Spectral Properties of an Operator-Fractal
Keri Kornelson
University of Oklahoma - Norman
NSA
Texas A&M University
July 18, 2012
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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 }ki=1 of contractive
maps on a complete metric space.
The map on the compact subsets given by
A 7→
k
[
τi (A)
i=1
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
[
τi (X ) = X .
(1)
i=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
[
τi (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)
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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
[
τi (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 A0 , successive iterations of our contraction
An+1 =
k
[
τi (An )
i=1
converge (in Hausdorff metric) to the attractor X .
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
K. Kornelson (U. Oklahoma)
1
x
3
τ1 (x ) =
Operator-Fractal
1
2
x+
3
3
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
1
x
3
τ1 (x ) =
1
2
x+
3
3
The Sierpinski gasket in R2 .
K. Kornelson (U. Oklahoma)
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
1
x
3
τ1 (x ) =
1
2
x+
3
3
The Sierpinski gasket in R2 .
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
1
x
3
τ1 (x ) =
1
2
x+
3
3
The Sierpinski gasket in R2 .
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
1
x
3
τ1 (x ) =
1
2
x+
3
3
The Sierpinski gasket in R2 .
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Bernoulli-Cantor Measures
Examples
The Cantor ternary set in R:
τ0 (x ) =
1
x
3
τ1 (x ) =
1
2
x+
3
3
The Sierpinski gasket in R2 .
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Bernoulli-Cantor Measures
IFS Measure
Hutchinson, 1981
{τi }ki=1 an IFS
{pi }ki=1 probability weights, i.e. pi ≥ 0,
Define a map on measures:
ν 7→
k
X
Pk
i=1
pi = 1
pi (ν ◦ τi−1 )
(2)
i=1
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
X
pi (µ ◦ τi−1 ).
(3)
i=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.
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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− = 12 ) — 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− = 12 ) — 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
P
k
k ±λ 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?
K. Kornelson (U. Oklahoma)
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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
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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
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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?
K. Kornelson (U. Oklahoma)
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Fourier bases
Getting started: µ
bλ
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
1
−1
(µλ ◦ τ+−1 + µλ ◦ τ−
).
2
Then the Fourier transform of µλ is:
µλ =
µ
bλ (t)
=
=
=
=
..
.
=
Z
e2πixt dµλ (x )
Z
Z
1
1
e2πi(λx+λ)t dµλ (x ) +
e2πi(λx−λ)t dµλ (x )
2
2
cos(2πλt)b
µλ (λt)
cos(2πλt) cos(2πλ2 t)b
µλ (λ2 t)
..
.
∞
Y
cos(2πλk t)
k =1
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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 ).
heγ , eγ̃ iL2
=
Z
eγ−γ̃ dµλ
= µ
bλ (γ − γ̃)
∞
Y
k
=
cos 2π λ (γ − γ̃)
k =1
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Fourier bases
Orthogonality Condition
Denote by eγ the exponential function e2πiγ· in L2 (µλ ).
heγ , eγ̃ iL2
=
Z
eγ−γ̃ dµλ
= µ
bλ (γ − γ̃)
∞
Y
k
=
cos 2π λ (γ − γ̃)
k =1
Lemma
The two exponentials eγ , eγ̃ are orthogonal if and only if one of the factors in
the infinite product above is zero. This is equivalent to
1 −k
λ (2m + 1) : k ∈ N, m ∈ Z =: Zλ .
γ − γ̃ ∈
4
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Fourier bases
Surprising first results
Theorem (Jorgensen, Pedersen 1998)
L2 (µ 1 ) has an ONB of exponential functions.
4
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Fourier bases
Surprising first results
Theorem (Jorgensen, Pedersen 1998)
L2 (µ 1 ) has an ONB of exponential functions.
4
Example
E(Γ 1 ) is an ONB for L2 (µ 1 ), where
4
Γ1 =
4
4

p
X

aj 4j : aj ∈ {0, 1}, p finite
j=0
K. Kornelson (U. Oklahoma)



= {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 = 31 , 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 = 31 , 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 = n1 , 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.
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Fourier bases
More recent progress
Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008)
For λ = ba , 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
orthonormal exponentials in L2 (µλ ).
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Fourier bases
More recent progress
Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008)
For λ = ba , 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
orthonormal exponentials in L2 (µλ ).
Theorem (Dutkay, Han, Jorgensen 2009)
If λ > 12 , 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


p

X
n no
Γ1 =
, p finite .
aj (2n)j : aj ∈ 0,
2n


2
j=0
We call Γ 1 the canonical spectrum and E(Γ 1 ) the canonical ONB for L2 (µ 1 ).
2n
2n
2n
Note: We will justify the nomenclature by describing alternate bases for the
same spaces.
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Families of ONBs
Families of ONBs
Jorgensen, K, Shuman 2011 JFAA
1
. We can construct alternate orthogonal families of exponentials
Let λ = 2n
from the canonical ONBs E(Γ 1 ). We then determine whether these alternate
2n
sets are ONBs.
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Families of ONBs
Families of ONBs
Jorgensen, K, Shuman 2011 JFAA
1
. We can construct alternate orthogonal families of exponentials
Let λ = 2n
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 L2 (µ 1 ).
8
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8
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Families of ONBs
Families of ONBs
Jorgensen, K, Shuman 2011 JFAA
1
. We can construct alternate orthogonal families of exponentials
Let λ = 2n
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 L2 (µ 1 ).
8
8
Theorem
If p <
2(2n−1)
,
π
then E(pΓ 1 ) is an ONB for L2 (µ 1 ).
K. Kornelson (U. Oklahoma)
2n
2n
Operator-Fractal
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Families of ONBs
Families of ONBs
Jorgensen, K, Shuman 2011 JFAA
1
. We can construct alternate orthogonal families of exponentials
Let λ = 2n
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 L2 (µ 1 ).
8
8
Theorem
If p <
2(2n−1)
,
π
then E(pΓ 1 ) is an ONB for L2 (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 41 case.
2n
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Operator-fractal
Operators on L2
Dutkay, Jorgensen, 2009:
When λ = 41 , both Γ 1 and 5Γ 1 are spectra for L2 (µ 1 ).
4
4
4
Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}
4
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Operator-fractal
Operators on L2
Dutkay, Jorgensen, 2009:
When λ = 41 , both Γ 1 and 5Γ 1 are spectra for L2 (µ 1 ).
4
4
4
Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}
4
For each γ ∈ Γ 1 , define
4
S0
S1
U
K. Kornelson (U. Oklahoma)
: eγ 7→ e4γ
: eγ →
7 e4γ+1
: eγ 7→ e5γ
Operator-Fractal
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Operator-fractal
Operators on L2
Dutkay, Jorgensen, 2009:
When λ = 41 , both Γ 1 and 5Γ 1 are spectra for L2 (µ 1 ).
4
4
4
Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}
4
For each γ ∈ Γ 1 , define
4
S0
S1
U
: eγ 7→ e4γ
: eγ →
7 e4γ+1
: eγ 7→ 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 = 41 , both Γ 1 and 5Γ 1 are spectra for L2 (µ 1 ).
4
4
4
Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}
4
For each γ ∈ Γ 1 , define
4
S0
S1
U
: eγ 7→ e4γ
: eγ →
7 e4γ+1
: eγ 7→ 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)
= 42 Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ)
..
.
= 4k Γ ∪ 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)
= 42 Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ)
..
.
= 4k Γ ∪ 4k −1 (1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ)
S0 (H) is the span of the exponentials E(4Γ)
S0k (H) is the span of E(4k Γ)
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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)
= 42 Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ)
..
.
= 4k Γ ∪ 4k −1 (1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ)
S0 (H) is the span of the exponentials E(4Γ)
S0k (H) is the span of E(4k Γ)
H ⊃ S0 (H) ⊃ S02 (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)
= 42 Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ)
..
.
= 4k Γ ∪ 4k −1 (1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ)
S0 (H) is the span of the exponentials E(4Γ)
S0k (H) is the span of E(4k Γ)
H ⊃ S0 (H) ⊃ S02 (H) ⊃ · · ·
If we define Wk = S0k (H) ⊖ S0k +1 (H), then
H = sp(e0 ) ⊕
∞
M
Wk
k =0
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Operator-fractal
The operator U
Recall U : eγ 7→ 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 7→ 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
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Operator-fractal
The operator U
Recall U : eγ 7→ 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!
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Operator-fractal
The operator U
Recall U : eγ 7→ 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 ) ⊕
∞
M
Me1 U.
k =1
We call U an operator-fractal.
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Operator-fractal
Matrix of U
0
Γ0
Γ1
Γ2
Γ3
K. Kornelson (U. Oklahoma)
0
1
0
0
0
0
..
.
Γ0
0
Me1 U
0
0
0
..
.
Γ1
0
0
Me1 U
0
0
..
.
Γ2
0
0
0
Me1 U
0
..
.
Operator-Fractal
Γ3
0
0
0
0
Me1 U
..
.
···
···
···
···
···
···
..
.
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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 )
K. Kornelson (U. Oklahoma)
Operator-Fractal
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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].
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
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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.
K. Kornelson (U. Oklahoma)
Operator-Fractal
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Operator-fractal
Nelson-like cyclic subspaces
Assume that Uv = v for some kv k = 1, v ∈
/ sp(e0 ).
We modify the cyclic subspaces of Nelson for use with our unitary operator
U.
Given v 6= 0, H(v ) := sp{U k v : k ∈ Z} is the U-cyclic subspace for v .
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
22 / 25
Operator-fractal
Nelson-like cyclic subspaces
Assume that Uv = v for some kv k = 1, v ∈
/ sp(e0 ).
We modify the cyclic subspaces of Nelson for use with our unitary operator
U.
Given v 6= 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 mv be the
real measure mv (A) = hE(A)v , v i.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
22 / 25
Operator-fractal
Nelson-like cyclic subspaces
Assume that Uv = v for some kv k = 1, v ∈
/ sp(e0 ).
We modify the cyclic subspaces of Nelson for use with our unitary operator
U.
Given v 6= 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 mv be the
real measure mv (A) = hE(A)v , v i.
If kv k = 1, mv is a probability measure.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
22 / 25
Operator-fractal
Nelson-like cyclic subspaces
Assume that Uv = v for some kv k = 1, v ∈
/ sp(e0 ).
We modify the cyclic subspaces of Nelson for use with our unitary operator
U.
Given v 6= 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 mv be the
real measure mv (A) = hE(A)v , v i.
If kv k = 1, mv is a probability measure.
Theorem
H(v ) = {φ(U)v : φ ∈ L2 (mv )}. Moreover, the map φ 7→ φ(U)v is an isometric
isomorphism between L2 (mv ) and H(v ).
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
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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
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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
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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 kv k = 1 then mv is a Dirac point mass at 1.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
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Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
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Operator-Fractal
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Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
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Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
24 / 25
Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
H(v2 ) ⊥ H(eγ ) by one of the lemmas above.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
24 / 25
Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
H(v2 ) ⊥ H(eγ ) by one of the lemmas above.
mv (A) = hE(A)v1 , v1 i + hE(A)v2 , v2 i = mv1 (A) + mv2 (A). The cross
terms are zero.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
24 / 25
Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
H(v2 ) ⊥ H(eγ ) by one of the lemmas above.
mv (A) = hE(A)v1 , v1 i + hE(A)v2 , v2 i = mv1 (A) + mv2 (A). The cross
terms are zero.
e v1 and m
e v2 , giving
Normalize to probability measures m
e v1 (A) + kv2 k2 m
e v2 (A).
mv (A) = kv1 k2 m
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
24 / 25
Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
H(v2 ) ⊥ H(eγ ) by one of the lemmas above.
mv (A) = hE(A)v1 , v1 i + hE(A)v2 , v2 i = mv1 (A) + mv2 (A). The cross
terms are zero.
e v1 and m
e v2 , giving
Normalize to probability measures m
e v1 (A) + kv2 k2 m
e v2 (A).
mv (A) = kv1 k2 m
Thus mv is a convex combination of probability measures.
K. Kornelson (U. Oklahoma)
Operator-Fractal
TAMU 07/18/2012
24 / 25
Operator-fractal
Proof of ergodicity of U:
Assume Uv = v for some v ∈
/ sp(e0 ), kv k = 1.
There exists some γ ∈ Γ \ {0} such that hv , eγ i =
6 0.
Let v1 be the projection of v onto H(eγ ) and v2 = v − v1 . Note that
v2 6= 0 and kv2 k2 = 1 − kv1 k2 .
H(v2 ) ⊥ H(eγ ) by one of the lemmas above.
mv (A) = hE(A)v1 , v1 i + hE(A)v2 , v2 i = mv1 (A) + mv2 (A). The cross
terms are zero.
e v1 and m
e v2 , giving
Normalize to probability measures m
e v1 (A) + kv2 k2 m
e v2 (A).
mv (A) = kv1 k2 m
Thus mv is a convex combination of probability measures.
But, by hypothesis, mv 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 07/18/2012
24 / 25
Thank You
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Operator-Fractal
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