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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