[PPT] - Spectral Properties of an Operator-Fractal Keri Kornelson PowerPoint Presentation

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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 07/18/2012 1 / 25

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Acknowledgements

This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College).

K. Kornelson (U. Oklahoma)

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Outline

1 Bernoulli-Cantor Measures

2 Fourier bases

3 Families of ONBs

4 Operator-fractal

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Bernoulli-Cantor Measures

Iterated Function Systems (IFSs)

We will construct an L2 space via an iterated function system. Definition An Iterated Function System (IFS) is a finite collection {τi}k

i=1 of contractive

maps on a complete metric space. The map on the compact subsets given by A →

k

i=1

τi(A) is a contraction in the Hausdorff metric.

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Bernoulli-Cantor Measures

IFS Attractor Set

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

i=1

τi(X) = X. (1)

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Bernoulli-Cantor Measures

IFS Attractor Set

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

i=1

τi(X) = X. (1) The set X is called the attractor of the IFS. We say (1) is an invariance held by X.

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Operator-Fractal TAMU 07/18/2012 5 / 25

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Bernoulli-Cantor Measures

IFS Attractor Set

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

i=1

τi(X) = X. (1) The set X is called the attractor of the IFS. We say (1) is an invariance held by X. Given any compact set A0, successive iterations of our contraction An+1 =

k

i=1

τi(An) converge (in Hausdorff metric) to the attractor X.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 5 / 25

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3 The Sierpinski gasket in R2.

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3 The Sierpinski gasket in R2.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 6 / 25

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3 The Sierpinski gasket in R2.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 6 / 25

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3 The Sierpinski gasket in R2.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 6 / 25

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Bernoulli-Cantor Measures

Examples

The Cantor ternary set in R: τ0(x) = 1 3x τ1(x) = 1 3x + 2 3 The Sierpinski gasket in R2.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 6 / 25

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Bernoulli-Cantor Measures

IFS Measure

Hutchinson, 1981

{τi}k

i=1 an IFS

{pi}k

i=1 probability weights, i.e. pi ≥ 0, k i=1 pi = 1

Define a map on measures: ν →

k

i=1

pi(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

r IFS measure for the IFS.

As a fixed point, µ satisfies the invariance property: µ =

k

i=1

pi(µ ◦ τ −1

i

). (3)

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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.

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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 — Xλ

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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 — Xλ Bernoulli convolution measure (p+ = p− = 1

2) — µλ— supported on Xλ.

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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 — Xλ Bernoulli convolution measure (p+ = p− = 1

2) — µλ— supported on Xλ.

Historical note: The Bernoulli measures date back to work of Erdös and others, long before this IFS approach came along. µλ is the distribution of the random variable

k ±λk where + and − have equal probability.
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Fourier bases

Fourier bases for Bernoulli measures?

Consider the Hilbert space L2(µλ).

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Fourier bases

Fourier bases for Bernoulli measures?

Consider the Hilbert space L2(µλ). Is it ever possible for L2(µλ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions?

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Fourier bases

Fourier bases for Bernoulli measures?

Consider the Hilbert space L2(µλ). Is it ever possible for L2(µλ) 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 07/18/2012 9 / 25

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Fourier bases

Fourier bases for Bernoulli measures?

Consider the Hilbert space L2(µλ). Is it ever possible for L2(µλ) 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 07/18/2012 9 / 25

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Fourier bases

Fourier bases for Bernoulli measures?

Consider the Hilbert space L2(µλ). Is it ever possible for L2(µλ) 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?

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Operator-Fractal TAMU 07/18/2012 9 / 25

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Fourier bases

Getting started: µλ

Recall that, given λ ∈ (0, 1), the Bernoulli IFS is: τ+(x) = λ(x + 1) and τ−(x) = λ(x − 1). The Bernoulli measure µλ satisfies the invariance µλ = 1 2(µλ ◦ τ −1

+

+ µλ ◦ τ −1

− ).

Then the Fourier transform of µλ is:

µλ(t)

=

e2πixt dµλ(x)

= 1 2

e2πi(λx+λ)t dµλ(x) + 1

2

e2πi(λx−λ)t dµλ(x)

= cos(2πλt) µλ(λt) = cos(2πλt) cos(2πλ2t) µλ(λ2t) . . . . . . =

∞

k=1

cos(2πλkt)

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ).

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ). eγ, e˜

γL2

=

eγ−˜

γ dµλ

=

µλ(γ − ˜

γ) =

∞

k=1

cos

2π
λ

k (γ − ˜ γ)

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ). eγ, e˜

γL2

=

eγ−˜

γ dµλ

=

µλ(γ − ˜

γ) =

∞

k=1

cos

2π
λ

k (γ − ˜ γ)

Lemma

The two exponentials eγ, e˜

γ are orthogonal if and only if one of the factors in

the infinite product above is zero. This is equivalent to γ − ˜ γ ∈ 1 4λ−k(2m + 1) : k ∈ N, m ∈ Z

=: Zλ.
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Fourier bases

Surprising first results

Theorem (Jorgensen, Pedersen 1998) L2(µ 1

4 ) has an ONB of exponential functions.

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Fourier bases

Surprising first results

Theorem (Jorgensen, Pedersen 1998) L2(µ 1

4 ) has an ONB of exponential functions.

Example E(Γ 1

4 ) is an ONB for L2(µ 1 4 ), where

Γ 1

4 =

  

p

j=0

aj4j : aj ∈ {0, 1}, p finite    = {0, 1, 4, 5, 16, 17, 20, 21, 64, . . .}.

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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:

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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

2n ) but

when n is odd, there can be only finitely many elements in any orthogonal collection of exponentials.

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Fourier bases

More recent progress

Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a

b, if b is odd, then any orthonormal collection of exponentials in

L2(µλ) must be finite. If b is even, then there exists countable collections of

rthonormal exponentials in L2(µλ).
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Fourier bases

More recent progress

Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a

b, if b is odd, then any orthonormal collection of exponentials in

L2(µλ) must be finite. If b is even, then there exists countable collections of

rthonormal exponentials in L2(µλ).

Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1

2, i.e. there is essential overlap, then L2(µλ) does not have an ONB (or

even a frame) of exponential functions.

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Fourier bases

“Canonical” ONBs

[Jorgensen, Pedersen 1998]

Definition Let λ =

1 2n and consider the set from Jorgensen & Pedersen

Γ 1

2n =

  

p

j=0

aj(2n)j : aj ∈

0, n

2

, p finite

   . We call Γ 1

2n the canonical spectrum and E(Γ 1 2n ) the canonical ONB for L2(µ 1 2n ).

Note: We will justify the nomenclature by describing alternate bases for the same spaces.

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Families of ONBs

Jorgensen, K, Shuman 2011 JFAA

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs.

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Families of ONBs

Jorgensen, K, Shuman 2011 JFAA

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Theorem The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

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Families of ONBs

Jorgensen, K, Shuman 2011 JFAA

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Theorem The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

Theorem If p < 2(2n−1)

π

, then E(pΓ 1

2n ) is an ONB for L2(µ 1 2n ).

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Families of ONBs

Jorgensen, K, Shuman 2011 JFAA

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Theorem The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

Theorem If p < 2(2n−1)

π

, then E(pΓ 1

2n ) is an ONB for L2(µ 1 2n ).

Laba/Wang and Dutkay/Jorgensen have described many other values of p for which pΓ 1

2n is a spectrum, particularly in the 1

4 case.

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ S0 and S1 map between ONB elements, so are both isometries.

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ S0 and S1 map between ONB elements, so are both isometries. U maps one ONB to another, so U is a unitary operator.

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ)

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

H ⊃ S0(H) ⊃ S2

0(H) ⊃ · · ·

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

H ⊃ S0(H) ⊃ S2

0(H) ⊃ · · ·

If we define Wk = Sk

0(H) ⊖ Sk+1

(H), then H = sp(e0) ⊕

∞

k=0

Wk

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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)

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U.

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U. With respect to the Wk ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same!

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U. With respect to the Wk 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: U = (e0 ⊗ e0) ⊕

∞

k=1

Me1U. We call U an operator-fractal.

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Operator-fractal

Matrix of U

Γ0 Γ1 Γ2 Γ3 · · · 1 · · · Γ0 Me1U · · · Γ1 Me1U · · · Γ2 Me1U · · · Γ3 Me1U · · · . . . . . . . . . . . . . . . ...

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Operator-fractal

Spectral properties of U

Jorgensen, K, Shuman 2012

Proposition (JKS 2012) U has the following properties.

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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. (Ue0 = e0)

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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. (Ue0 = e0) U is not spatially implemented; i.e. is not of the form Uf = f ◦ τ for τ a point transformation on [0, 1].

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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. (Ue0 = e0) 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 = ce0 for some c ∈ C.

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Operator-fractal

Nelson-like cyclic subspaces

Assume that Uv = v for some v = 1, v / ∈ sp(e0). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v = 0, H(v) := sp{Ukv : k ∈ Z} is the U-cyclic subspace for v.

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Operator-fractal

Nelson-like cyclic subspaces

Assume that Uv = v for some v = 1, v / ∈ sp(e0). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v = 0, H(v) := sp{Ukv : k ∈ Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let mv be the real measure mv(A) = E(A)v, v.

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Operator-fractal

Nelson-like cyclic subspaces

Assume that Uv = v for some v = 1, v / ∈ sp(e0). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v = 0, H(v) := sp{Ukv : k ∈ Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let mv be the real measure mv(A) = E(A)v, v. If v = 1, mv is a probability measure.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 22 / 25

SLIDE 59

Operator-fractal

Nelson-like cyclic subspaces

Assume that Uv = v for some v = 1, v / ∈ sp(e0). We modify the cyclic subspaces of Nelson for use with our unitary operator U. Given v = 0, H(v) := sp{Ukv : k ∈ Z} is the U-cyclic subspace for v. Let E be the projection-valued measure for U and, given v, let mv be the real measure mv(A) = E(A)v, v. If v = 1, mv is a probability measure. Theorem H(v) = {φ(U)v : φ ∈ L2(mv)}. Moreover, the map φ → φ(U)v is an isometric isomorphism between L2(mv) and H(v).

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 22 / 25

SLIDE 60

Operator-fractal

Lemmas

The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L2(mv). Lemma If Uv = v, then v is not in H(eγ) for any γ ∈ Γ \ {0}.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 23 / 25

SLIDE 61

Operator-fractal

Lemmas

The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L2(mv). Lemma If Uv = v, then v is not in H(eγ) for any γ ∈ Γ \ {0}. Lemma If v1 ⊥ H(v2), then H(v1) ⊥ H(v2).

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 23 / 25

SLIDE 62

Operator-fractal

Lemmas

The following lemmas make heavy use of the isometric isomorphism between cyclic subspaces H(v) and L2(mv). Lemma If Uv = v, then v is not in H(eγ) for any γ ∈ Γ \ {0}. Lemma If v1 ⊥ H(v2), then H(v1) ⊥ H(v2). Lemma If Uv = v for v = 1 then mv is a Dirac point mass at 1.

K. Kornelson (U. Oklahoma)

Operator-Fractal TAMU 07/18/2012 23 / 25

SLIDE 63

Operator-fractal