m Isometric block Toeplitz operators

R E S E A R C H

Open Access

m

-Isometric block Toeplitz operators

Eungil Ko

_{and Jongrak Lee}

*_{Correspondence:} [email protected]

2_{Institute of Mathematical Sciences,}

Ewha Womans University, Seoul, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, we studym-isometric block Toeplitz operators with trigonometric symbols. In addition, we give a necessary and sufficient condition for block Toeplitz operators with trigonometric polynomial symbols to bem-contractive.

MSC: 47A62; 47B15; 47B20

Keywords: m-isometric operators; Expansive operators; Contractive operators; Block Toeplitz operators

1 Introduction

LetHbe a complex Hilbert space and letL(H) be the algebra of all bounded linear op-erators onH. In the 1990s, Agler and Stankus [1] studied the following operator. For an operatorT∈L(H) and a positive integerm, define

Bm(T) =

m

j=0

(–1)m–j

m

j

T∗jTj.

We say thatTism-contractive (respectively,m-expansive andm-isometric) ifBm_(T)≥0

(respectively,Bm_{(T)≤0 andB}m_{(T) = 0) for some positive integerm.}

Them-isometric operators have been widely investigated in recent years. The theory of

m-isometric operators was investigated especially by Agler and Stankus [1–3]. Agler and Stankus [1–3] developed a rich theory ofm-isometric operators and highlighted its con-nections to Toeplitz operators and function theory. Agler [4] illustrated the connection be-tweenm-isometric operators and the classical disconjugacy theory. In recent years, there have been studies on products ofm-isometric operators [5], andm-isometric composition operators [6]. In [7], the authors characterizedm-isometric Toeplitz operators by prop-erties of the rational symbols and gave some results form-expansive andm-contractive Toeplitz operators with trigonometric polynomial symbols.

Let L2≡L2(T) be the set of square integrable measurable functions onTandH2≡ H2(T) be the corresponding Hardy space. LetL∞≡L∞(T) be the set of bounded mea-surable functions onTand let H∞ ≡H∞(T) :=L∞∩H2_{. We introduce the notion of}

block Toeplitz operators. LetMn×rdenote the set of alln×rcomplex matrices and write Mn=Mn×n. We observe thatL∞Mn=L

∞_⊗M

nandH_C2n=H2⊗Cn. For the matrix-valued

functionΦ∈L∞_Mn, theblock Toeplitz operator with symbolΦ is the operatorTΦ on the

vector-valued Hardy spaceH2

Cnof the unit disc defined by

TΦf :=Pn(Φf)

f ∈H_C2n

,

wherePndenotes the orthogonal projection ofL2_Cn(=L2⊗Cn) ontoH_C2n. If we setH_C2n=

H2_{(T)⊕ · · · ⊕H}2_{(T), then we see that if}

Φ=

⎡ ⎢ ⎢ ⎣

ϕ11 · · · ϕ1n

.. .

ϕn1 · · · ϕnn ⎤ ⎥ ⎥

⎦, thenTΦ=

⎡ ⎢ ⎢ ⎣

Tϕ11 · · · Tϕ1n

.. .

Tϕn1 · · · Tϕnn

⎤ ⎥ ⎥ ⎦.

Gu, Hendricks, and Rutherford [8] studied the hyponormality of block Toeplitz oper-ators and characterized the hyponormality of block Toeplitz operoper-ators in terms of their symbols. In particular, they showed that the hyponormality of the block Toeplitz operator

TΦwill forceΦto be normal, that is,Φ∗Φ=ΦΦ∗.

This paper is organized as follows. In Sect.2, we present some preliminary knowledge of block Toeplitz operators andm-isometric operators. In Sect.3, we give several results for them-isometric (respectively,m-expansive andm-contractive) block Toeplitz opera-tors with rational symbols. We give a concrete description ofm-isometric block Toeplitz operators in terms of the coefficients of the matrix-valued rational symbols.

2 Preliminary

We first review the results of m-isometric Toeplitz operators with rational symbols. The properties of Toeplitz operators enable us to establish several consequences ofm -isometric operators. Given a positive integerm, it follows from the definition that an op-eratorT∈L(H) is anm-isometry if and only if

m

j=0

(–1)m–j

m

j

Tjx2= 0 for allx∈H. (2.1)

Using the identity (2.1) and the block Toeplitz operator with matrix-valued trigonometric polynomial symbolΦ, we consider the following equation:

m

j=0

(–1)m–j

m

j

TΦjK

2

= 0 (2.2)

for allK∈H2

Cn. A matrix functionΘ∈H_M∞

m×nis calledinnerifΘ

∗_Θ=I

na.e. onT. From

the well-known results for Toeplitz operators and isometric operators, we can extend the results to block versions. As the idea of the proof is completely the same as in that of [9], we omit the proof of the following lemma.

Lemma 2.1 A necessary and sufficient condition that a Toeplitz operator TΦ be an

iso-metric operator is thatΦis inner.

For a matrix-valued functionΦ∈H_M2_n×r, writeΦ(z) :=Φ∗(z). We say that∈H_M2_n×m

A∈H2

Mm×r (m≤n). We also say thatΦ∈H

2

Mn×r andΨ ∈H

2

Mn×m areleft coprimeif the

only common left inner divisor ofΦandΨ is a unitary constant, and thatΦ∈H2

Mn×rand

Ψ ∈H2

Mm×rareright coprimeifΦandΨare left coprime. Two matrix functionsΦandΨ

inH2

Mnarecoprimeif they are both left and right coprime.

ForΦ∈L∞_Mn, we write

Φ+:=PnΦ∈HM2n and Φ–:=

P⊥_nΦ∗∈H_M2_n.

Thus we can writeΦ=Φ++Φ–∗. Recall that a functionϕ∈L∞ is said to be ofbounded

type(or in the Nevanlinna class) if there are analytic functionsψ1,ψ2∈H∞such that

ϕ(z) = ψ1(z)

ψ2(z)

for almost allz∈T.

For a matrix-valued functionΦ= [ϕij]∈L∞Mn, we say thatΦ is ofbounded typeif each

entryϕijis of bounded type and thatΦisrationalif each entryϕijis a rational function.

SupposeΦ= [ϕij]∈L∞Mnis such thatΦ

∗_{is of bounded type. Then we can writeϕ}

ij=θijbij,

whereθijis inner andθijandbijare coprime. Thus ifθis the least common multiple of the θij, then we write

Φ= [ϕij] = [θijbij] = [θaij] =ΘA∗ (2.3)

whereΘ=θInandA∈HM2n. We note that Eq. (2.3) is minimal, in the sense that ifωIn(ωis

inner) is a common inner divisor ofΘandA, thenωis constant. LetΦ≡Φ++Φ–∗∈L∞Mn

be such thatΦandΦ∗are of bounded type. Then we can write

Φ+=Θ0A∗ and Φ–=Θ1B∗,

whereΘi=θiInwith an inner functionθifori= 0, 1 andA,B∈HM2n. In particular, ifΦ∈

L∞_Mnis rational then theθican be chosen as finite Blaschke products. LetΩbe the greatest

common left inner divisor ofAandΘ0. ThenA=ΩAandΘ0=ΩΩ0for someA∈HM2n

and some inner matrixΩ0. Therefore we can write

Φ+=A∗Ω0

whereΩ0andAare left coprime. In this caseΩ0A∗is called aleft coprime factorization

ofΦ+. Similarly,

Φ+=0A∗r

where0andArare right coprime. In this case0A∗ris called aright coprime factorization

ofΦ+.

We writeZ(θ) is the set of zeros of an inner functionθ. The following lemma will be useful in the sequel.

Lemma 2.2([10]) Let B∈H_M2_n andΘ:=θInwith a finite Blaschke productθ.Then the

(i) B(α)is invertible for eachα∈Z(θ); (ii) BandΘare right coprime; (iii) BandΘare left coprime.

3 Main results

In this section, we give several results for m-isometric block Toeplitz operators with trigonometric polynomial symbols.

Theorem 3.1 LetΦ:=Φ++Φ–∗is a rational function of the form

Φ+=Θ0A∗ and Φ–=Θ1B∗ (coprime factorization),

whereΘi=θiInwith a finite Blaschke productθifor i= 0, 1and A,B∈HM2n andZ(θ0)∩

Z(θ1)=∅.If TΦis an m-isometry,thenΦis analytic.

Proof SinceTΦis anm-isometry,

m

j=0

(–1)m–j

m

j

TΦ∗jT j ΦK= 0

holds for allK∈H2

Cn. PutK=Θ₀mΘ₁m. Then

0 =

m

j=0

(–1)m–j

m j

T_Φ∗jT_ΦjK=

m

j=0

(–1)m–j

m j

Φ∗jΦjΘ₀mΘ₁m

=

m

j=0

(–1)m–j

m

j

Φ₊∗+Φ–

j Φ++Φ–∗

j Θ₀mΘ₁m

=AmBm+H,

for someH∈θH_M∞_nwhereθ∈Z(θ0)∩Z(θ1). ThereforeAm(α)Bm(α) = 0 forα∈Z(θ). By

Lemma2.2,A(α) andB(α) are invertible, a contradiction. So eitherΦ+ orΦ–is zero. If

Φ+= 0, i.e.,Φ=Φ–∗, then, forΘ1m∈HC2n,

0 =

m

j=0

(–1)m–j

m

j

TΦ∗jT j ΦΘ1m=

m

j=0

(–1)m–j

m

j

TΦj–T j Φ–∗Θ

m 1 = m j=0

(–1)m–j

m

j

Φ_–jPn

Φ_–∗jΘ₁m= (–1)mΘ₁m+KB

for some nonzeroK∈H_M∞_n. SinceBandΘ1are coprime, we have a contradiction.

There-foreΦ–= 0 and henceΦis analytic. This completes the proof.

Example3.2 Suppose that

Φ(z) =

⎡ ⎣

2z–1 1–1₂z 0

0 z–13

ThenΦis analytic, but

TΦ2∗TΦ2 – 2TΦ∗TΦ+I=

9I 0

0 I

= 0.

HenceTΦis not 2-isometric.

Next, we show that everym-isometric block Toeplitz operators with trigonometric poly-nomial symbol is an isometry.

Theorem 3.3 LetΦ:=Φ++Φ–∗be normal and rational of the form

Φ+=Θ0A∗ and Φ–=Θ1B∗ (coprime factorization),

whereΘi=θiInwith a finite Blaschke productθifor i= 0, 1and A,B∈HM2n andZ(θ0)∩

Z(θ1)=∅.Then the Toeplitz operator TΦis an m-isometry if and only ifΦis inner.

Proof Sufficiency is obvious. To prove necessity, ifTΦis anm-isometry, then

m

j=0

(–1)m–j

m

j

TΦ∗jT j ΦK= 0

holds for allK∈H_C2n. PutK=Θ₀mΘ₁m. SinceΦis normal

m

j=0

(–1)m–j

m

j

T_Φ∗jT_ΦjΘ₀mΘ₁m=

m

j=0

(–1)m–j

m

j

Φ∗jΦjΘ₀mΘ₁m

=Φ∗Φ–ImΘ₀mΘ₁m.

Thus (Φ∗Φ –I)m _{= 0 and so, Φ}∗_{Φ –I is nilpotent of order m. Hence the spectrum} σ(Φ∗Φ) ={1}. ThusΦ∗Φ=I. From Lemma2.1and Theorem3.1,Φis inner. This

com-pletes the proof.

Corollary 3.4 Suppose thatΦ,Ψ are normal and rational of the form

Φ+=Θ0A∗ and Φ–=Θ1B∗ (coprime factorization)

and

Ψ+=Θ2C∗ and Ψ–=Θ3D∗ (coprime factorization),

whereΘi=θiInwith a finite Blaschke productθifor i= 0, 1, 2, 3and A,B,C,D∈HM2nand

Z(θ0)∩Z(θ1)=∅andZ(θ2)∩Z(θ3)=∅.Then the following hold.

(i) IfTΦandTΨ arem-isometric operators,thenTΦTΨ andTΨTΦarem-isometric operators.

Proof By Theorem3.3,TΦ andTΨ are isometric operators. ThereforeTΨ∗TΦ∗TΦTΨ =I,

i.e.,TΦTΨ is isometric and soΦΨ is normal. HenceTΦTΨ is anm-isometric operator. Similarly,TΨTΦis also anm-isometric operator. For (ii), suppose thatTΦ–TΨ is anm -isometric operator. By Theorem3.3,TΦ–TΨ is an isometry. Hence

0 =TΦ∗ –TΨ∗

(TΦ–TΨ) –I=TΦ∗TΦ–TΦ∗TΨ–TΨ∗TΦ+TΨ∗TΨ –I

=I–TΦ∗Ψ+Ψ∗Φ.

Thus, we haveTΦ∗_Ψ+Ψ∗_Φ=Ior equivalently,Φ∗Ψ+Ψ∗Φ=I. Conversely, ifΦ∗Ψ+Ψ∗Φ=

I, thenTΦ–TΨ is isometry. HenceTΦ–TΨ is anm-isometric operator.

Next, we study contractive and expansive Toeplitz operators. We considerm-contractive Toeplitz operators with trigonometric polynomial symbols whenm= 1 and 2.

The next lemma plays a key role in finding necessary and sufficient conditions form -expansive andm-contractive Toeplitz operators.

Lemma 3.5([11]) Let A,B,C be complex matrices where A and C are square matrices.

Then

A B

B∗ C

≥0 ⇔

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

A≥0,

B=AW (for some W),

C≥W∗AW.

Moreover,rankA B

B∗C

=rankA⇔C=W∗AW.

First, we consider the 1-contractive Toeplitz operators.

Theorem 3.6 Suppose thatΦis a matrix-valued function in L∞_C2 of the form

Φ(z) =

a(z) b(z)

c(z) d(z)

. (3.1)

(i) Ifa∞+c∞> 1,thenTΦis contractive if and only if

T_b∗_,dTb,d–I≥Tb∗,dTa,c

T_a∗_,cTa,c–I –1

T_a∗_,cTb,d,

whereTx,y= Tx

Ty

.

(ii) Ifa∞+c∞= 1,thenTΦis contractive if and only ifb∞+d∞≥1.

(iii) Ifa∞+c∞< 1,thenTΦis never contractive.

Proof From the definition,TΦis contractive if and only ifTΦ∗TΦ–I≥0 or equivalently,

TaTa+TcTc–I TaTb+TcTd T_bTa+TdTc TbTb+TdTd–I

≥0.

By Lemma3.5,TΦis contractive if and only if

and

T_bTb+TdTd–I≥W∗(TaTb+TcTd) (3.3)

for someW. Ifa∞+c∞> 1 thenTaTa+TcTc–Iis invertible. From Eq. (3.2), we

deduce that

W= (TaTa+TcTc–I)–1(TaTb+TcTd).

Therefore, by the inequality (3.3),

T_bTb+TdTd–I≥

T_b∗T_a∗+T_d∗T_c∗(TaTa+TcTc–I)–1(TaTb+TcTd)

or equivalently,

T_b∗_,dTb,d–I≥Tb∗,dTa,c

T_a∗_,cTa,c–I –1

T_a∗_,cTb,d,

whereTx,y= Tx

Ty

. Ifa∞+c∞= 1, thenTaTa+TcTc=Iand from Eqs. (3.2) and (3.3), T_bTb+TdTd–I≥0.

Ifa∞+c∞< 1, thenTaTa+TcTc<I. By Eq. (3.2),TΦis never contractive. This

com-pletes the proof.

Using the same arguments of Theorem3.6that we can check the following consequence.

Corollary 3.7 Suppose thatΦis a matrix-valued function in L∞_C2of the form(3.1).

(i) IfTaTa+TcTc<I,thenTΦis expansive if and only if

T_bTb+TdTd–I≤

T_b∗T_a∗+T_d∗T_c∗(TaTa+TcTc–I)–1(TaTb+TcTd).

(ii) IfTaTa+TcTc=I,thenTΦis expansive if and only ifTbTb+TdTd–I≤0.

(iii) IfTaTa+TcTc>I,thenTΦis never expansive.

Corollary 3.8 Suppose thatΦ is a matrix-valued function in L∞_C2 of the form(3.1).Then

the following statements hold.

(i) Ifb= 0andc= 0,thenTΦis contractive if and only ifTaandTdare contractive.

(ii) Ifa= 0andd= 0,thenTΦis contractive if and only ifTbandTcare contractive.

Proof Ifb=c= 0, then from Eqs. (3.2) and (3.3),TΦis contractive if and only ifTaTa≥I

andT_dTd≥I.

Corollary 3.9 Suppose thatΦis a matrix-valued function in L∞_C2of the form(3.1)where

Ta,Tb,Tc,and Tdare isometric.Then TΦis contractive if and only if Tab+cdis expansive.

Proof From Lemma2.1and Theorem3.6, we haveT_ab+cdTab+cd≤Ior equivalently,Tab+cd

Next, we study 2-contractive Toeplitz operators. Suppose thatΦis a matrix-valued func-tion inL∞_Cnof the form

Φ(z) =

A(z) B(z)

C(z) D(z)

.

Then by a direct calculation we deduce that

B2(TΦ) =TΦ2∗TΦ2– 2TΦ∗TΦ+I=

M1 M2

M∗₂ M3

, (3.4)

where

M1=TA2∗TA2+TA∗TC∗TCTA+TC∗TB∗TA2+TC∗TD∗TCTA+TA2∗TBTC

+TA∗TC∗TDTC+TC∗TB∗TBTC+TC∗TD∗TDTC– 2TA∗TA– 2TC∗TC+I,

M2=TA2∗TATB+TA∗TC∗TCTB+TC∗TB∗TATB+TC∗TD∗TCTB+TA2∗TBTD

+TA∗TC∗TD2+TC∗TB∗TBTD+TC∗TD∗TD2– 2TA∗TB– 2TC∗TD,

and

M3=TB∗TA∗TATB+TB∗TC∗TCTB+TD∗TB∗TATB+TD2∗TCTB+TB∗TA∗TBTD

+TB∗TC∗TD2+TD∗TB∗TBTD+TD2∗TD2– 2TB∗TB– 2TD∗TD+I.

Since the induced relations are very complicated, in order to consider the 2-contractive or 2-expansive Toeplitz operators, we consider the case with a simple symbol. Recently,

m-isometric Toeplitz operators with single-valued symbols were studied in [7].

Lemma 3.10([7]) Letϕbe a rational function.A Toeplitz operator Tϕis an m-isometry if

and only if Tϕis an isometry.

Theorem 3.11 Suppose thatΦis a matrix-valued rational function in L∞_Cnof the form

Φ(z) =

A(z) B(z)

C(z) D(z)

,

where TA,TB,TC,and TDare m-isometric operators.Then TΦis2-contractive if and only if

M1≥0, M2=M1W, and M3≥W∗M2 (3.5)

for some W where

M1=TC∗B∗A2+T_C∗_D∗_CA+T_A∗2_BC+TA∗C∗DC+I,

M3=TD∗B∗AB+TD∗2CB+TB∗A∗BD+TB∗C∗D2+I.

In particular,suppose that A,B,C,and D are single-valued trigonometric polynomials.

If TΦis2-contractive then

degA=degD, 2degA=degB+degC. (3.6)

Proof From Lemmas2.1,3.10, and Eq. (3.4), if follows thatTΦis 2-contractive if and only if the relations (3.5) hold for someW. In particular, ifA,B,C, andDare single-valued trigonometric polynomials, then from Lemma3.10, we can setA(z) =λ1zka,B(z) =λ2zkb,

C(z) =λ3zkc, andD(z) =λ4zkdwith|λ1|=|λ2|=|λ3|=|λ4|= 1. Hence

M1=Tλ2₁λ2λ3z2ka–kb–kc+Tλ1λ4zka–kd+Tλ12λ2λ3zkb+kc–2ka+Tλ1λ4zkd–ka+I.

From Eq. (3.5),

M1≥0 ⇐⇒ T_λ2

1λ2λ3z2ka–kb–kc+λ1λ4zka–kd+λ12λ2λ3zkb+kc–2ka+λ1λ4zkd–ka+I≥0

⇐⇒ T_2Re{λ2

1λ2λ3z2ka–kb–kc+λ1λ4zka–kd}+1≥0,

and henceka=kdand 2ka=kb+kc. This completes the proof.

The next result is a necessary and sufficient condition forTΦto be 2-contractive, under the same hypotheses as Theorem3.11but with the additional condition that each entry in the matrix is a single-valued function.

Corollary 3.12 Suppose thatΦis a matrix-valued function in L∞_C2of the form

Φ(z) =

azm _bzl

cz2m–l _dzm

where|a|=|b|=|c|=|d|= 1.Then TΦis2-contractive if and only if

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(2Re{a2bc+ad}+ 1)(2Re{bcd2+ad}+ 1)≥ |ac+bd+a2bd+acd2|2

if2Re{a2_{bc+ad}+ 1 > 0,}

2Re{bcd2+ad}+ 1≥0 if2Re{a2bc+ad}+ 1 = 0.

Furthermore,if2Re{a2_{bc+ad}+ 1 < 0,then TΦis not2-contractive.}

Proof From the proof of Theorem3.11, we deduced thatTΦis 2-contractive if and only if

M1≥0, M2=M1W, and M3≥W∗M2 (3.7)

for someW where

M2=Taczl–m+T_bdzl–m+T_a2_bdzl–m+T_acd2_zl–m,

M3=Tad+Tbcd2+Tad+Tbcd2+I.

From Eq. (3.7),M1≥0 if and only if 2Re{a2bc+ad}+ 1≥0. There are two cases to

con-sider. If 2Re{a2_{bc+ad}+ 1 > 0, then}

W=2Rea2bc+ad+ 1–1T_(ac+bd+a2_bd+acd2_)zl–m.

HenceM3≥W∗M2if and only if

2Rea2bc+ad+ 12Rebcd2+ad+ 1≥ac+bd+a2bd+acd22.

If 2Re{a2_{bc+ad}+ 1 = 0, thenM}

1=M2= 0 and henceM3≥0 or equivalently, 2Re{bcd2+

ad}+ 1≥0. This completes the proof.

Example3.13 Suppose thatΦ(z) =–_{z z}z–z. Then Eq. (3.6) holds, butM1= –3I< 0.

There-foreTΦis not 2-contractive. Hence the converse of Theorem3.11does not hold.

4 Conclusion

In [7],m-isometric Toeplitz operators were studied for the case of single trigonometric polynomial symbols. In this paper, we study the properties of Toeplitz operators with matrix-valued trigonometric polynomial symbols and we obtain a necessary and suffi-cient condition for block Toeplitz operators with trigonometric polynomial symbols to be

m-isometric orm-contractive.

Funding

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03931937). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2019R1A6A1A11051177) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Ewha Womans University, Seoul, Korea.}2_{Institute of Mathematical Sciences, Ewha}

Womans University, Seoul, Korea.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 29 October 2018 Accepted: 20 June 2019

References

1. Agler, J., Stankus, M.: m-isometric transformations of Hilbert space I. Integral Equ. Oper. Theory21, 383–429 (1995) 2. Agler, J., Stankus, M.: m-isometric transformations of Hilbert space II. Integral Equ. Oper. Theory23, 1–48 (1995) 3. Agler, J., Stankus, M.: m-isometric transformations of Hilbert space III. Integral Equ. Oper. Theory24, 379–421 (1995) 4. Agler, J.: A disconjugacy theorem for Toeplitz operators. Am. J. Math.112, 1–14 (1990)

5. Bermúdez, T., Martinón, A., Noda, J.A.: Products ofm-isometries. Linear Algebra Appl.438, 80–86 (2013) 6. Patton, L., Robins, M.: Composition operators that arem-isometries. Houst. J. Math.31(1), 255–266 (2005) 7. Ko, E., Lee, J.: Onm-isometric Toeplitz operators. Bull. Korean Math. Soc.55, 367–378 (2018)

9. Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math.213, 89–102 (1963) 10. Curto, R.E., Hwang, I.S., Lee, W.Y.: Which subnormal Toeplitz operators are either normal or analytic? J. Funct. Anal.263,

2333–2354 (2012)