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Spectral properties of
Toeplitz+Hankel Operators
Torsten Ehrhardt
University of California, Santa Cruz
ICM Satellite Conference on
Operator Algebras and Applications
Cheongpung, Aug. 8-12, 2014
Overview
1. Toeplitz operators T (a):
Invertibility/Fredholmness ↔ factorization theory (L∞-symbols)
Invertibility/Fredholm theory for PC symbols
Spectral theory for PC symbols
2. Toeplitz+Hankel operators T (a) +H(b):
Fredholm theory for PC symbols
For Special classes of Toeplitz+Hankel operators:
Invertibility/Fredholmness ↔ factorization theory
Invertibility theory for PC symbols
Spectral theory for PC symbols
1
Notation
T = { z ∈ C : |z| = 1 } unit circle
Hardy spaces
Hp = { f ∈ Lp(T) : fn = 0 for all n < 0 }
Hp = { f ∈ Lp(T) : fn = 0 for all n > 0 }
Some operators on Lp(T)N , 1 < p <∞:
P :∞∑
n=−∞fnt
n 7→∞∑n=0
fntn Riesz projection
J : f(t) 7→ t−1f(t−1) flip operator
L(a) : f(t) 7→ a(t)f(t) multiplication operator
(a ∈ L∞(T)N×N)
2
Multiplication operator
matrix representation of L(a) with respect to basis {tn}∞n=−∞ in Lp(T)
L(a) ∼= [aj−k] =
. . . . . . . . .
. . . a0 a−1 a−2. . .
. . . a1 a0 a−1 a−2a2 a1 a0 a−1
. . .. . . a2 a1 a0
. . .. . . . . . . . .
(Laurent matrix)
an =1
2π
∫ 2π
0a(eix)e−inx dx Fourier coefficients
3
Toeplitz and Hankel operators
Given a ∈ L∞(T)N×N , define
T (a) = PL(a)P |(Hp)N , H(a) = PL(a)JP |(Hp)N
acting on the vector-valued Hardy space (Hp(T))N , 1 < p <∞.
Matrix representation with respect to standard basis {tn}∞n=0 in Hp:
T (a) ∼= [aj−k] H(a) ∼= [aj+k+1]a0 a−1 a−2 . . .a1 a0 a−1a2 a1 a0
. . .... . . . . . .
a1 a2 a3 . . .a2 a3 a4a3 a4 a5...
4
PART I
Toeplitz operators
(classical and well known)
5
Toeplitz, Hankel operators: identities
For a, b ∈ L∞(T),
T (ab) = T (a)T (b) +H(a)H (b)
H(ab) = T (a)H(b) +H(a)T (b)
where b(t) := b(t−1).
Note:
• If a ∈ H∞ or b ∈ H∞, then T (ab) = T (a)T (b)
(because H(a) = 0 or H (b) = 0).
• T (a) lower triangular Toeplitz for a ∈ H∞
• T (b) upper triangular Toeplitz for b ∈ H∞
6
Toeplitz operators: factorization (Motivation)
Assume a = a−da+ with a− ∈ H∞ and a+ ∈ H∞. Then
T (a) = T (a−)T (d)T (a+).
If, in addition, a−1− ∈ H∞, d = 1, a−1
+ ∈ H∞, then
T (a)−1 = T (a+)−1T (a−)−1 = T (a−1+ )T (a−1
− ).
Note:
• T (a+)−1 = T (a−1+ ) and T (a−)−1 = T (a−1
− )
• T (a)−1 = PL(a−1+ )PL(a−1
− )P = L(a−1+ )PL(a−1
− )
i.e., T (a−1) : f 7→ a−1+ · P (a−1
− · f)
7
Wiener-Hopf factorization of smooth functions
Consider
W := { a ∈ L∞(T) :∞∑
n=−∞|an| <∞ } Wiener algebra
W+ := W ∩H∞, W− := W ∩H∞.
Assume that a ∈WN×N has a Wiener-Hopf factorization in W , i.e.,
a(t) = a−(t)d(t)a+(t)
with
• a+, a−1+ ∈WN×N
+
• a−, a−1− ∈W
N×N−
• d(t) = diag(tκ1, . . . , tκN), t ∈ T,
κ1, . . . , κN ∈ Z partial indices.
8
Then:
T (a) is a Fredholm operator and
dim ker T (a) = −∑κk<0
κk, dim ker T (a)∗ =∑κk>0
κk.
In particular:
T (a) is invertible iff κ1 = · · · = κN = 0.
Reason: T (a) = T (a−)T (d)T (a+) with T (a±) invertible
Moreover, for a ∈WN×N :
T (a) Fredholm ⇒ det a(t) 6= 0 ⇒ a(t) possesses factorization
9
Generalized Factorization
Generalized or Φ-factorization in Lp of a ∈ (L∞)N×N :
a(t) = a−(t)d(t)a+(t)
with
• a+ ∈ (Hq)N×N , a−1+ ∈ (Hp)N×N ,
• a− ∈ (Hp)N×N , a−1− ∈ (Hq)N×N ,
• d(t) = diag(tκ1, . . . , tκN), t ∈ T, κ1, . . . , κN ∈ Z partial indices,
• the mapping
f 7→ a−1+ · P (a+ · f) (∗)
extends to a bounded linear operator on (Hp)N .
(Here 1 < p <∞, 1/p+ 1/q = 1.)
10
Remarks:
• existence of factorization, factors and part. indices depend on p
• partial indices are unique, up to change of order
• factors a± are unique up to “simple” modification
• condition (∗) is equivalent to an Ap-condition or
Hunt-Muckenhoupt-Wheeden condition
• for N = 1 and, e.g., functions a ∈ PC:
factorization can be constructed explicitly
• for N > 1, except in special cases, no explicit procedure for the
construction of the factorization (or determination of the partial indices)
is known.
(PC = set of piecewise continuous functions on T)
11
Construction in the scalar case (illustration)
Construction of factorization in case a ∈W :
Assume a(t) 6= 0 and wind(a) = κ.
Then
a(t) = a−(t)tκa+(t)
with
a+ = exp(Pb), a− = exp((I − P )b),
and
b(t) = log[a(t)t−κ
].
12
Theorem: (Gohberg, Krupnik)
Let a ∈ (L∞)N×N . Then T (a) is a Fredholm operator on (Hp)N if and only
if a admits a Φ-factorization in Lp. In this case,
dim ker T (a) = −∑κk<0
κk, dim ker T (a)∗ =∑κk>0
κk.
Hence: T (a) is invertible iff, in addition, κ1 = · · · = κN = 0.
13
Toeplitz operators: Fredholm theory for PC-symbols
Theorem: Let a ∈ PCN×N .
Then T (a) is Fredholm on (Hp)N iff
• a(t+ 0) and a(t− 0) are invertible matrices for all t ∈ T,
• for each t ∈ T, the arguments of the eigenvalues of the matrix
a(t− 0)a(t+ 0)−1 are all different from 2π/p+ 2πZ, i.e.,
1
2πarg EVk
[a(t− 0)a(t+ 0)−1
]/∈
1
p+ Z, 1 ≤ k ≤ N.
Here: one-sided limits of a ∈ PCN×N at t ∈ T:
a(t± 0) = limε→+0
a(te±iε)
14
Essential spectrum of T (a), a ∈ PC:
λ belongs to the essential spectrum of T (a) (acting on Hp)
m
T (a− λ) is not Fredholm on Hp
m
for some t ∈ T:
a(t+ 0)− λ = 0 or a(t− 0)− λ = 0 or
1
2πarg
(a(t− 0)− λa(t+ 0)− λ
)∈
1
p+ Z
15
Corollary:
The essential spectrum of T (a) with a ∈ PC is
spess(T (a) = R(a) ∪⋃t∈JA1/p[a(t− 0), a(t+ 0)].
Here:
• R(a) = essential range of the function a ∈ PC ⊆ L∞,
• A1/p[u, v] ={z ∈ C : 1
2π arg u−zv−z ∈
1p + Z
}line segment (p = 2) or certain circular arc (p 6= 2)
connecting the points u and v,
• J = { t ∈ T : a(t+ 0) 6= a(t− 0)} set of jumps of a
16
Example
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
image of some a ∈ PC
-1 1
-i
i
spess(T (a)); p = 2
17
-1 1
-i
i
p = 2
-1 1
-i
i
p = 2.4
-1 1
-i
i
p = 2.8
-1 1
-i
i
p = 3.218
Spectrum of T (a), a ∈ PC
Coburn’s lemma:
If a ∈ L∞(T), a 6≡ 0, then T (a) or T (a)∗ has a trivial kernel.
Corollary: Let a ∈ L∞(T). Then:
T (a) is invertible ⇔ T (a) is Fredholm and has Fredholm index zero.
For a ∈ PC, the spectrum of T (a) can be described geometrically:
sp(T (a)) = im(a#,p) ∪{λ /∈ im(a#,p) : wind(a#,p;λ) 6= 0
}
19
PART II
Toeplitz+Hankel operators
20
Toeplitz+Hankel operators
The goal is to develop, if possible, a Fredholm and invertibility theory
(more generally: spectral theory) for Toeplitz + Hankel operators
T (a) +H(b),
a, b ∈ L∞(T).
Questions:
• Is there relation to factorization theory ?
• Can one establish explicit invertibility criteria, say, for a, b ∈ PC ?
• Is there a geometric charaterization of the spectrum ?
21
Fredholm theory for T (a) +H(b) with a, b ∈ PCN×N
One can establish explicit criteria for the Fredholmness of T (a) +H(b) with
a, b ∈ PCN×N . The conditions can be expressed in terms of
a(t± 0) and b(t± 0)
exclusively, similar as, but more complicated than in the Toeplitz case.
Moreover, available:
• Fredholm index of T (a) +H(b)
• essential spectrum of T (a) +H(b)
Difficult (open) problem:
• invertibility and spectrum
22
Equivalence after extension
Notion:
A ∈ L(X) and B ∈ L(Y ) are called equivalent after extension if there exist
Banach spaces X1 and Y1 and invertible operators
E : X ⊕X1 → Y ⊕ Y1, F : Y ⊕ Y1 → X ⊕X1
such that
E
(A 00 IX1
)F =
(B 00 IY1
).
In this case:
A invertible iff B invertible,
A Fredholm iff B Fredholm,
dim kerA = dim kerB, etc.
23
T+H operators ∼ block T-operators
Let a, b ∈ L∞(T) and assume a−1 ∈ L∞(T). Then(T (a) +H(b) 0
0 T (a)−H(b)
)acting of Hp ⊕Hp
is equivalent after extension to
T (Φ) acting of Hp ⊕Hp
with
Φ =
(a b0 1
)(1 0b a
)−1
=
(a− ba−1b ba−1
−ba−1 a−1
)
Remarks:
• Φ is triangular if aa = bb (important special case)
• Triangular matrix functions can (under certain conditions) be
factored explicitly.
• Special case: a = b, i.e., T (a) +H(a).
24
Special Toeplitz+Hankel operators:
T (a) +H(a)
[Basor/E. 2004]
25
Identities for T (a) +H(a)
Let us consider
M(a) := T (a) +H(a).
Recall
T (ab) = T (a)T (b) +H(a)H (b)
H(ab) = T (a)H(b) +H(a)T (b)
to derive the identity
M(ab) = M(a)M(b) +H(a)M (b− b)
Hence
M(ab) = M(a)M(b)
if a ∈ H∞ or b = b (i.e., b(t) = b(t−1)).
26
This suggest to consider the “asymmetric” factorization:
a = a−da0
where
• a−, a−1− ∈ H∞
• a0 = a0 and a0, a−10 ∈ L∞
In this case:
M(a) = M(a−)M(d)M(a0)
where
• M(a−) = T (a−) is invertible: T (a−)−1 = T (a−1− )
• M(a0) is invertible: M(a0)−1 = M(a−10 )
27
Trivial observation: even symbols
Suppose a ∈ L∞(T) is even.
Then M(a) is Fredholm (invertible) iff a−1 ∈ L∞(T).
Hence
sp(M(a)) = spess(M(a)) = R(a).
Consequence:
(Essential) spectrum of M(a) can be disconnected.
(contrasting the case of T (a))
28
Asymmetric factorization vs. Wiener-Hopf factorization
(Heuristic)
Assume
a = a−da0
with a±1− ∈ H∞ and a±1
0 ∈ L∞ and a0 = a0.
Then
aa−1 = a−da0a−10 d−1a−1
− = a−dd−1a−1
−
i.e.,
aa−1 = a−ra−1−
with middle factor r = dd−1 and a±1− ∈ H∞.
29
Asymmetric factorization and Fredholmness of M(a)
We say that a ∈ L∞ possesses an asymmetric Φ-factorization in Lp if
a(t) = a−(t)tκa0(t)
with
• (1 + t−1)a−(t) ∈ Hp, (1− t−1)a−1− (t) ∈ Hq,
• |1− t|a0(t) ∈ Lqeven, |1 + t|a−10 (t) ∈ Lpeven,
• κ ∈ Z (index),
• the mapping
f 7→ a−10 · (I + J)P (a−1
− · f) (∗∗)
extends to a bounded linear operator on Hp.
Here
Lpeven :=
{b ∈ Lp(T) : b = b
}and 1 < p <∞, 1/p+ 1/q = 1.
30
Theorem (Basor/ E. 2004):
Let a ∈ L∞. Then M(a) = T (a) +H(a) is Fredholm in Hp iff
a admits an asymmetric Φ-factorization in Hp.
In this case
dim kerM(a) = −min{0, κ}, dim kerM(a)∗ = max{0, κ}
In particular: M(a) is invertible iff κ = 0.
Remark:
Condition (∗∗) is equivalent to an Ap-condition for the function σ defined on
[−1,1] by
σ(cosx) = |a−10 (eix)|
(1 + cosx)1/2q
(1− cosx)1/2p
31
Asymmetric vs. Antisymmetric factorization
Assume a, a−1 ∈ L∞. Then:
a admits an asymmetric factorization in Lp,
a(t) = a−(t) tκ a0(t),
• (1 + t−1)a−(t) ∈ Hp, (1− t−1)a−1− (t) ∈ Hq,
• |1− t|a0(t) ∈ Lqeven, |1 + t|a−10 (t) ∈ Lpeven,
m
aa−1 admits an anti-symmetric factorization in Lp,
a(t)a−1(t) = a−(t) t2κ a−1− (t),
• (1 + t−1)a−(t) ∈ Hp, (1− t−1)a−1− (t) ∈ Hq.
32
Fredholmness of M(a), a ∈ PC
Theorem. Let a ∈ PC. Then M(a) = T (a) +H(a) is Fredholm on Hp iff
a(t± 0) 6= 0 for all t ∈ T and
•1
2πarg
a(1− 0)
a(1 + 0)/∈
1
2p+ Z
•1
2πarg
a(−1− 0)
a(−1 + 0)/∈
1
2p+
1
2+ Z
•1
2πarg
a(t− 0)a(t− 0)
a(t+ 0)a(t+ 0)/∈
1
p+ Z
for all t ∈ T, Im(t) > 0.
33
Essential spectrum of M(a)
Corollary:
The essential spectrum of T (a) +H(a) on Hp with a ∈ PC is equal to
spess(T (a) =R(a)
∪ A1/2p[a(1− 0), a(1 + 0)]
∪ A1/2p+1/2[a(−1− 0), a(−1 + 0)]
∪⋃t∈JH1/p[a(t− 0), a(t+ 0), a(t− 0), a(t+ 0)].
where
H1/p(u, v, x, y) ={λ ∈ C :
1
2πarg
(u− λ)(x− λ)
(v − λ)(y − λ)∈
1
p+ Z
}
34
Example
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
image of some a ∈ PC
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
spess(M(a)); p = 2
35
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
p = 2
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
p = 1.34
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
p = 1.32...
-1 1 2 3
-i
i
-i-
-1+
-i+
1-
i-
1+
-1-
i+
Fig. 1: image of �
-1 1 2 3
-i
i
Fig. 2: p = 1.3
-1 1 2 3
-i
i
Fig. 3: p = 1.32324434438365901
-1 1 2 3
-i
i
Fig. 4: p = 1.34
-1 1 2 3
-i
i
Fig. 5: p = 2
-1 1 2 3
-i
i
Fig. 6: p = 5
25
p = 1.336
Spectrum of M(a), a ∈ PC
• For a ∈ PC and M(a) Fredholm on Hp, the Fredholm index of M(a)
can be determined “geometrically”.
• An analogue of Coburn’s lemma holds:
Theorem: Let a ∈ L∞, a−1 ∈ L∞. Then M(a) or M(a)∗ has a trivial kernel.
Corollary: Let a ∈ L∞. Then M(a) is invertible iff M(a) is Fredholm and
Fredholm index of T (a) is zero.
• For a ∈ PC, the spectrum of M(a) can be described geometrically.
37
Generalizations
Same methods as for
T (a) +H(a) = (aj−k + aj+k+1)
also work for operators
T (a)−H(a) = (aj−k − aj+k+1)
and
T (a) +H(at) = (aj−k + aj+k)
and
T (a)−H(at−1) = (aj−k − aj+k+2)
(Results are similar)
38
Still special, but more general operators:
T (a) +H(b)
with aa = bb
39
Fredholmness of T (a) +H(b), a, b ∈ PC
Theorem:
Let a, b ∈ PC satisfy aa = bb and assume (wlog) a−1, b−1 ∈ PC.Let 1 < p <∞, 1/p+ 1/q = 1. Put
c =a
b=b
a, d =
a
b=b
a.
Then T (a) +H(b) is Fredholm on Hp if and only if
1
2πarg c−(1) /∈
1
2+
1
2p+ Z,
1
2πarg d−(1) /∈
1
2+
1
2q+ Z,
1
2πarg c−(−1) /∈
1
2p+ Z,
1
2πarg d−(−1) /∈
1
2q+ Z,
1
2πarg
(c−(τ)
c+(τ)
)/∈
1
p+ Z,
1
2πarg
(d−(τ)
d+(τ)
)/∈
1
q+ Z,
for each τ ∈ T, Im(τ) > 0.
Notation: c±(τ) = c(τ ± 0), d±(τ) = d(τ ± 0).
40
Geometric Interpretation
The function c ∈ PC satisfies cc = 1.
Construct a curve c#,p as follows:
• consider the image of c(eix), 0 < x < π
⇒ “curve with jumps”
• fill in the arc A1/2+1/2p(1, c+(1)) (if 1 6= c+(1))
• fill in the arcs A1/p(c−(τ), c+(τ)) (whenever c−(τ) 6= c+(τ), Im(τ) > 0)
• fill in the arc A1/2p(c−(−1),1) (if 1 6= c−(1))
⇒ This gives a closed oriented curve c#,p.
Similar construction to obtain a closed oriented curve d#,q.
41
Under the assumptions on a, b as in the theorem:
T (a) +H(b) is Fredholm on Hp iff
0 does not lie on any of the curves c#,p and d#,q.
Moreover,
ind(T (a) +H(b)) = wind(d#,q)−wind(c#,p)
.
42
Example:
Consider function c ∈ PC, cc = 1 whose image c(eix), 0 < x < π is as follows:
-1 1
-i
i
cH1+0L
cHi-0L
cHi+0L
cH-1-0L
43
-1 1
-i
i
p = 1.16
-1 1
-i
i
p = 1.13
-1 1
-i
i
p = 4/3
-1 1
-i
i
cH1+0L
cHi-0L
cHi+0L
cH-1-0L
c(eix), 0 < x < π
-1 1
-i
i
p = 2
-1 1
-i
i
p = 1.5
44
Factorization results:
Theorem (Basor/ E. 2013):
Let a, b ∈ L∞ satisfy aa = bb, and assume a−1, b−1 ∈ L∞.
Define the auxiliary functions
c =a
b=b
a, d =
a
b=b
a.
Now assume that c and d have factorizations of the form
c = c+t2nc−1
+ , d = d+t2md−1
+
with n,m ∈ Z and
(1 + t)c+ ∈ Hq, (1− t)c−1+ ∈ Hp
(1 + t)d+ ∈ Hp, (1− t)d−1+ ∈ Hq
1 < p <∞, 1/p+ 1/q = 1.
45
Now also assume that T (a) +H(b) are Fredholm operators on Hp.
Then
dim ker(T (a) +H(b)) =
0 if n > 0, m ≤ 0−n if n ≤ 0, m ≤ 0dim kerAn,m if n > 0, m > 0m− n if n ≤ 0, m > 0,
dim ker(T (a) +H(b))∗ =
0 if m > 0, n ≤ 0−m if m ≤ 0, n ≤ 0dim ker(An,m)T if m > 0, n > 0n−m if m ≤ 0, n > 0.
Therein, in case n > 0, m > 0,
An,m :=[ρi−j + ρi+j
]n−1
i=0
m−1
j=0.
and
ρ := t−m−n(1 + t)(1 + t−1)c+d+b−1 ∈ L1(T).
In particular, the Fredholm index of T (a) +H(b) is equal to m− n.
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Remarks:
• Factorizations of c, d exist if T (a) +H(b) is Fredholm and a, b ∈ PC.
They can be constructed explicitly.
n = wind(c#,p), m = wind(d#,q)
• In general (a, b /∈ PC) such factorizations need not exist,
even if T (a) +H(b) is invertible !!!
Corollary:
Assume all of the above (in particular Fredholmness of T (a) +H(b)).
Then T (a) +H(b) is invertible on Hp iff
• n = m = 0, or
• n = m > 0 and An,n is an invertible matrix.
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What’s missing ... ?
Nonetheless, we have no description of spectrum of
T (a) +H(b) with aa = bb, a, b ∈ PC
yet !!!
Reason:
T (a) +H(b)− λI = T (a− λ) +H(b− λ),
but (a− λ)(a− λ) 6= (b− λ)(b− λ) in general.
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Thank you!
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