Toral and Spherical Aluthge Transforms (joint work with ...adonsig1/NIFAS/1711-Curto.pdf · Toral...

Toral and Spherical Aluthge Transforms(joint work with Jasang Yoon)

Raul E. Curto

University of Iowa

INFAS, Des Moines, IA

November 11, 2017

(our first report on this research appeared in

Comptes Rendus Acad. Sci. Paris (2016))

Raul E. Curto (Univ. of Iowa) Multivariable Aluthge Transforms INFAS, Nov 11, 2017

Overview

1 Unilateral Weighted Shifts

2 Aluthge Transforms

3 Multivariable Weighted Shifts

4 Toral Aluthge Transform

5 Spherical Aluthge Transform

6 Invariance of Hyponormality

7 Spherically Quasinormal Pairs


Hyponormality and Subnormality

L(H): algebra of operators on a Hilbert space HT ∈ L(H) is

normal if T ∗T = TT ∗

subnormal if T = N|H, where N is normal and NH ⊆ H (We say

that N is a lifting of T , or an extension of T .

hyponormal if T ∗T ≥ TT ∗

normal ⇒ subnormal ⇒ hyponormal


For S ,T ∈ B(H), [S ,T ] := ST − TS .

An n-tuple T ≡ (T1, ...,Tn) is (jointly) hyponormal if

[T∗,T] :=

[T ∗1 ,T1] [T ∗

2 ,T1] · · · [T ∗n ,T1]

[T ∗1 ,T2] [T ∗

2 ,T2] · · · [T ∗n ,T2]

...... · · · ...

[T ∗1 ,Tn] [T ∗

2 ,Tn] · · · [T ∗n ,Tn]

≥ 0.

For k ≥ 1, an operator T is k-hyponormal if (T , ...,T k) is (jointly)

hyponormal, i.e.,

[T ∗,T ] · · · [T ∗k ,T ]...

. . ....

[T ∗,T k ] · · · [T ∗k ,T k ]

≥ 0

(Bram-Halmos):

T subnormal ⇔ T is k-hyponormal for all k ≥ 1.


Unilateral Weighted Shifts

α ≡ {αk}∞k=0 ∈ ℓ∞(Z+), αk > 0 (all k ≥ 0)

Wα : ℓ2(Z+) → ℓ2(Z+)

Wαek := αkek+1 (k ≥ 0)

When αk = 1 (all k ≥ 0), Wα = U+, the (unweighted) unilateral shift

In general, Wα = U+Dα (polar decomposition)

‖Wα‖ = supk αk

W nαek = αkαk+1 · · ·αk+n−1ek+n, so

W nα∼=

n−1⊕

i=0

Wβ(i) ,


Weighted Shifts and Berger’s Theorem

Given a bounded sequence of positive numbers (weights)

α ≡ α0, α1, α2, ..., the unilateral weighted shift on ℓ2(Z+) associated with

α is

Wαek := αkek+1 (k ≥ 0).

The moments of α are given as

γk ≡ γk(α) :=

{1 if k = 0

α20 · ... · α2

k−1 if k > 0

}.

Wα is never normal

Wα is hyponormal ⇔ αk ≤ αk+1 (all k ≥ 0)


Berger Measures

(Berger; Gellar-Wallen) Wα is subnormal if and only if there exists a

positive Borel measure ξ on [0, ‖Wα‖2] such that

γk =

∫tk dξ(t) (all k ≥ 0).

ξ is the Berger measure of Wα.

For 0 < a < 1 we let Sa := shift (a, 1, 1, ...).

The Berger measure of U+ is δ1.

The Berger measure of Sa is (1− a2)δ0 + a2δ1.

The Berger measure of B+ (the Bergman shift) is Lebesgue measure

on the interval [0, 1]; the weights of B+ are αn :=√

n+1n+2 (n ≥ 0).


Spectral Pictures of Hyponormal U.W.S.

WLOG, assume ‖Wα‖ = 1. Observe that r(Wα) = 1 = supα. Thus,

σ(Wα) = D

σe(Wα) = T

ind(Wα − λ) = −1 for |λ| < 1.

Therefore, all norm-one hyponormal weighted shifts are spectrally

equivalent.


(0, 0)

index = −1

0 1

index = −1

σe(Wα)

Figure 1. Spectral picture of a norm-one hyponormal weighted shift


(RC, 1990) Wα is k-hyponormal if and only if the following Hankel

moment matrices are positive for m = 0, 1, 2, . . . :

γm γm+1 γm+2 . . . γm+k

γm+1 γm+2 . . . γm+k+1

γm+2 . . . . . . γm+k+2

......

...

γm+k γm+k+1 . . . γm+2k

≥ 0.

(Thus, an operator matrix condition is replaced by a scalar matrix

condition.)


Aluthge Transform

Let T be a Hilbert space operator, let P := |T | be its positive part, and

let T = VP denote the canonical polar decomposition of T , with V a

partial isometry and ker V = ker T = ker P .

We define the Aluthge transform of T as

T :=√PV

√P .

The iterates are

T n+1 := (T )n (n ≥ 1).

The Aluthge transform has been extensively studied, in terms of algebraic,

structural and spectral properties.


For instance,

(i) T = T ⇔ T is quasinormal;

(ii) (Aluthge, 1990) If 0 < p < 12 and T is p-hyponormal, then T is

(p + 12)-hyponormal;

(iii) (Jung, Ko & Pearcy, 2000) If T has a n.i.s., then T has a n.i.s.

(iv) (Kim-Ko, 2005; Kimura, 2004) T has property (β) if and only if

T has property (β); and

(v) (Ando, 2005)∥∥(T − λ)−1

∥∥ ≥∥∥∥(T − λ)−1

∥∥∥ (λ /∈ σ(T )).

(vi) Observe that if A :=√P and B := V

√P , then T = AB and

T = BA, and therefore

σ(T ) \ {0} = σ(T ) \ {0}.


On the other hand,

G. Exner (IWOTA 2006 Lecture): subnormality is not preserved under the

Aluthge transform. Concretely, Exner proved that the Aluthge transform

of the weighted shift in the following example is not subnormal.

Example

(RC, Y. Poon and J. Yoon, 2005) Let

α ≡ αn :=

√12 , if n = 0√2n+ 1

22n+1 , if n ≥ 1

,

Then Wα is subnormal, with 3-atomic Berger measure

µ =1

3(δ0 + δ1/2 + δ1).


(S.H. Lee, W.Y. Lee and J. Yoon, 2012) For k ≥ 2, the Aluthge

transform, when acting on weighted shifts, need not preserve

k-hyponormality.

Note that the Aluthge transform of a weighted shift is again a weighted

shift.

Concretely, the weights of Wα are

√α0α1,

√α1α2,

√α2α3,

√α3α4, · · · .

Define

W√α := shift (

√α0,

√α1,

√α2, · · · ) .

Then Wα is the Schur product of W√α and its restriction to the subspace

∨{e1, e2, · · · }. Thus, a sufficient condition for the subnormality of Wα is

the subnormality of W√α.


Agler Shifts

For j = 2, 3, . . ., the j-th Agler shift Aj is given by

αj :=

√1

j,

√2

j + 1,

√3

j + 2, . . . .

It is well known that Aj is subnormal, with Berger measure

dµj(t) = (j − 1)(1− t)j−2dt.

Clearly, A2 is the Bergman shift, and the remaining Agler shifts are the

upper row shifts of the Drury-Arveson 2-variable weighted shift, which

incidentally is a spherical complete hyperexpansion.


Weight Diagram of the Drury-Arveson Shift

T1

T2

(0, 1)

(0, 2)

(0, 3)

(0, 0) (1, 0) (2, 0) (3, 0)

1 1 1 · · · δ1

√

1

2

√

2

3

√

3

4· · · dt

√

1

3

√

2

4

√

3

5· · · 2(1− t)dt

√

1

4

√

2

5

√

3

6· · · 3(1− t)2dt· · · · · · · · · · · ·

1

1

1

...

√

1

2

√

2

3

√

3

4

...

√

1

3

√

2

4

√

3

5

...


Multivariable Weighted Shifts

αk, βk ∈ ℓ∞(Z2+), k ≡ (k1, k2) ∈ Z2

+ := Z+ × Z+

ℓ2(Z2+)

∼= ℓ2(Z+)⊗

ℓ2(Z+).

We define the 2-variable weighted shift T ≡ (T1,T2) by

T1ek := αkek+ε1 T2ek := βkek+ε2 ,

where ε1 := (1, 0) and ε2 := (0, 1). Clearly,

T1T2 = T2T1 ⇐⇒ βk+ε1αk = αk+ε2βk (all k).

(k1, k2) (k1 + 1, k2)

αk1,k2

αk1,k2+1(k1, k2 + 1) (k1 + 1, k2 + 1)

βk1,k2βk1+1,k2


(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01 α11 α21 · · ·

α02 α12 α22 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

β01

β02

...

β10

β11

β12

...

β20

β21

β22

...


Recall the definition of joint hyponormality.

An n-tuple T ≡ (T1, ...,Tn) is (jointly) hyponormal if

[T∗,T] :=

[T ∗1 ,T1] [T ∗

2 ,T1] · · · [T ∗n ,T1]

[T ∗1 ,T2] [T ∗

2 ,T2] · · · [T ∗n ,T2]

...... · · · ...

[T ∗1 ,Tn] [T ∗

2 ,Tn] · · · [T ∗n ,Tn]

≥ 0.


To detect hyponormality, there is a simple criterion:

Theorem

(RC, 1988) (Six-point Test) Let T ≡ (T1,T2) be a 2-variable weighted

shift, with weight sequences α and β. Then

T is hyponormal ⇔(

α2k+ε1

− α2k αk+ε2βk+ε1 − αkβk

αk+ε2βk+ε1 − αkβk β2k+ε2

− β2k

)≥ 0

(all k ∈ Z2+).


(k1, k2) (k1 + 1, k2) (k1 + 2, k2)

αk1,k2αk1+1,k2

αk1,k2+1

(k1, k2 + 1)

(k1, k2 + 2)

(k1 + 1, k2 + 1)

βk1,k2

βk1,k2+1

βk1+1,k2

Figure 3. Weight diagram used in the Six-point Test


We now recall the notion of moment of order k for a commuting pair

(α, β). Given k ∈ Z2+, the moment of (α, β) of order k is γk ≡ γk(α, β)

:=

1 if k = 0

α2(0,0) · ... · α2

(k1−1,0) if k1 ≥ 1 and k2 = 0

β2(0,0) · ... · β2

(0,k2−1) if k1 = 0 and k2 ≥ 1

α2(0,0) · ... · α2

(k1−1,0) · β2(k1,0)

· ... · β2(k1,k2−1) if k1 ≥ 1 and k2 ≥ 1.

By commutativity, γk can be computed using any nondecreasing path from

(0, 0) to (k1, k2).


(Jewell-Lubin)

Wα is subnormal ⇔ γk :=

k1−1∏

i=0

α2(i ,0) ·

k2−1∏

j=0

β2(k1−1,j)

=

∫tk11 tk22 dµ(t1, t2) (all k ≥ 0).

Thus, the study of subnormality for multivariable weighted shifts is

intimately connected to multivariable real moment problems.


The Spectral Picture of Subnormal

2-variable Weighted Shifts

For subnormal 2-variable weighted shifts, RC-K. Yan gave in 1995 a

complete description of the spectral picture, by exploiting the groupoid

machinery in Muhly-Renault and RC-Muhly, and the presence of the

Berger measure, which was used to analyze the asymptotic behavior of

sequences of weights.

Notation

µ : compactly supported finite positive Borel measure on Cn (n > 1)

P2(µ) : norm closure in L2(µ) of C[z1, · · · , zn]Mz ≡ M

(µ)z := (M

(µ)z1 , · · · ,M(µ)

zn ) : multiplication operators acting on

P2(µ)

Mz on P2(µ) is the universal model for cyclic subnormal n-tuples


Theorem

(RC-K. Yan, 1995) Let µ be a Reinhardt measure on C2, and let

C := log∣∣∣K∣∣∣. Assume that ∂K

⋂(z1z2 = 0) contains no 1-dimensional

open disks. Then

(i) σT (Mz,P2(µ)) = σr (Mz,P

2(µ)) = K

(ii) σTe(Mz,P2(µ)) = σre (Mz,P

2(µ)) = ∂K

(iii) index(Mz − λ) =

{1 if λ ∈ int.(K )

0 if λ /∈ int.(K )

(vi) kerD1Mz−λ = ranD0

Mz−λ for all λ ∈ int.(K ).


(0, 0) (a, 0) (1, 0)

|z2|

(0, b)

(0, 1)

’

-

index = 1

σT e(Mz)

|z1|

Figure 5. Spectral picture of a typical subnormal 2-variable weighted shift


Toral Aluthge Transform

We introduce the toral Aluthge transform of 2-variable weighted shifts

W(α,β) ≡ (T1,T2).

For i = 1, 2, consider the polar decomposition Ti ≡ Ui |Ti |. Then for a

2-variable weighted shift W(α,β) ≡ (T1,T2), we define the toral Aluthge

transform of W(α,β) as follows:

W(α,β) := ˜(T1,T2) :=(|T1|

12U1|T1|

12 , |T2|

12U2|T2|

12

). (1)

Observe: commutativity of W(α,β) does not automatically follow from the

commutativity of W(α,β); actually, the necessary and sufficient condition to

preserve commutativity is

α(k1,k2+1)α(k1+1,k2+1) = α(k1+1,k2)α(k1,k2+2) (for all k1, k2 ≥ 0).


(k1, k2) (k1 + 1, k2) (k1 + 2, k2)

α(k1,k2+1) α(k1+1,k2+1)

· · · · · · · · ·

T1

T2

(k1, k2 + 1)

(k1, k2 + 2)


(k1, k2) (k1 + 1, k2) (k1 + 2, k2)

α(k1+1,k2)

αk1,k2+2

· · · · · · · · ·

T1

T2

(k1, k2 + 1)

(k1, k2 + 2)


(k1, k2) (k1 + 1, k2) (k1 + 2, k2)

α(k1,k2) α(k1+1,k2)

α(k1,k2+1) α(k1+1,k2+1)

αk1,k2+2

· · · · · · · · ·

T1

T2

(k1, k2 + 1)

(k1, k2 + 2)

β(k1,k2)

β(k1,k2+1)

β(k1+1,k2)

β(k1+1,k2+1)


Spherical Aluthge Transform

Consider a (joint) polar decomposition of the form

(T1,T2) ≡ (U1P ,U2P) .

where P :=√T ∗1T1 + T ∗

2T2. Now let

W(α,β) :=(√

PU1

√P ,

√PU2

√P), (2)

One can prove that U∗1U1 + U∗

2U2 is a (joint) partial isometry, and that

W(α,β) is commutative whenever W(α,β) is commutative.


The toral Aluthge transform does not preserve hyponormality:

x0 ω0 ω1 · · · ξ

a ω0 ω1 · · ·

a ω0 ω1 · · ·

· · · · · · · · ·

x0

ω0

ω1

...

a

ω0

ω1

...

a

ω0

ω1

...

Let ξ be the Berger measure of shift (ω0, ω1, · · · ) and let ρ :=∫

1sdξ(s).

Theorem

Assume ω21ρ < 2. Then: (i) (T1,T2) is subnormal; (ii) ˜(T1,T2) is not

hyponormal.

(The condition ω21ρ < 2 can be satisfied with a 2-atomic measure ξ.)


Question

When is hyponormality invariant under the toral Aluthge transform?

Given a 1-variable weighted shift Wω, let Θ (Wω) ≡ W(α,β) be the

2-variable weighted shift given by

α(k1,k2) = β(k1,k2) := ωk1+k2 for k1, k2 ≥ 0.

We will say that Θ (Wω) is a 2-variable embedding of the unilateral

weighted shift Wω.


(0, 0) (1, 0) (2, 0) (3, 0)

ω0 ω1 ω2 · · ·

ω1 ω2 ω3 · · ·

ω2 ω3 ω4 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

ω0

ω1

ω2

...

ω1

ω2

ω3

...

ω2

ω3

ω4

...

(i)


(0, 0) (1, 0) (2, 0) (3, 0)

ω0 ω1 ω2 · · ·

ω1 ω2 ω3 · · ·

ω2 ω3 ω4 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

ω0

ω1

ω2

...

ω1

ω2

ω3

...

ω2

ω3

ω4

...

(i) (ii) T1

T2

(0, 0) (1, 0) (2, 0)

√ω0ω1

√ω1ω2

√ω1ω2

√ω2ω3

√ω2ω3

√ω3ω4

· · · · · ·

√ω0ω1

√ω1ω2

√ω2ω3

...

√ω1ω2

√ω2ω3

√ω3ω4

...

√ω

√ω

√ω

...


Proposition

Consider Θ(Wω) ≡ (T1,T2) given as above. Then for k ≥ 1

Wω is k-hyponormal if and only if Θ(Wω) is k-hyponormal.

Theorem

Suppose that Θ(Wω) is hyponormal. Then the toral Aluthge transform

Θ (Wω) ≡ Θ(Wω

)is also hyponormal. The same result holds for the

spherical Aluthge transform.


Spherically quasinormal pairs

Let T be a commuting pair with joint polar decomposition

Ti ≡ UiP , (i = 1, 2). Recall that the spherical Aluthge transform

preserves commutativity for 2-variable weighted shifts. We say that T is

spherically quasinormal if T = T.

Lemma

Assume P injective. Then T is spherically quasinormal if and only if

TiP = PTi (i = 1, 2) if and only if UiP = PUi (i = 1, 2). As a

consequence, if T is spherically quasinormal then (U1,U2) is commuting.

Proposition

(RC-J. Yoon; 2015) A 2-variable weighted shift T is spherically

quasinormal if and only if there exists C > 0 such that 1CT is a spherical

isometry, that is, T ∗1T1 + T ∗

2T2 = I .Raul E. Curto (Univ. of Iowa) Multivariable Aluthge Transforms INFAS, Nov 11, 2017

Definition

A commuting pair T is a spherical isometry if T ∗1T1 + T ∗

2T2 = I .

Lemma

(RC-J. Yoon; 2015) Wα,β is a spherical isometry if and only if

α2k + β2

k = 1 for all k ∈ Z2+.

Theorem

(Athavale; JOT, 1990) A spherical isometry is always subnormal.

Corollary

(RC-J. Yoon; 2016) A spherically quasinormal 2-variable weighted shift is

subnormal.


Corollary

(RC-J. Yoon; 2016) Let T be a spherically quasinormal pair, and assume

that P is injective. Then T is hyponormal.

Theorem

(A. Athavale - S. Poddar; 2015 and S. Chavan - V. Sholapurkar; 2013)

Let T be a spherically quasinormal pair. Then T is subnormal.

Theorem

(V. Muller - M. Ptak; 1999) Spherical isometries are hyperreflexive.

Theorem

(J. Eschmeier - M. Putinar; 2000) For every n ≥ 3 there exists a

non-normal spherical isometry T such that the polynomially convex hull of

σT (T) is contained in the unit sphere.


Construction of spher. isom. 2-var. w.s.

(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00 β10 β20



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

...

β10 β20



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01 α11 α21 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

...

β10

...

β20

...



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01 α11 α21 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

β01

...

β10

β11

...

β20

β21

...



(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01 α11 α21 · · ·

α02 α12 α22 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

β01

β02

...

β10

β11

β12

...

β20

β21

β22

...


Recall that a unilateral weighted shift is recursively generated if the

sequence of moments satisfy a linear relation

γn+k = ϕ0γn + ϕ1γn+1 + · · ·+ ϕk−1γn+k−1 (k ≥ 1, n ≥ 0).

Theorem

(RC-Fialkow; IEOT, 1993) A subnormal weighted shift is recursively

generated if and only if its Berger measure is finitely atomic.


σ

τ

(0, 0) (1, 0) (2, 0) (3, 0)

α00 α10 α20 · · ·

α01 α11 α21 · · ·

α02 α12 α22 · · ·

· · · · · · · · ·

T1

T2

(0, 1)

(0, 2)

(0, 3)

β00

β01

β02

...

β10

β11

β12

...

β20

β21

β22

...


Theorem

(RC-Yoon; 2016) Let W(α,β) be a spherical isometry, and assume that the

zero-th row is subnormal with finitely atomic Berger measure σ.

(i) Each horizontal row is recursively generated, and its moments satisfy

the same linear relation as the zero-th row.

(ii) Each vertical column is recursively generated, and its moments satisfy

the linear relation obtained from (ii) which appropriately reflects the

condition α2k + β2

k = 1 (k ∈ Z2+).

(iii) The Berger measure of W(α,β) is finitely atomic, with support

contained in the Cartesian product of σ and τ , where τ is the Berger

measure of the zero-th column of W(α,β).


Theorem (cont.)

(iv) If Λ(0) and (0)Λ are the Riesz functional of the zero-th row and

zero-th column of W(α,β), resp., then

(0)Λ(p(t)) = Λ(0)(p(1− t))

for every polynomial p. As a result,

supp τ = 1− supp σ.


Spectral Properties

We aim to calculate the spectral picture of Wα,β ≡ (T1,T2) and of its

toral and spherical Aluthge transforms. This entails finding the Taylor

spectrum, the Taylor essential spectrum, and the Fredholm index. We

focus on the case when the pair has a core of tensor form.

The class of pairs with core of tensor form is large, and has been used to

exhibit structural and spectral behavior of 2-variable weighted shifts, not

found in the classical theory of unilateral weighted shifts.


Special Case (tensor core): Given ξ, η, σ, τ ,a, study sp. picture of Wα,β .

T1 ≈ σ

T2 ≈ τ

(T1,T2)|M∩N ≈ ξ × η

M∩N

(0, 0) (1, 0) (2, 0) (3, 0)

x0 x1 x2 · · ·

a α1 α2 · · ·

aβ1

y1 α1 α2 · · ·

· · · · · · · · · · · ·

y0

y1

y2

...

ay0

x0

β1

β2

...

ay0α1

x0x1

β1

β2

...


Spectral Properties, cont.

Lemma

(i) Let H1 and H2 be Hilbert spaces, and let Ai ∈ B(H1), Ci ∈ B(H2)

and Bi ∈ B(H1,H2), (i = 1, · · · , n) be such that

(A 0

B C

):=

((A1 0

B1 C1

), . . . ,

(An 0

Bn Cn

))

is commuting. Assume that A and

(A 0

B C

)are Taylor invertible.

Then, C is Taylor invertible. Furthermore, if A and C are Taylor

invertible, then

(A 0

B C

)is Taylor invertible.


Lemma (cont.)

(ii) For A and B two commuting n-tuples of bounded operators on

Hilbert space, we have:

σT (A⊗ I , I ⊗ B) = σT (A)× σT (B)

and

σTe(A⊗ I , I ⊗ B) = σTe (A)× σT (B) ∪ σT (A)× σTe (B) .


To use the Lemma, we split ℓ2(Z2+) as the orthogonal direct sum of the

0-th row and the rest. For the Taylor essential spectrum, we use the fact

that compressions of W(α,β) and W(α,β) differ by a compact perturbation,

when W(α,β) is hyponormal.

(0, 0) (1, 0) (2, 0) (3, 0)

H1

H2


Theorem

Consider a hyponormal 2-variable weighted shift W(α,β) ≡ (T1,T2). Then

σT(W(α,β)

)=(‖Wω‖ · D× ‖Wτ‖ · D

)and

σTe(W(α,β)

)=(‖Wω‖ · T× ‖Wτ‖ · D

)∪(‖Wω‖ · D× ‖Wτ‖ · T

).

(3)

Here D denotes the closure of the open unit disk D and T the unit circle.


We next consider the Taylor essential spectrum σTe(T1,T2) of

W(α,β) ≡ (T1,T2). To prove the result for σTe , observe that Wω(2) is a

compact perturbation of Wω(1) and Wω(0) .ω0y0x0x1

I and τ0I are also compact

perturbations of I1 and I2, respectively.

Theorem

Consider a hyponormal 2-variable weighted shift W(α,β) ≡ (T1,T2).

Then, we have

σT

(W(α,β)

)=(‖Wω‖ · D× ‖Wτ‖ · D

)and

σTe

(W(α,β)

)=(‖Wω‖ · T× ‖Wτ‖ · D

)∪(‖Wω‖ · D× ‖Wτ‖ · T

).

(4)

A similar result holds for the spherical Aluthge transform.


Consider now the Drury-Arveson 2-shift, denoted by DA. As usual, DA is

the toral Aluthge transform of DA and DA is the spherical Aluthge

transform of DA. Also, it is well known that DA is essentially normal.

Theorem

(i) DA is a compact perturbation of DA.

(ii) DA is a compact perturbation of DA.

Corollary

DA, DA and DA all share the same Taylor spectral picture; that is,

(i) σT (DA) = B2, (ii) σTe(DA) = ∂B2, and

(iii) index DA = index DA = index DA.

(Here B2 denotes the open unit ball in C2, and ∂B2 its topological

boundary.)Raul E. Curto (Univ. of Iowa) Multivariable Aluthge Transforms INFAS, Nov 11, 2017

Thank you!


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