Spectral Properties of an Operator-Fractal

Transcription

1 Spectral Properties of an Operator-Fractal Keri Kornelson University of Oklahoma - Norman NSA Texas A&M University July 18, 2012 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

2 Acknowledgements This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

3 Outline 1 Bernoulli-Cantor Measures 2 Fourier bases 3 Families of ONBs 4 Operator-fractal K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

4 Bernoulli-Cantor Measures Iterated Function Systems (IFSs) We will construct an L 2 space via an iterated function system. Definition An Iterated Function System (IFS) is a finite collection {τ i } k i=1 of contractive maps on a complete metric space. The map on the compact subsets given by k A τ i (A) i=1 is a contraction in the Hausdorff metric. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

5 IFS Attractor Set Bernoulli-Cantor Measures By the Banach Contraction Mapping Theorem, there exists a unique fixed point of the map. In other words, there is a compact set X satisfying the invariance relation: k τ i (X) = X. (1) i=1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

6 IFS Attractor Set Bernoulli-Cantor Measures By the Banach Contraction Mapping Theorem, there exists a unique fixed point of the map. In other words, there is a compact set X satisfying the invariance relation: k τ i (X) = X. (1) i=1 The set X is called the attractor of the IFS. We say (1) is an invariance held by X. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

7 IFS Attractor Set Bernoulli-Cantor Measures By the Banach Contraction Mapping Theorem, there exists a unique fixed point of the map. In other words, there is a compact set X satisfying the invariance relation: k τ i (X) = X. (1) i=1 The set X is called the attractor of the IFS. We say (1) is an invariance held by X. Given any compact set A 0, successive iterations of our contraction A n+1 = k τ i (A n ) i=1 converge (in Hausdorff metric) to the attractor X. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

8 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

9 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x The Sierpinski gasket in R 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

10 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x The Sierpinski gasket in R 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

11 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x The Sierpinski gasket in R 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

12 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x The Sierpinski gasket in R 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

13 Examples Bernoulli-Cantor Measures The Cantor ternary set in R: τ 0 (x) = 1 3 x τ 1(x) = 1 3 x The Sierpinski gasket in R 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

14 Bernoulli-Cantor Measures IFS Measure Hutchinson, 1981 {τ i } k i=1 an IFS {p i } k i=1 probability weights, i.e. p i 0, k i=1 p i = 1 Define a map on measures: ν k i=1 p i (ν τ 1 i ) (2) The Banach Theorem yields again a unique fixed point, in this case a probability measure supported on X. This measure µ is called an equilibrium or IFS measure for the IFS. As a fixed point, µ satisfies the invariance property: µ = k i=1 p i (µ τ 1 i ). (3) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

15 Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ on R of the form τ + (x) = λ(x + 1) τ (x) = λ(x 1) for 0 < λ < 1. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

16 Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ on R of the form τ + (x) = λ(x + 1) τ (x) = λ(x 1) for 0 < λ < 1. Bernoulli attractor set X λ K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

17 Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ on R of the form τ + (x) = λ(x + 1) τ (x) = λ(x 1) for 0 < λ < 1. Bernoulli attractor set X λ Bernoulli convolution measure (p + = p = 1 2 ) µ λ supported on X λ. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

18 Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ on R of the form τ + (x) = λ(x + 1) τ (x) = λ(x 1) for 0 < λ < 1. Bernoulli attractor set X λ Bernoulli convolution measure (p + = p = 1 2 ) µ λ supported on X λ. Historical note: The Bernoulli measures date back to work of Erdös and others, long before this IFS approach came along. µ λ is the distribution of the random variable k ±λk where + and have equal probability. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

19 Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 (µ λ ). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

20 Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 (µ λ ). Is it ever possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

21 Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 (µ λ ). Is it ever possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

22 Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 (µ λ ). Is it ever possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ? Given a spectral measure, what are the possible spectra? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

23 Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 (µ λ ). Is it ever possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ? Given a spectral measure, what are the possible spectra? Could a non-spectral measure have a frame of exponential functions? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

24 Fourier bases Getting started: µ λ Recall that, given λ (0, 1), the Bernoulli IFS is: τ + (x) = λ(x + 1) and τ (x) = λ(x 1). The Bernoulli measure µ λ satisfies the invariance µ λ = 1 2 (µ λ τ µ λ τ 1 ). Then the Fourier transform of µ λ is: µ λ (t) = e 2πixt dµ λ (x) = 1 e 2πi(λx+λ)t dµ λ (x) e 2πi(λx λ)t dµ λ (x) = cos(2πλt) µ λ (λt) = cos(2πλt) cos(2πλ 2 t) µ λ (λ 2 t). =. cos(2πλ k t) k=1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

25 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

26 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). e γ, e γ L 2 = e γ γ dµ λ = µ λ (γ γ) ( ( ) ) k(γ = cos 2π λ γ) k=1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

27 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). e γ, e γ L 2 = e γ γ dµ λ = µ λ (γ γ) ( ( ) ) k(γ = cos 2π λ γ) k=1 Lemma The two exponentials e γ, e γ are orthogonal if and only if one of the factors in the infinite product above is zero. This is equivalent to { } 1 γ γ 4 λ k (2m + 1) : k N, m Z =: Z λ. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

28 Surprising first results Fourier bases Theorem (Jorgensen, Pedersen 1998) L 2 (µ 1 ) has an ONB of exponential functions. 4 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

29 Surprising first results Fourier bases Theorem (Jorgensen, Pedersen 1998) L 2 (µ 1 ) has an ONB of exponential functions. 4 Example E(Γ 1 4 Γ 1 = 4 ) is an ONB for L 2 (µ 1 ), where 4 p a j 4 j j=0 : a j {0, 1}, p finite = {0, 1, 4, 5, 16, 17, 20, 21, 64,...}. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

30 Fourier bases Another surprise: λ = 1 3 Theorem (Jorgensen, Pedersen 1998) Not only is there no Fourier basis when λ = 1 3, but orthogonal collections of exponential functions can have at most 2 elements. There is also a more general version in the [JP1998] paper: K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

31 Fourier bases Another surprise: λ = 1 3 Theorem (Jorgensen, Pedersen 1998) Not only is there no Fourier basis when λ = 1 3, but orthogonal collections of exponential functions can have at most 2 elements. There is also a more general version in the [JP1998] paper: Theorem (Jorgensen, Pedersen 1998) Given λ = 1 n, if n is even, there is an ONB of exponentials for L2 (µ 1 ) but 2n when n is odd, there can be only finitely many elements in any orthogonal collection of exponentials. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

32 More recent progress Fourier bases Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a b, if b is odd, then any orthonormal collection of exponentials in L 2 (µ λ ) must be finite. If b is even, then there exists countable collections of orthonormal exponentials in L 2 (µ λ ). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

33 More recent progress Fourier bases Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a b, if b is odd, then any orthonormal collection of exponentials in L 2 (µ λ ) must be finite. If b is even, then there exists countable collections of orthonormal exponentials in L 2 (µ λ ). Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1 2, i.e. there is essential overlap, then L2 (µ λ ) does not have an ONB (or even a frame) of exponential functions. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

34 Canonical ONBs [Jorgensen, Pedersen 1998] Fourier bases Definition Let λ = 1 2n and consider the set from Jorgensen & Pedersen Γ 1 2n = p a j (2n) j j=0 { : a j 0, n } 2, p finite. We call Γ 1 the canonical spectrum and E(Γ 1 ) the canonical ONB for L 2 (µ 1 ). 2n 2n 2n Note: We will justify the nomenclature by describing alternate bases for the same spaces. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

35 Families of ONBs Jorgensen, K, Shuman 2011 JFAA Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

36 Families of ONBs Jorgensen, K, Shuman 2011 JFAA Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Theorem The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

37 Families of ONBs Jorgensen, K, Shuman 2011 JFAA Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Theorem The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 Theorem If p < 2(2n 1) π, then E(pΓ 1 ) is an ONB for L 2 (µ 1 ). 2n 2n K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

38 Families of ONBs Jorgensen, K, Shuman 2011 JFAA Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Theorem The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 Theorem If p < 2(2n 1) π, then E(pΓ 1 ) is an ONB for L 2 (µ 1 ). 2n 2n Laba/Wang and Dutkay/Jorgensen have described many other values of p for which pγ 1 is a spectrum, particularly in the 1 2n 4 case. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

39 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

40 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

41 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ S 0 and S 1 map between ONB elements, so are both isometries. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

42 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ S 0 and S 1 map between ONB elements, so are both isometries. U maps one ONB to another, so U is a unitary operator. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

43 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

44 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

45 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) H S 0 (H) S 2 0 (H) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

46 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) H S 0 (H) S 2 0 (H) If we define W k = S0 k (H) Sk+1 0 (H), then H = sp(e 0 ) k=0 W k K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

47 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2011) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

48 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2011) Each subspace W k is invariant under U. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

49 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2011) Each subspace W k is invariant under U. With respect to the W k ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same! K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

50 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2011) Each subspace W k is invariant under U. With respect to the W k ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same! Even more, in the ( 4, 5) case U actually has a self-similar structure: We call U an operator-fractal. U = (e 0 e 0 ) M e1 U. k=1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

51 Matrix of U Operator-fractal 0 Γ 0 Γ 1 Γ 2 Γ Γ 0 0 M e1 U Γ M e1 U 0 0 Γ M e1 U 0 Γ M e1 U K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

52 Operator-fractal Spectral properties of U Jorgensen, K, Shuman 2012 Proposition (JKS 2012) U has the following properties. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

53 Operator-fractal Spectral properties of U Jorgensen, K, Shuman 2012 Proposition (JKS 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

54 Operator-fractal Spectral properties of U Jorgensen, K, Shuman 2012 Proposition (JKS 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) U is not spatially implemented; i.e. is not of the form Uf = f τ for τ a point transformation on [0, 1]. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

55 Operator-fractal Spectral properties of U Jorgensen, K, Shuman 2012 Proposition (JKS 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) U is not spatially implemented; i.e. is not of the form Uf = f τ for τ a point transformation on [0, 1]. Theorem (JKS 2012) U is an ergodic operator; i.e. if Uv = v then v = ce 0 for some c C. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

56 Operator-fractal Nelson-like cyclic subspaces Assume that Uv = v for some v = 1, v / sp(e 0 ). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v 0, H(v) := sp{u k v : k Z} is the U-cyclic subspace for v. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

57 Operator-fractal Nelson-like cyclic subspaces Assume that Uv = v for some v = 1, v / sp(e 0 ). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v 0, H(v) := sp{u k v : k Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let m v be the real measure m v (A) = E(A)v, v. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

58 Operator-fractal Nelson-like cyclic subspaces Assume that Uv = v for some v = 1, v / sp(e 0 ). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v 0, H(v) := sp{u k v : k Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let m v be the real measure m v (A) = E(A)v, v. If v = 1, m v is a probability measure. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

59 Operator-fractal Nelson-like cyclic subspaces Assume that Uv = v for some v = 1, v / sp(e 0 ). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v 0, H(v) := sp{u k v : k Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let m v be the real measure m v (A) = E(A)v, v. Theorem If v = 1, m v is a probability measure. H(v) = {φ(u)v : φ L 2 (m v )}. Moreover, the map φ φ(u)v is an isometric isomorphism between L 2 (m v ) and H(v). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

60 Operator-fractal Lemmas The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L 2 (m v ). Lemma If Uv = v, then v is not in H(e γ ) for any γ Γ \ {0}. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

61 Operator-fractal Lemmas The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L 2 (m v ). Lemma If Uv = v, then v is not in H(e γ ) for any γ Γ \ {0}. Lemma If v 1 H(v 2 ), then H(v 1 ) H(v 2 ). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

62 Operator-fractal Lemmas The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L 2 (m v ). Lemma If Uv = v, then v is not in H(e γ ) for any γ Γ \ {0}. Lemma If v 1 H(v 2 ), then H(v 1 ) H(v 2 ). Lemma If Uv = v for v = 1 then m v is a Dirac point mass at 1. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

63 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

64 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

65 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

66 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. H(v 2 ) H(e γ ) by one of the lemmas above. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

67 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. H(v 2 ) H(e γ ) by one of the lemmas above. m v (A) = E(A)v 1, v 1 + E(A)v 2, v 2 = m v1 (A) + m v2 (A). The cross terms are zero. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

68 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. H(v 2 ) H(e γ ) by one of the lemmas above. m v (A) = E(A)v 1, v 1 + E(A)v 2, v 2 = m v1 (A) + m v2 (A). The cross terms are zero. Normalize to probability measures m v1 and m v2, giving m v (A) = v 1 2 m v1 (A) + v 2 2 m v2 (A). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

69 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. H(v 2 ) H(e γ ) by one of the lemmas above. m v (A) = E(A)v 1, v 1 + E(A)v 2, v 2 = m v1 (A) + m v2 (A). The cross terms are zero. Normalize to probability measures m v1 and m v2, giving m v (A) = v 1 2 m v1 (A) + v 2 2 m v2 (A). Thus m v is a convex combination of probability measures. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

70 Operator-fractal Proof of ergodicity of U: Assume Uv = v for some v / sp(e 0 ), v = 1. There exists some γ Γ \ {0} such that v, e γ 0. Let v 1 be the projection of v onto H(e γ ) and v 2 = v v 1. Note that v 2 0 and v 2 2 = 1 v 1 2. H(v 2 ) H(e γ ) by one of the lemmas above. m v (A) = E(A)v 1, v 1 + E(A)v 2, v 2 = m v1 (A) + m v2 (A). The cross terms are zero. Normalize to probability measures m v1 and m v2, giving m v (A) = v 1 2 m v1 (A) + v 2 2 m v2 (A). Thus m v is a convex combination of probability measures. But, by hypothesis, m v is a Dirac point mass, hence is an extreme point in the space of probability measures. This gives a contradiction. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25

71 Thank You K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/ / 25