When does the k-hyponormality of a 2-variable weighted shift become subnormality?

(1)

Contents lists available atScienceDirect

Journal of Mathematical Analysis and

Applications

www.elsevier.com/locate/jmaa

When does the

k

-hyponormality of a 2-variable weighted shift become

subnormality?

✩

Jasang Yoon

Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539, United States

a r t i c l e

i n f o

a b s t r a c t

Article history:

Received 15 November 2010 Available online 18 January 2011 Submitted by R. Curto

Keywords:

Jointly hyponormal pairs Subnormal pairs k-Hyponormal 2-variable weighted shift Tensor core

In this article we construct a sequence of nontrivial classes of 2-variable weighted shifts {Gk}∞k=2such that thek-hyponormality of an arbitrary power of a memberW(α,β)fromGk is equivalent to its subnormality.

1. Introduction

Let

H

be a complex Hilbert space and let

B

(

H

)

denote the algebra of bounded linear operators on

H

. For S

,

T

∈

B

(

H

)

let

[

S

,

T

] :=

S T

−

T S. We say that ann-tupleT

=

(

T1

, . . . ,

Tn

)

of operators on

H

is (jointly) hyponormalif the operator matrix

T∗

,

T

:=

⎛

⎜

⎝

[

T₁∗

,

T1

] [

T₂∗

,

T1

] · · · [

Tn∗

,

T1

]

[

T₁∗

,

T2

] [

T2∗

,

T2

] · · · [

Tn∗

,

T2

]

..

.

..

.

. .

.

..

.

[

T₁∗

,

Tn

] [

T₂∗

,

Tn

] · · · [

Tn∗

,

Tn

]

⎞

⎟

⎠

is positive on the direct sum ofn copies of

H

(cf. [1,10]). Then-tupleTis said to benormal ifT is commuting and each Ti is normal, andT is subnormal ifT is the restriction of a normal n-tuple to a common invariant subspace. For k

1, a commuting pairT

≡

(

T1

,

T2

)

is said to bek-hyponormalif

T(k

)

:=

T1

,

T2

,

T12

,

T2T1

,

T22

, . . . ,

T1k

,

T2T1k−1

, . . . ,

Tk2

is hyponormal, or equivalently

T(k

)

∗

,

T(k

)

=

T₂qT₁p

∗

,

T₂mT₁n

₁n+mk 1p+qk

0

.

Clearly, normal

⇒

subnormal

⇒

k-hyponormal. The Bram–Halmos criterion states that an operatorT

∈

B

(

H

)

is subnormal if and only if thek-tuple

(

T

,

T2

, . . . ,

Tk

)

is hyponormal for allk

1.

✩ _{This work was partially supported by a Faculty Research Council Grant at The University of Texas-Pan American.} E-mail address:[email protected].

URL:http://www.math.utpa.edu/~yoonj/.

(2)

For

α

≡ {

α

n

}

n∞=0 a bounded sequence of positive real numbers (calledweights), let Wα

:

2

(

Z

+

)

→

2

(

Z

+

)

be the asso-ciated unilateral weighted shift, deﬁned by Wαen

:=

α

nen+1 (alln

0), where

{

en

}

∞n=0 is the canonical orthonormal basis in

2

(

Z

₊

)

. We letWα

≡

shift

(

α

0

,

α

1

, . . .)

andU+

:=

shift

(

1

,

1

, . . .)

(the (unweighted) unilateral shift). For 0

<

a

1 we also let Sa

:=

shift

(

a

,

1

,

1

, . . .)

. The moments of Wα are given as

γ

k

≡

γ

k

(

Wα

)

:=

1

,

ifk

=

0

α

2 0

· · ·

α

k2−1

,

ifk

>

0

.

It is easy to see that Wα is never normal, and that it is hyponormal if and only if

α

0

α

1

· · ·

.

Similarly, consider double-indexed positive bounded sequences

α

k

, β

k

∈

∞

(

Z

2₊

)

,k

≡

(

k1

,

k2

)

∈

Z

2₊

:=

Z

+

×

Z

+ and let

2

₍

_Z

2

+

)

be the Hilbert space of square-summable complex sequences indexed by

Z

2₊. (Recall that

2

(

Z

2₊

)

is canonically isometrically isomorphic to

2

(

Z

+

)

⊗

2

(

Z

+

)

.) We deﬁne the 2-variable weighted shiftW(α,β)

≡

(

T1

,

T2

)

by

T1ek

:=

α

kek+ε1

,

T2ek

:=

β

kek+ε2

,

where

ε

1

:=

(

1

,

0

)

and

ε

2

:=

(

0

,

1

)

. Clearly,

T1T2

=

T2T1

⇔

β

k+ε1

α

k

=

α

k+ε2

β

k

allk

∈

Z

2₊

.

(1.1)

In an entirely similar way one can deﬁne multivariable weighted shifts. Trivially, a pair of unilateral weighted shiftsWaand Wβ gives rise to a 2-variable weighted shiftW(α,β), if we let

α

(k1,k2)

:=

α

k1 and

β

(k1,k2)

:=

β

k2 (allk1

,

k2

∈

Z

+). In this case,

W(α,β)is subnormal (resp. hyponormal) if and only if so areT1andT2; in fact, under the canonical identiﬁcation of

2

(

Z

2₊

)

with

2

(

Z

₊

)

⊗

2

(

Z

₊

)

, we haveT1

∼

=

I

⊗

Wa andT2

∼

=

Wβ

⊗

I, and W(α,β) is also doubly commuting. For this reason, we

do not focus attention on shifts of this type, and use them only when the above mentioned triviality is desirable or needed. Givenk

≡

(

k1

,

k2

)

∈

Z

2₊, the moments ofW(α,β)of orderk are

γ

k

≡

γ

k

(

W(α,β)

)

:=

⎧

⎪

⎨

⎪

⎩

1

,

if

(

k1

,

k2

)

=

(

0

,

0

),

α

2 (0,0)

· · ·

α

2 (k1−1,0)

,

ifk1

1 andk2

=

0

,

β

₍2₀_,₀₎

· · ·

β

₍2₀_,_k 2−1)

,

ifk1

=

0 andk2

1

,

α

2 (0,0)

· · ·

α

(2k1−1,0)

β

2 (k1,0)

· · ·

β

2 (k1,k2−1)

,

ifk1

1 andk2

1

.

(1.2)

We remark that, due to the commutativity condition (1.1),

γ

kcan be computed using any nondecreasing path from

(

0

,

0

)

to

(

k1

,

k2

)

. We now recall a well-known characterization of subnormality for multivariable weighted shifts [20], due to C. Berger (cf. [2, III.8.16]) and independently established by R. Gellar and L.J. Wallen [18] in the single variable case: W(α,β)admits a

commuting normal extension if and only if there is a probability measure

μ

(which we call theBerger measureofW(α,β))

deﬁned on the 2-dimensional rectangle R

= [

0

,

a1

] × [

0

,

a2

]

(whereai

:=

Ti

2) such that

γ

k

=

Rtkd

μ

(t)

:=

Rt k1 1t k2 2 d

μ

(t),

for allk

∈

Z

2

+. Observe thatU+andSaare subnormal, with Berger measures

δ

1and

(

1

−

a2

)δ

0

+

a2

δ

1, respectively, where

δ

p denotes the point-mass probability measure with support the singleton set

{

p

}

. Single and multivariable weighted shifts have played an important role in the study of the problems of existence of commuting normal extensions (cf. [5–7,9, 13–15,21,26]). They have also played a signiﬁcant role in the study of cyclicity and reﬂexivity, in the study of C∗-algebras generated by multiplication operators on Bergman spaces, as fertile ground to test new hypotheses, and as canonical models for theories of dilation and positivity (cf. [12,19,22]). We need some further notation to describe our results. We use

H

0 (resp.

H

∞

)

to denote the set of commuting pairs of subnormal operators (resp. subnormal pairs) on Hilbert space. Fork

1, we let

H

k denote the class ofk-hyponormal pairs in

H

0. Clearly,

H

∞

⊆ · · · ⊆

H

k

⊆ · · · ⊆

H

2

⊆

H

1

⊆

H

0. The main results in [5,13] show that these inclusions are all proper. For an arbitrary 2-variable weighted shiftW(α,β), we let

M

i(resp.

N

j) be the subspace of

2

(

Z

2₊

)

which is spanned by the canonical orthonormal basis associated to indicesk

≡

(

k1

,

k2

)

withk1

0 andk2

i (resp.k1

j andk2

0). We will often write

M

1 simply as

M

and

N

1 as

N

. The core c

(

W(α,β)

)

of W(α,β)

is the restriction of W(α,β) to the invariant subspace

M

∩

N

. A 2-variable weighted shift W(α,β) is said to beof tensor

formif it is of the form

(

I

⊗

Wα

,

Wβ

⊗

I

)

. The class of all 2-variable weighted shiftsW(α,β)

∈

H

0 whose core is of tensor form will be denoted by

T C

; in symbols,

T C

:= {

W(α,β)

∈

H

0: c

(

W(α,β)

)

is of tensor form

}

. Note that ifW(α,β)

∈

T C

, then

W(α,β)

∈

H

∞. Given a subnormal 2-variable weighted shiftW(α,β)with Berger measure

μ

, we letWα(j)

(

j

0

)

(resp.W_β(i)

(

i

0

)

) denote the associated j-th horizontal (resp.i-th vertical) slice of W(α,β). Clearly,Wα(j) (resp.W_β(i)) is subnormal,

and we let

ξ

j (resp.

η

i) denote its Berger measure. Fork

1 we let

G

k

:= {

W(α,β)

∈

A

: W(α,β)

|

M

≡

(

I

⊗

Sa

,

U+

⊗

I

)

and card

(

supp

ξ

0

)

(

k

+

1

)

}

, where

A

:= {

W(α,β)

∈

T C

: c

(

W(α,β)

)

has 1-atomic Berger measure

}

and card

(

supp

ξ

0

)

means the cardinality of the support of

ξ

0 (cf. see Fig. 1(ii)). For the meaning of set inclusion, we clearly have

G

1

=

S

1

G

2

· · ·

G

k

· · ·

A

T C

. Observe that a 2-variable weighted shiftW(α,β)

∈

S

1 has a core with Berger measure

δ

{1,1}

=

δ

1

×

δ

1. Fork

1, we note that if W(α,β)

∈

G

k, then W(α,β)can be fully determined by 3 parameters: the weight a

:=

α(

0,1), the

weight y

:=

β

(0,0)and the Berger measure

ξ

0 of the 0-th horizontal subnormal slice shift

(

x0

,

x1

,

x2

, . . .)

ofW(α,β). Thus we

(3)

Fig. 1.Weight diagram of a generic 2-variable weighted shift inS1and the 2-variable weighted shift in Theorem 2.5, respectively. 2. Main results

For a general operator T on Hilbert space, it is well known that the subnormality of T implies the subnormality of Th

(

h

2

)

. The converse implication, however, is false; in fact, the subnormality of all powers Th

(

h

2

)

does not necessarily imply the subnormality of T, even if T

≡

Wα is a unilateral weighted shift [24, p. 378]. Thus, it is natural

to ask when the subnormality of all powers Wh

α

(

h

2

)

does imply the subnormality of Wα. More general, we might

ask when the k-hyponormality

(

k

1

)

of all powers Wh_α

(

h

2

)

does imply the subnormality of Wα. Fork

2, we let Wα

≡

shift

(

a

,

b

,

1

,

1

, . . .)

where 0

<

a

<

b

<

1. ThenWαh

(

h

2

)

is subnormal, butWα is notk-hyponormal

(

k

2

)

, because

the k-hyponormality of Wα

(

k

2

)

⇔

b

=

1. For k

=

1, we consider Wα

≡

shift

(

1

,

1

−

x

,

y

,

y

, . . .)

where 0

<

x

<

1

<

y.

Then a simple calculation shows that Wh

α

(

h

2

)

is subnormal, but Wα is not hyponormal. In the multivariable case, we

can consider these analogous results. The standard assumption on a pairT

≡

(

T1

,

T2

)

is that each componentTiis subnor-mal (i

=

1

,

2). With this in mind, the analogous questions are highly nontrivial. In [8,17], we identiﬁed a large nontrivial class

S

1 (cf. see Fig. 1(i)) of 2-variable weighted shifts for which the 2-hyponormality of an arbitrary power of the initial pair is equivalent to subnormality of the initial pair. Thus, it is natural to consider

Problem 2.1.(See [17, Problem 6.8].) Is

S

1the largest class in

A

for which the implication

W(h0,0) (α,β)

:=

Th0 1

,

T 0 2

∈

H

2 for someh0

,

0

1

⇒

W(α,β)

∈

H

∞ holds?

In this paper we give a concrete answer for Problem 2.1 above and build a class

G

k

(

k

2

)

in

A

such that ifW(α,β)

∈

G

k with card

(

supp

ξ

0

)

=

k

+

1, then for someh0

,

0

1

W(h0,0) (α,β)

∈

H

k

⇔

W(α,β)

∈

H

∞

.

For this, we ﬁrst recall that, in one variable, then-th power of a weighted shift is unitarily equivalent to the direct sum ofn weighted shifts. Something similar happens in two variables, as we will see in the proof of Theorem 2.5 below. We let

H

≡

2

(

Z

₊

)

=

∞_j₌₀

{

ej

}

. Given integers i andh(h

1, 0

i

h

−

1), deﬁne

H

i

:=

∞j=0

{

ehj+i

}

; clearly,

H

=

hi=−01

H

i. Following the notation in [11], for a weight sequence

α

≡ {

α

n

}

∞n=0 we let

Wα(h:i)

:=

shift

_h₋₁

n=0

α

hj+i+n

∞ j=0

;

(2.1)

that is,Wα(h:i)denotes the sequence of products of weights in adjacent packets of sizeh, beginning with

α

i

· · ·

α

i+h−1. For example, given a weighted shift Wα

≡

shift

(

α

0

,

α

1

, . . .)

, we have Wα(2:0)

=

shift

(

α

0

α

1

,

α

2

α

3

, . . .)

, Wα(2:1)

=

shift

(

α

1

α

2

,

α

3

α

4

, . . .)

and Wα(3:2)

=

shift

(

α

2

α

3

α

4

,

α

5

α

6

α

7

, . . .)

. For h

1

,

0

i

h

−

1

,

we note that Wα(h:i) is

unitar-ily equivalent toWh

α

|

Hi

.

Therefore,W h

α is unitarily equivalent to

h−1

(4)

Berger measure

μ

, thenWα(h,i)is subnormal with the Berger measure

μ

i, where d

μ

0

(

s

)

=

d

μ

s1h

_and _d

μ

_i

₍

_s

₎

=

s 1 h

γ

i d

μ

s1h

_{for 1}

_i

_h

−

₁

_.

_(2.2) Furthermore, we have

Lemma 2.2.(See [11, Corollary 2.8].) (i) Let k

1. Then Wh

αis k-hyponormal

⇔

Wα(h:i)is k-hyponormal for0

i

h

−

1.

(ii) W_αhis subnormal

⇔

Wα(h:i)is subnormal for0

i

h

−

1.

We now introduce a key family of examples for our main results. Fork

2, 0

<

ai

(

1

i

k

−

1

)

, 0

c0

,

c1

, . . . ,

ck

1 (with

k_i₌₀ci

=

1), we letWx

≡

shift

(

x0

,

x1

, . . .)

be given by

xn

:=

⎧

⎪

⎨

⎪

⎩

k−1 i=1ciai

+

ck

,

ifn

=

0

,

k−1 i=1ciani+1+ck k−1 i=1ciani+ck

,

ifn

1

.

(2.3) We now consider

ξ

0

:=

c0

δ

0

+

k−1

i=1 ci

δ

ai

+

ck

δ

1

.

(2.4)

Then

ξ

0is a probability measure. Forn

=

0, we denote

γ

0

(

Wx

)

by 1. We note that forn

1 the moments associated with Wx are

γ

n

(

Wx

)

≡

x20x21x22

· · ·

xn2−1

=

_k₋₁

i=1 ciai

+

ck

·

k−1 i=1cia2_i

+

ck

k−1 i=1ciai

+

ck

· · ·

k−1 i=1cian_i

+

ck

k−1 i=1ciani−1

+

ck

=

k−1

i=1 ciani

+

ck

=

snd

ξ

0

(

s

)

(

n

1

).

Thus, it follows that Wx is subnormal, with Berger measure

d

ξ

0

(

s

)

=

c0d

δ

0

(

s

)

+

k−1

i=1

cid

δ

ai

(

s

)

+

ckd

δ

1

(

s

).

Lemma 2.3.For k

2,0

<

ai

(

1

i

k

−

1

)

,0

c0

,

c1

, . . . ,

ck

1 (with

k i=0ci

=

1), h0

1, k1

0, we let G

(

h0

,

k1

,

k

)

:=

⎛

⎜

⎝

k−1 i=1ciak_i1+2h0

+

ck

ki=−11ciak_i1+3h0

+

ck

· · ·

ki=−11cia_ik1+(k+1)h0

+

ck

k−1 i=1ciak_i1+3h0

+

ck

ki=−11ciak_i1+4h0

+

ck

· · ·

ki=−11cia_ik1+(k+2)h0

+

ck

..

.

..

.

. .

.

..

.

k−1 i=1cia k1+(k+1)h0 i

+

ck

k−1 i=1cia k1+(k+2)h0 i

+

ck

· · ·

k−1 i=1cia k1+2kh0 i

+

ck

⎞

⎟

⎠

.

Then G

(

h0

,

k1

,

k

)

is invertible and

detG

(

h0

,

k1

,

k

)

=

ck k−1

i=1 ciak_i1+2h0

1

−

ah0 i

2

·

k

−1 i<j

ah0 i

−

a h0 j

2

,

(2.5) where k_i_<−1_j

(

ah0 i

−

a h0 j

)

2

:=

1, if k

=

2.

(5)

Lemma 2.4.Under the conditions of Lemma2.3and for0

<

a

,

y

1, we let D

(

a

,

y

)

:=

_y2 _a2_y2 _y2 a2y2 a2y2 a2y2 y2 _a2_y2 ₁

,

F

(

a

,

y

,

h0

,

0

,

k

)

:=

⎛

⎝

a 2_y2 _a2_y2

_{· · ·}

_a2_y2 a2y2 a2y2

· · ·

a2y2

k−1 i=1ciahi0

+

ck

k−1 i=1cia2ih0

+

ck

· · ·

k−1 i=1ciakhi 0

+

ck

⎞

⎠

,

P

(

a

,

y

,

h0

,

0

,

k

)

:=

D

(

a

,

y

)

F

(

a

,

y

,

h0

,

0

,

k

)

F∗

(

a

,

y

,

h0

,

0

,

k

)

G

(

h0

,

0

,

k

)

.

Then we have P

(

a

,

y

,

h0

,

0

,

k

)

0

⇔

y

⎧

⎨

⎩

min

{

ck a2

,

c0 1−a2

}

,

if0

<

a

<

1

,

√

ck

,

if a

=

1

.

(2.6)

Proof. By Lemma 2.3, since D

(

h0

)

is invertible, if we apply Lemma A.5 to P

(

a

,

y

,

h0

,

0

,

k

)

, we have

P

(

a

,

y

,

h0

,

0

,

k

)

0

⇔

D

(

a

,

y

)

−

W

(

a

,

y

,

h0

,

0

,

k

)

∗G

(

h0

,

0

,

k

)

W

(

a

,

y

,

h0

,

0

,

k

)

0

,

where W

(

a

,

y

,

h0

,

0

,

k

)

:=

⎛

⎜

⎝

k−1 i=1 a2_y2_ah0 i (ah_i0−1)ck k−1 i=1 a2_y2_ah0 i (ah_i0−1)ck g1

(

h0

,

k

)

a2_y2₍_ah0 1 a h0 2 ···a h0 k−2+···+a h0 2 a h0 3 ···a h0 k−1) (−1) k_i₌−₁1(ah0 i −1)ck a2_y2₍_ah0 1 a h0 2 ···a h0 k−2+···+a h0 2 a h0 3 ···a h0 k−1) (−1) k_i₌−₁1(ah0 i −1)ck g2

(

h0

,

k

)

a2_y2₍_ah0 1 a h₀ 2 ···a h₀ k−3+···+a h₀ 3 a h₀ 4 ···a h₀ k−1) (−1)2 k−1 i=1(a h0 i −1)ck a2_y2₍_ah0 1 a h₀ 2 ···a h₀ k−3+···+a h₀ 3 a h₀ 4 ···a h₀ k−1) (−1)2 k−1 i=1(a h0 i −1)ck g3

(

h0

,

k

)

..

.

..

.

..

.

(

−

1

)

k−1a 2_y2₍k−1 i=1a h0 i ) k−1 i=1(a h₀ i −1)ck

(

−

1

)

k−1a 2_y2₍k−1 i=1a h0 i ) k−1 i=1(a h₀ i −1)ck gk−1

(

h0

,

k

)

(−1)ka2_y2 k−1 i=1(a h₀ i −1)ck (−1)ka2_y2 k−1 i=1(a h₀ i −1)ck gk

(

h0

,

k

)

⎞

⎟

⎠

and g1

(

h0

,

k

)

:=

1

+

k−1

i=1 a−h0 i

,

g2

(

h0

,

k

)

:=

(

−

1

)

_k₋₁

i=1 a−h0 i

+

k−1

i<j a−h0 i a− h0 j

,

g3

(

h0

,

k

)

:=

(

−

1

)

2

_k₋₁

i<j a−h0 i a− h0 j

+

k−1

i<j< a−h0 i a− h0 j a− h0

,

gk−1

(

h0

,

k

)

:=

(

−

1

)

k−1

a−h0 1 a− h0 2

· · ·

a− h0 k−2

+ · · · +

a −h0 2 a− h0 3

· · ·

a− h0 k−1

+

k

−1 i=1 a−h0 i

,

gk

(

h0

,

k

)

:=

(

−

1

)

k k

−1 i=1 a−h0 i

.

A direct calculation shows that

D

(

a

,

y

)

−

W

(

a

,

y

,

h0

,

0

,

k

)

∗G

(

h0

,

0

,

k

)

W

(

a

,

y

,

h0

,

0

,

k

)

0

⇔

⎛

⎜

⎝

y2

−

a4_cy4 k a 2_y2

₋

a4_y4 ck

(

1

−

a 2

₎

_y2 a2_y2

₋

a4_y4 ck a 2_y2

₋

a4_y4 ck 0

(

1

−

a2

₎

_y2 ₀ _c 0

⎞

⎟

⎠

0

⇔

y2

₋

a4_y4 ck

+

(1−a2₎2_y4 c0 a 2_y2

₋

a4_y4 ck a2y2

−

a4_cy4 k a 2_y2

₋

a4_y4 ck

=:

a11 a12 a21 a22

=:

A

0

.

(6)

To check A

0, it is suﬃcient to checka22

0 and detA

0, because (a22

0 and detA

0

)

⇒

a11

0. Thus, a straight-forward calculation shows thata22

0

⇔

a2y2

ckand

detA

0

⇔

−

y2

+

a2y2

+

c0

ck

−

a2y2

0

.

Thus, it follows that

P

(

a

,

y

,

h0

,

0

,

k

)

0

⇔

!

a2y2

ck−1andy2

1

−

a2

c0

"

⇔

⎧

⎨

⎩

min

{

ck a2

,

c0 1−a2

}

,

if 0

<

a

<

1

,

√

ck

,

ifa

=

1

.

2

For our main results, we recall that fork

1

,

G

k

= {

W(α,β)

∈

A

: W(α,β)

|

M

≡

(

I

⊗

Sa

,

U+

⊗

I

)

and card

(

supp

ξ

0

)

(

k

+

1

)

}

(cf. see Fig. 1(ii)). We then have

Theorem 2.5.For k

2

,

we let W(α,β)

≡

(

T1

,

T2

)

≡

a

,

y

, ξ

0

∈

G

k(where the0-th horizontal slice Wx

≡

shift

(

x0

,

x1

, . . .)

is as

in(2.3)with0

<

c0

,

c1

, . . . ,

ck

<

1). Then the following statements are equivalent: (i) W(α,β)

∈

H

k;

(ii) W(α,β)

∈

H

∞;

(iii) for some h0

,

0

1, W₍(_αh0_,β),0)

≡

(

T₁h0

,

T₂0

)

∈

H

k.

Proof. (i)

⇒

(ii): From Lemma A.2, we recall that a 2-variable weighted shiftW(α,β)isk-hyponormal if and only if

Mk

(

k

)

=

(

γ

k+(m,n)+(p,q)

)

0n+mk 0p+qk

0

,

for all k

∈

Z

2₊. It is straightforward to verify that W(α,β)

|

_M

∼

=

(

I

⊗

Sa

,

U+

⊗

I

)

, so that W(α,β)

|

_M is subnormal. Since

W(α,β)

∈

H

k, if we apply Lemma A.2(ii) at k

=

(

k1

,

0

)

(allk1

0), we haveM(k1,0)

(

k

)

0. We note that forh0

,

0

1 the moments

γ

k

(k

∈

Z

2₊

)

associated withW((αh0,β),0)are

γ

k

W(h0,0) (α,β)

=

⎧

⎪

⎨

⎪

⎩

1

,

ifk1

=

0 andk2

=

0

,

γ

k1h0

(

Wα

),

ifk1

1 andk2

=

0

,

y2

_,

_if_k 1

=

0 andk2

1

,

a2_y2

_,

_if_k 1

1 andk2

1

.

(2.7)

Thus, by (2.7), we have that

W₍(_α1,_,β)1)

≡

W(α,β)

∈

H

k

⇒

M(k1,0)

(

k

)(

W(α,β))

0

⇒

M(0,0)(k

)(

W(α,β))

0

.

From Lemma 2.4, a direct computation (i.e., interchanging rows and columns, discarding a redundant row and column in the moment matrix of M(0,0)

(

k

)(

W(α,β)

)

) shows that

M(0,0)(k

)(

W(α,β))

0

⇔

P

(

a

,

y

,

1

,

0

,

k

)

0

⇔

y

⎧

⎨

⎩

min

{

ck a2

,

c0 1−a2

}

,

if 0

<

a

<

1

,

√

_c k

,

ifa

=

1

.

Thus, we have W(α,β)

∈

H

k

⇒

y

⎧

⎨

⎩

min

{

ck a2

,

c0 1−a2

}

,

if 0

<

a

<

1

,

√

ck

,

ifa

=

1

.

(2.8)

We now characterize the subnormality of W(α,β)using its parametric characterizations. Lemma A.3 will help us

charac-terize W(α,β)

∈

H

∞. Since W(α,β)

|

M

≡

(

I

⊗

Sa

,

U+

⊗

I

)

is subnormal with Berger measure

μ

_M

≡ [

(

1

−

a2

)δ

0

+

a2

δ

1

] ×

δ

1, we can think of W(α,β) as a backward extension of W(α,β)

|

M (in the t direction) and apply Lemma A.3. Note that

d

(

μ

_M

)

ext

(

s

,

t

)

≡ [

(

1

−

a2

)δ

0

+

a2

δ

1

] ×

δ

1and

α

₀₀2

1_t

L1₍_μ

(7)

P

(

a

,

y

,

1

,

0

,

k

)

0

⇔

y2

1

−

a2

δ

0

+

a2

δ

1

c0

δ

0

+

k−1

i=1 ci

δ

ai

+

ck

δ

1

⇔

α

2₀₀

##

1 t

##

L1₍_μ M)

(

μ

_M

)

_extX

ξ

0

,

where

ξ

0is the Berger measure ofWxas in (2.3). Thus, by Lemma A.3, we have

W(α,β)

∈

H

∞

⇔

P

(

a

,

y

,

1

,

0

,

k

)

0

.

Therefore, by Lemma 2.4 and (2.8), it follows thatW(α,β)

∈

H

k

⇒

W(α,β)

∈

H

∞.

(ii)

⇒

(iii): Since W(α,β)

∈

H

∞, for allh0

,

0

1, we have W₍(_αh0_,β),0)

∈

H

∞ by the functional calculus. Thus we get for someh0

,

0

1,W₍(_αh0_,β),0)

∈

H

k.

(iii)

⇒

(i): For some h0

,

0

1, we suppose that W₍(_αh0_,β),0)

∈

H

k. Fixed h0

,

0

1, we ﬁrst let

H

(m,n)

:=

∞

i,j=0

{

e(h0i+m,0j+n): h0

,

0

1}, for 0

m

h0

−

1 and 0

n

0

−

1. Then we have

2

Z

2₊

≡

h

0−1 m=0

0−1 n=0

H

(m,n). Observe that

H

(m,n)reducesT1h0 andT

0

2 . Thus, if a 2-variable weighted shiftW(α,β)is given in Fig. 1(ii), then forh0

,

0

1, we can write W(h0,0) (α,β)

≡

Th0 1

,

T 0 2

∼

=

W_α₍h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

⊕

h

0−1 i=1

W_α₍h0:i)

⊕

(

I

⊗

U+

),

T2

|

Hi cf. see Fig. 2(i)

,

where Wα(h0:i)

=

shift

$

γ

(i+1)h0

γ

ih0

,

$

γ

(i+2)h0

γ

(i+1)h0

, . . .

and

H

i

:=

0−1 n=0

H

(i,n)

(

0

i

k

−

1

).

Thus, we observe thatW(h0,0) (α,β)

∈

H

kis equivalent to

(

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

)

∈

H

kand

(

Wα(h0:i)

⊕

(

I

⊗

U+

),

T2

|

Hi

)

∈

H

k, for 1

i

h0

−

1. To show W(α,β)

∈

H

k, by Lemma A.2, it is enough to show that W(α,β)

|

M

∈

H

k, W(α,β)

|

N

∈

H

k and M(0,0)

(

k

)(

W(α,β)

)

0. Since W(α,β)

|

M

∼

=

(

I

⊗

Sa

,

U+

⊗

I

)

is subnormal, we need to show W(α,β)

|

N

∈

H

k and M(0,0)

(

k

)(

W(α,β)

)

0. ForW(α,β)

|

_N

∈

H

k, we ﬁrst want to show that for someh0

,

0

1

(

W(α,β)

|

N

)

(h0,0)

∈

_H

k

⇔

W(α,β)

|

N

∈

H

k

.

(2.9) SinceW(α,β)

|

M∩N

∼

=

(

I

⊗

U+

,

U+

⊗

I

)

is subnormal, to show (2.9), we need to show that

M₍k1,0)

(

k

)

W(h0,0) (α,β)

0

⇔

M₍k1,0)

(

k

)(

W(α,β))

0 fork1

1

.

SinceG

(

h0

,

k1

,

k

)

is invertible, by (2.7), Fig. 2(ii) and Lemma A.5, fork1

1 and someh0

,

0

1

,

we have that

M(k1,0)

(

k

)

W(h0,0) (α,β)

0

⇔

M

0

⇔

a2y2

ck

⇔

M(k1,0)

(

2

)(

W(α,β)

)

0

,

where M

:=

⎛

⎜

⎝

a2_y2 _a2_y2 a2y2

k_i₌−₁1ciahi0

+

ck

a2_y2

_{· · ·}

_a2_y2

k−1 i=1ciaki1+h0

+

ck

· · ·

k−1 i=1ciaki1+kh0

+

ck

⎛

⎜

⎝

a2y2

k_i₌−₁1ciak_i1+h0

+

ck

..

.

..

.

a2y2

k_i₌−₁1ciaki1+kh0

+

ck

⎞

⎟

⎠

G

(

h0

,

k1

,

k

)

⎞

⎟

⎠

.

Thus, we have

(

W(α,β)

|

N

)

(h0,0)

∈

H

k

⇔

W(α,β)

|

N

∈

H

k

.

(8)

Fig. 2.Weight diagram of the 2-variable weighted shift W(h0,0)

(α,β) in Theorem 2.5 and weight diagram of the 2-variable weighted shift in Lemma A.4, respectively.

ForM(0,0)

(

k

)(

W(α,β)

)

0, we note that

W(h0,0) (α,β)

∈

H

k

⇒

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

∈

H

k

.

We also observe that

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

_∼

=

0−1 n=0

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T 0 2

%%

_H₍₀_,_n₎

and

0−1 n=0

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T 0 2

%%

_H₍₀_,_n₎

_∼

=

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T 0 2

%%

_H₍₀_,₀₎

⊕

0−1 n=0

(

I

⊗

Sa

,

U+

⊗

I

).

Note that the second summand is clearly subnormal; thus, forh0

,

0

1, thek-hyponormality of

(

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

)

is equivalent to thek-hyponormality of the ﬁrst summand,

(

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T 0

2

|

H(0,0)

)

. Observe also that

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T 0 2

%%

H(0,0)

_∼

=

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H(0,0)

.

Thus we have

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H0

∈

H

k

⇔

W_α(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H(0,0)

∈

H

k

.

To check the k-hyponormality of

(

Wα(h0:0)

⊕

(

I

⊗

Sa

),

T2

|

H(0,0)

)

, we observe that it suﬃces to apply Lemma A.2(ii) at k

=

(

0

,

0

)

. From Lemma 2.4 and (2.7), after we apply Lemma A.2 to

(

0

,

0

)

of

(

Wα(h0:0)

⊕

(

I

⊗

U+

),

T2

|

H0

)

, we have

M(0,0)(k

)

W_α₍h0:0)

⊕

(

I

⊗

U+

),

T2

|

H0

0

⇔

P

(

a

,

y

,

h0

,

0

,

k

)

0

⇔

y2

c2

,

where P

(

a

,

y

,

h0

,

0

,

k

)

is as in Lemma 2.4. Thus we get

W(h0,0) (α,β)

∈

H

k

⇒

y2

ck

⇔

M(0,0)

(

k

)(

W(α,β)

)

0

.

(2.10)

Therefore, we have for someh0

,

0

1

,

W(h0,0) (α,β)

∈

H

k

⇒

W(α,β)

∈

H

k and our proof is now complete.

2

(9)

Remark 2.6.

(i) We note that card

(

supp

ξ

0

)

=

(

k

+

1

)

, where

ξ

0 is as in Theorem 2.5.

(ii) In Theorem 2.5, we show that for givenk

2, there exists a 2-variable weighted shiftW(α,β)

∈

G

k

A

for which some h0

,

0

1

W(h0,0) (α,β)

∈

H

k

⇔

W(α,β)

∈

H

∞

.

(iii) By Theorem 2.5, we note that there exists a 2-variable weighted shift W(α,β)

∈

G

2in

A

for which someh0

,

0

1

W(h0,0) (α,β)

∈

H

k

⇔

W(α,β)

∈

H

∞

.

Thus, Theorem 2.5 gives an answer for Problem 2.1 and more.

Observe that if W(α,β)

≡

a

,

y

, ξ

0

∈

G

k

(

k

2

)

, then by (2.4) the measure

ξ

0 can be completely determined by the

(

2k

−

1

)

parameters

{

ai

}

ki=−11 and

{

ci

}

ki=0. Thus we can also denote a 2-variable weighted shift W(α,β)

≡

a

,

y

, ξ

0

∈

G

k by

a

,

y

,

{

ai

}

k_i₌−₁1

,

{

ci

}

k_i₌₀

(cf. see Fig. 1(ii)). We now assumeay

ki=−11ciah_i

+

ck (allh

1), because we need to ensure that W(α,β)

∈

H

0. We now obtain a canonical representation for the powers

a

,

y

,

{

ai

}

k_i₌−₁1

,

{

ci

}

k_i₌₀

(h,) as an orthogonal direct sum of 2-variable weighted shifts in

G

k. In what follows, we abbreviate the orthogonal direct sums of hcopies of a shift

a

,

y

,

{

ai

}

k_i₌−₁1

,

{

ci

}

k_i₌₀

byh

·

a

,

y

,

{

ai

}

_ik₌−₁1

,

{

ci

}

k_i₌₀

. Then we have

Proposition 2.7.We let

a

,

y

,

{

ai

}

ki=−11

,

{

ci

}

ki=0

≡

a

,

y

,

{

ai

}

ki=−11

,

c0

,

{

ci

}

ki=−11

,

ck

∈

G

k. Then for h

,

1, we have

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'

(h,)

∼

=

&

a

,

y

,

!

a 1 h i

"

k−1 i=1

,

c0

,

{

ci

}

k−1 i=1

,

ck

'

⊕

(

1

,

ay k−1 i=1ciamhi

+

ck

,

!

a 1 h i

"

k−1 i=1

,

0

,

_a1 h ici

k−1 i=1ciami

+

ck

k−1 i=1

,

ck k−1 i=1ciami

+

ck

)

⊕

(

−

1

)

· &

a

,

1

,

0

,

1

−

a2

,

0

,

a2

'

⊕

(

h

−

1

)(

−

1

)

·

1

,

1

,

0

,

0

,

0

,

1

,

so that the class

G

kis invariant under all powers.

Proof. We recall that we decompose the space

2

(

Z

2₊

)

as the orthogonal direct sum of h

subspaces

H

(m,n), each

iso-metrically isomorphic to

2

(

Z

2₊

)

, namely

H

(m,n)

:=

_∞

i,j=0

{

e(hi+m,j+n)

}

(0

m

h

−

1, 0

n

−

1). This particular

decomposition allows us to write the power

&

a

,

y

,

{

ai

}

ki=−11

,

c0

,

{

ci

}

ki=−11

,

ck

'

(h,)

as the orthogonal direct sum

(h,)

0mh−1,0n−1

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'%%

_H (m,n)

.

From (2.2), we will now identify each of the summands

a

,

y

,

b

,

c0

,

c1

,

c2

(h,)

|

_H(m,n) (0

m

h

−

1, 0

n

−

1).

Case 1:(m

=

0,n

=

0) Direct inspection of the weight families

α

and

β

shows that

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'

(h,) e(hi,j)

=

&

a

,

y

,

!

a 1 h i

"

k−1 i=1

,

c0

,

{

ci

}

k−1 i=1

,

ck

'

e(hi,j), and therefore

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'

(h,)

%%

H(0,0)

∼

=

&

a

,

y

,

!

a 1 h i

"

k−1 i=1

,

c0

,

{

ci

}

k−1 i=1

,

ck

'

.

(10)

Case 2:(m

>

0,n

=

0) In this case the generic basis vector of

H

(m,0)ise(hi+m,j), so that

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'

(h,) e(hi+m,j)

=

(

1

,

ay k−1 i=1ciamhi

+

ck

,

!

a 1 h i

"

k−1 i=1

,

0

,

a 1 h ici

k−1 i=1ciami

+

ck

k−1 i=1

,

ck k−1 i=1ciami

+

ck

)

e₍hi+m,j)

.

It follows that

&

a

,

y

,

{

ai

}

ki=−11

,

c0

,

{

ci

}

ki=−11

,

ck

'

(h,)

%%

H(m,0)

∼

=

(

1

,

ay k−1 i=1ciamhi

+

ck

,

!

a 1 h i

"

k−1 i=1

,

0

,

_a1 h ici

k−1 i=1ciami

+

ck

k−1 i=1

,

ck k−1 i=1ciami

+

ck

)

.

Case 3:(m

=

0,n

>

0) In this case the generic basis vector of

H

(0,n)ise(hi,j+n), and therefore

&

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

k_i₌−₁1

,

ck

'

(h,) e(hi,j+n)

=

&

a

,

1

,

0

,

1

−

a2

,

0

,

a2

'

e(hi,j+n)

.

It follows that

a

,

y

,

{

ai

}

k_i₌−₁1

,

c0

,

{

ci

}

_ik₌−₁1

,

ck

(h,)

|

H(0,n)

∼

=

a

,

1

,

0

,

1

−

a 2

_,

₀

_,

_a2

_.

Case 4:(m

>

0,n

>

0) Since

H

(m,n)

⊆

M

∩

N

, and the core ofW(α,β)is trivial, it is clear that all relevant weights are equal

to 1, so

&

a

,

y

,

{

ai

}

ki=−11

,

c0

,

{

ci

}

ki=−11

,

ck

'

(h,) e(hi+m,j+n)

=

1

,

1

,

0

,

0

,

0

,

1

e(hi+m,j+n), and therefore

a

,

y

,

{

ai

}

ki=−11

,

c0

,

{

ci

}

ki=−11

,

ck

(h,)

|

H(m,n)

∼

=

1

,

1

,

0

,

0

,

0

,

1

.

Therefore, our proof is now complete.

2

Corollary 2.8.For

a

,

y

,

{

ai

}

ki=−11

,

{

ci

}

ki=0

∈

G

k

(

k

2

)

, the following statements are equivalent: (i) for some h0

,

0

1,

a

,

y

,

{

ai

}

ik=−11

,

{

ci

}

ki=0

(h0,0)

∈

H

k; (ii) for all h

,

1,

a

,

y

,

{

ai

}

ki=−11

,

{

ci

}

ki=0

(h,)

∈

H

k; (iii)

a

,

y

,

{

ai

}

ik=−11

,

{

ci

}

ki=0

∈

H

k;

(iv) for some h0

,

0

1,

a

,

y

,

{

ai

}

ik=−11

,

{

ci

}

ki=0

(h0,0)

∈

H

∞; (v) for all h

,

1,

a

,

y

,

{

ai

}

ki=−11

,

{

ci

}

ki=0

(h,)

∈

H

∞; (vi)

a

,

y

,

{

ai

}

ik=−11

,

{

ci

}

ki=0

∈

H

∞.

Proof. This is straightforward from Theorem 2.5, Proposition 2.7 and the functional calculus.

2

Acknowledgments

The author is deeply indebted to the referee for a number of comments and observations that helped improve the content and presentation of this paper. The portions of the proof of some results were obtained using calculations with the software toolMathematica[25].

Appendix A

For the reader’s convenience, in this section we gather several well-known auxiliary results which are needed for the proofs of the main results in this article. First, to detect hyponormality for 2-variable weighted shifts we use a simple criterion involving a base pointk in

Z

2₊and its ﬁve neighboring points ink

+

Z

2₊at path distance at most 2.

Lemma A.1(Six-point Test). (See [3, Theorem 6.1].) Let W(α,β)

≡

(

T1

,

T2

)

be a2-variable weighted shift, with weight sequences

α

and

β

. Then

W₍∗_α_,β)

,

W(α,β)

0

⇔

H

(

k1

,

k2

)(

1

)

:=

α

_k2₊_ε 1

−

α

2 k

α

k+ε2

β

k+ε1

−

α

k

β

k

α

k+ε2

β

k+ε1

−

α

k

β

k

β

2 k+ε2

−

β

2 k

0

for allk

∈

Z

2₊

.