a Banach space and let \( S \in \mathcal{L}\left( E\right) \) be a power-bounded mean ergodic operator: Let \( T \) be a \( {k}^{\text{th }} \) root of \( S \), i.e., \( {T}^{k} = S \) for some \( k \in \mathbb{N} \) . Then \( T \) is also mean ergodic. | Proof. Denote by \( {P}_{S} \) the mean ergodic projection of \( S \) . Define \( P \mathrel{\text{:=}} \left( {\frac{1}{k}\mathop{\sum }\limits_{{j = 0}}^{{k - 1}}{T}^{j}}\right) {P}_{S} \) and observe that \( {Pf} \in \overline{\operatorname{conv}}\left\{ {{T}^{j}f : j \in {\mathbb{N}}_{0}}\right\} \) for all \( f \i... | Yes |
Take \( E = \mathrm{c} \), the space of convergent scalar sequences, and the multiplication operator\n\n\[ M : \mathrm{c} \rightarrow \mathrm{c},\;{\left( {x}_{n}\right) }_{n \in \mathbb{N}} \mapsto {\left( {a}_{n}{x}_{n}\right) }_{n \in \mathbb{N}} \]\n\nfor some sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}}... | Consider now a \( {k}^{\text{th }} \) root of unity \( 1 \neq b \in \mathbb{T} \) and define \( {T}_{k} \mathrel{\text{:=}} {bM} \) . Then it is easy to see that \( \operatorname{fix}\left( {T}_{k}^{\prime }\right) = \{ 0\} \), and hence, again by Theorem \( {8.20}\left( \mathrm{v}\right) ,{T}_{k} \) is mean ergodic. I... | No |
Theorem 8.28 (Kakutani). Let \( E \) be a Banach space. Then the identity operator \( {\mathrm{I}}_{E} \) is an extreme point of the closed unit ball in \( \mathcal{L}\left( E\right) \) . | Proof. Suppose that \( {\mathrm{I}}_{E} = \frac{1}{2}\left( {R + S}\right) \) with \( \parallel S\parallel ,\parallel R\parallel \leq 1 \) . Then \( {\mathrm{I}}_{{E}^{\prime }} = \frac{1}{2}\left( {{R}^{\prime } + {S}^{\prime }}\right) \) . Let \( {f}^{\prime } \) be an extreme point of the dual unit ball \( {\mathrm{... | Yes |
Lemma 8.29. Let \( R, S \) be two commuting power-bounded operators, and for \( t \in \) \( \left( {0,1}\right) \) let \( T \mathrel{\text{:=}} {tR} + \left( {1 - t}\right) S \) . Then \( \operatorname{fix}\left( T\right) = \operatorname{fix}\left( R\right) \cap \operatorname{fix}\left( S\right) \) . | Proof. Only the inclusion \( \operatorname{fix}\left( T\right) \subseteq \operatorname{fix}\left( R\right) \cap \operatorname{fix}\left( S\right) \) is not obvious. Endow \( E \) with an equivalent norm \( \parallel f{\parallel }_{1} \mathrel{\text{:=}} \sup \left\{ {\begin{Vmatrix}{{R}^{n}{S}^{m}f}\end{Vmatrix} : n, m... | Yes |
Theorem 8.30. Let \( {T}_{1},{T}_{2},\ldots ,{T}_{m} \) be commuting power-bounded, mean ergodic operators. Then every convex combination\n\n\[ T \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 1}}^{m}{t}_{j}{T}_{j} \]\n\nwith all \( {t}_{j} > 0 \), is mean ergodic. Denoting by \( {P}_{j} \) the mean ergodic projection... | Proof. It suffices to prove the statement for the case of \( m = 2 \), the general case can then be established by induction. So let \( S = {T}_{1}, R = {T}_{2} \), let \( 0 < t < 1 \), and let \( T \mathrel{\text{:=}} {tR} + \left( {1 - t}\right) S \) . By Lemma 8.29 we have \( \operatorname{fix}\left( T\right) = \ope... | Yes |
Theorem 8.32 (Contraction Semigroups on Hilbert Spaces). Let \( \mathcal{T} \) be a semigroup of linear contractions on a Hilbert space \( H \), and let \( P \in \mathcal{L}\left( H\right) \) be the orthogonal projection onto \( \operatorname{fix}\left( \mathcal{T}\right) \) . Then \( \mathcal{T} \) is mean ergodic wit... | Proof. Let \( T \in \mathcal{T} \) . Then \( {TP} = P \) by definition of \( P \), and \( \operatorname{fix}\left( T\right) = \operatorname{fix}\left( {T}^{ * }\right) \) by Corollary 8.7. Hence, \( {T}^{ * }P = P \), and taking adjoints yields \( {PT} = P \) .\n\nBy Theorem D.2, the closed convex set \( C \mathrel{\te... | Yes |
Theorem 8.34. Let \( E \) be a strictly convex Banach space \( E \) which has a strictly convex dual space. Then each relatively weakly compact contraction semigroup on \( E \) is mean ergodic. | Proof. We only sketch the proof. Let \( \mathcal{T} \) be a relatively weakly compact semigroup of contractions on \( E \), and let \( f \in E \) . Then by weak compactness, \( \overline{\operatorname{conv}}\left( {\mathcal{T}f}\right) \) contains an element \( g \) with minimal norm. This element is unique because of ... | No |
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