Journal of Functional Analysis 263 (2012) 2333–2354
www.elsevier.com/locate/jfa
Which subnormal Toeplitz operators are either normal
or analytic?
✩Raúl E. Curto
a,∗, In Sung Hwang
b, Woo Young Lee
c aDepartment of Mathematics, University of Iowa, Iowa City, IA 52242, USA bDepartment of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of KoreacDepartment of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea Received 27 September 2010; accepted 11 July 2012
Available online 24 July 2012 Communicated by D. Voiculescu
Abstract
We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos’s Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse’s theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factoriza-tion”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.
©2012 Elsevier Inc. All rights reserved.
Keywords:Block Toeplitz operators; Subnormal; Hyponormal; Bounded type functions
✩ The work of the first named author was partially supported by NSF Grant DMS-0801168. The work of the second
named author was supported by NRF grant funded by the Korea government (MEST) (2009-0075890). The work of the third author was supported by the Basic Science Research Program through the NRF grant funded by the Korea government (MEST) (2010-0001983).
* Corresponding author.
E-mail addresses:[email protected](R.E. Curto),[email protected](I.S. Hwang),[email protected] (W.Y. Lee).
0022-1236/$ – see front matter ©2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2012.07.002
1. Introduction
Toeplitz operators arise naturally in several fields of mathematics and in a variety of prob-lems in physics (in particular, in the field of quantum mechanics). On the other hand, the theory of subnormal operators is an extensive and highly developed area, which has made important contributions to a number of problems in functional analysis, operator theory, and mathematical physics. Thus, it becomes of central significance to describe in detail subnormality for Toeplitz operators. This paper focuses on subnormality forblockToeplitz operators and more precisely, the case of block Toeplitz operators with bounded type symbols. Our main result is an appro-priate generalization of Abrahamse’s theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic.
Naturally, this research is closely related to the study of subnormal operators with finite rank self-commutator, a class that has been extensively researched by many authors. However, un-til now a complete description of that class has proved elusive. Recently, D. Yakubovich[30]
has shown that if S is a pure subnormal operator with finite rank self-commutator and ad-mits a normal extension with no nonzero eigenvectors, thenSis unitarily equivalent to a block Toeplitz operator with analytic rational normal matrix symbol. A corollary of our main result illustrates, in a certain sense, the case of subnormal Toeplitz operators with finite rank self-commutator.
To describe our results in more detail, we first need to review a few essential facts about (block) Toeplitz operators, and for that we will use[10,11,14,28]. LetHbe a complex Hilbert space and let B(H) be the algebra of bounded linear operators acting on H. An operator
T ∈B(H)is said to behyponormalif its self-commutator [T∗, T] :=T∗T −T T∗is positive (semi-definite), andsubnormalif there exists a normal operatorNon some Hilbert spaceK⊇H such thatHis invariant underN andN|H=T. Let T≡∂Dbe the unit circle in the complex plane. Let L2≡L2(T)be the set of all square-integrable measurable functions on Tand let
H2≡H2(T)be the corresponding Hardy space. LetH∞≡H∞(T):=L∞(T)∩H2(T), that is,
H∞is the set of bounded analytic functions onD. Givenφ∈L∞, the Toeplitz operatorTφand
the Hankel operatorHφare defined by
Tφg:=P (φg) and Hφg:=J P⊥(φg)
g∈H2,
where P and P⊥ denote the orthogonal projections that map fromL2 ontoH2 and (H2)⊥, respectively, and whereJ denotes the unitary operator onL2defined byJ (f )(z)=zf (z).
In the early 1960s, normal Toeplitz operators were characterized by a property of their sym-bols by A. Brown and P.R. Halmos[3]. On the other hand, the exact nature of the relationship between the symbolφ∈L∞and the hyponormality ofTφ was understood much later, in 1988,
via Cowen’s theorem[6].
Cowen’s theorem.(See[6,27].) For eachφ∈L∞, let
E(φ)≡k∈H∞: k ∞1andφ−kφ∈H∞.
The elegant and useful theorem of C. Cowen has been used in the works[8,9,12,15–17,21–27, 31], which have been devoted to the study of hyponormality for Toeplitz operators onH2. When one studies hyponormality (also, normality and subnormality) of the Toeplitz operatorTφ one
may, without loss of generality, assume thatφ(0)=0; this is because hyponormality is invariant under translation by scalars. We now recall that a functionφ∈L∞is said to be ofbounded type
(or in the Nevanlinna class) if there are analytic functionsψ1, ψ2∈H∞(D)such that
φ(z)=ψ1(z) ψ2(z)
for almost allz∈T.
It is well known[1, Lemma 3]that ifφ /∈H∞then
φis of bounded type ⇔ kerHφ= {0}. (1)
Ifφ∈L∞, we write
φ+≡P φ∈H2 and φ−≡P⊥φ∈zH2.
Assume now that bothφ andφ are of bounded type. Since TzHψ =HψTz for all ψ ∈L∞,
it follows from Beurling’s theorem that kerHφ
−=θ0H
2 and kerH
φ+ =θ+H
2for some inner functionsθ0, θ+. We thus haveb:=φ−θ0∈H2, and hence we can write
φ−=θ0b and similarly φ+=θ+a for somea∈H2. In particular, ifTφis hyponormal andφ /∈H∞, and since
Tφ∗, Tφ =H∗ φHφ−Hφ∗Hφ=Hφ∗ +Hφ+−Hφ∗−Hφ−, it follows that Hφ
+f Hφ−f for allf ∈H2, and hence
θ+H2=kerHφ
+⊆kerHφ−=θ0H2,
which implies thatθ0 dividesθ+, i.e.,θ+=θ0θ1for some inner function θ1. We write, for an inner functionθ,
Hθ:=H2θ H2.
Note that iff =θ a∈L2, thenf ∈H2 if and only ifa∈Hzθ; in particular, iff (0)=0 then
a∈Hθ. Thus, ifφ=φ−+φ+∈L∞is such thatφandφare of bounded type such thatφ+(0)=0
andTφis hyponormal, then we can write
φ+=θ0θ1a¯ and φ−=θ0b,¯ wherea∈Hθ0θ1 andb∈Hθ0.
By Kronecker’s lemma[28, p. 183], iff ∈H∞ thenf is a rational function if and only if rankHf <∞, which implies that
On the other hand, M.B. Abrahamse [1, Lemma 6]also showed that if Tφ is hyponormal, if
φ /∈H∞, and ifφorφis of bounded type then bothφandφare of bounded type.
We now introduce the notion of block Toeplitz operators. For a Hilbert space X, letL2X ≡ L2X(T)be the Hilbert space of X-valued norm square-integrable measurable functions onT and let HX2 ≡HX2(T) be the corresponding Hardy space. We observe that L2Cn =L2⊗Cn
andHC2n=H2⊗Cn. IfΦ is a matrix-valued function inLM∞n≡L∞Mn(T)(=L∞⊗Mn) then
TΦ:HC2n→HC2ndenotes the block Toeplitz operator with symbolΦdefined by
TΦF:=Pn(ΦF ) forF ∈HC2n,
wherePnis the orthogonal projection ofL2Cn ontoHC2n. A block Hankel operator with symbol
Φ∈L∞M
nis the operatorHΦ:H
2
Cn→HC2ndefined by
HΦF:=JnPn⊥(ΦF ) forF ∈HC2n,
whereJn denotes the unitary operator from(HC2n)⊥ toHC2n given byJn(F )(z):=zInF (z)for
F ∈HC2n, and whereIn is then×nidentity matrix. If we setHC2n:=H2⊕ · · · ⊕H2then we
see that TΦ= ⎡ ⎣ Tφ11 . . . Tφ1n .. . Tφn1 . . . Tφnn ⎤ ⎦ and HΦ= ⎡ ⎣ Hφ11 . . . Hφ1n .. . Hφn1 . . . Hφnn ⎤ ⎦, where Φ= ⎡ ⎣ φ11 . . . φ1n .. . φn1 . . . φnn ⎤ ⎦∈L∞M n. ForΦ∈L∞M n, write Φ(z):=Φ∗(z). A matrix-valued functionΘ∈HM∞ n×m (=H ∞⊗M
n×m) is calledinnerifΘ(z)∗Θ(z)=Imfor
almost allz∈T. The following basic relations can be easily derived:
TΦ∗=TΦ∗, HΦ∗=HΦ Φ∈L∞M n ; TΦΨ −TΦTΨ =HΦ∗∗HΨ Φ, Ψ∈L∞M n ; (3) HΦTΨ =HΦΨ, HΨ Φ=TΨ∗HΦ Φ∈L∞M n, Ψ ∈H ∞ Mn ; (4) HΦ∗HΦ−HΘΦ∗ HΘΦ=HΦ∗HΘ∗HΘ∗∗HΦ Θ∈HM∞ n is inner, Φ∈L ∞ Mn .
For a matrix-valued functionΦ= [φij] ∈L∞Mn, we say thatΦ is ofbounded typeif each entry
φij is of bounded type and thatΦ isrationalif each entryφij is a rational function. Theshift
operatorSonHC2nis defined by S:= n j=1 ⊕Tz.
The following fundamental result known as the Beurling–Lax–Halmos theorem is useful in the sequel.
Beurling–Lax–Halmos theorem. A nonzero subspace M ofHC2n is invariant under the shift
operatorS onHC2n if and only ifM=ΘHC2m, whereΘ is an inner matrix function inHM∞n×m
(mn).
In view of (4), the kernel of a block Hankel operatorHΦ is an invariant subspace of the shift
operator onHC2n. Thus if kerHΦ= {0}then by the Beurling–Lax–Halmos theorem,
kerHΦ=ΘHC2m
for some inner matrix functionΘ. But we don’t guarantee that Θ is a square matrix. In fact, as we will refer in the sequel,Θ is square if and only ifΦ is of bounded type. Recently, Gu, Hendricks and Rutherford[18]considered the hyponormality of block Toeplitz operators and characterized the hyponormality of block Toeplitz operators in terms of their symbols. In partic-ular they showed that ifTΦ is a hyponormal block Toeplitz operator onHC2n, thenΦ is normal,
i.e.,Φ∗Φ=ΦΦ∗. Their characterization for hyponormality of block Toeplitz operators resem-bles the Cowen’s theorem except for an additional condition – the normality condition of the symbol.
Hyponormality of block Toeplitz operators. (See Gu, Hendricks and Rutherford[18].) For eachΦ∈L∞M n, let E(Φ):=K∈HM∞ n: K ∞1andΦ−KΦ ∗∈H∞ Mn .
ThenTΦis hyponormal if and only ifΦ is normal andE(Φ)is nonempty.
For a matrix-valued functionΦ∈HM2
n×r, we say that ∈H
2
Mn×m is aleft inner divisorof
Φ if is an inner matrix function such thatΦ= Afor someA∈HM2
m×r (mn). We also
say that two matrix functionsΦ∈HM2
n×r andΨ ∈H
2
Mn×m areleft coprimeif the only common
left inner divisor of bothΦandΨ is a unitary constant and thatΦ∈HM2
n×r andΨ ∈H
2
Mm×r are
right coprimeifΦandΨare left coprime. Two matrix functionsΦ andΨ inHM2
nare said to be
coprimeif they are both left and right coprime. We remark that ifΦ∈HM2
nis such that detΦ is
not identically zero then any left inner divisor of Φ is square, i.e., ∈HM2
n. IfΦ∈H
2
Mn is
such that detΦ is not identically zero then we say that ∈HM2
n is aright inner divisorofΦ if
is a left inner divisor ofΦ.
Lemma 1.1.(See[18].) ForΦ∈L∞M
n, the following statements are equivalent:
(i) Φis of bounded type;
(ii) kerHΦ=ΘHC2nfor some square inner matrix functionΘ;
(iii) Φ=AΘ∗, whereA∈HM∞
n andAandΘare right coprime.
ForΦ∈L∞M nwe write Φ+:=PnΦ∈HM2n and Φ−:= Pn⊥Φ∗∈HM2 n.
Thus we can writeΦ=Φ−∗ +Φ+. For an inner matrix functionΘ∈HM∞
n, write HΘ:= ΘHC2n ⊥ ≡HC2nΘHC2n.
SupposeΦ= [φij] ∈L∞Mn is such thatΦ∗is of bounded type. Then we may writeφij=θijbij,
whereθijis an inner function andθijandbij are coprime. Thus ifθis the least common multiple
of θij’s (i.e., theθij divideθ and if they divide an inner functionθthenθ in turn dividesθ),
then we can write
Φ= [φij] = [θijbij] = [θ aij] =ΘA∗
Θ=θ In, A∈HM2n
. (5)
We note that the representation (5) is “minimal”, in the sense that ifωIn(ωis inner) is a common
inner divisor ofΘandA, thenωis constant. LetΦ≡Φ−∗ +Φ+∈L∞M
n be such thatΦandΦ
∗
are of bounded type. Then in view of (5) we can write
Φ+=Θ1A∗ and Φ−=Θ2B∗,
whereΘi=θiInwith an inner functionθifori=1,2 andA, B∈HM2n. In particular, ifΦ∈L∞Mn
is rational then theθi are chosen as finite Blaschke products as we observed in (2).
We would remark that, in (5), by contrast with scalar-valued functions,ΘandAneed not be (right) coprime: indeed, ifΦ:=z zz zthen we can write
Φ=ΘA∗= z 0 0 z 1 1 1 1 , butΘ:=z00zandA:=1 1 1 1
are not right coprime because √1
2 z−z
1 1
is a common right inner factor, i.e., Θ=√1 2 1 z −1 z ·√1 2 z −z 1 1 and A=√2 0 1 0 1 ·√1 2 z −z 1 1 . (6)
In this paper we consider the subnormality of block Toeplitz operators and in particular, the block version of Halmos’s Problem 5: Which subnormal block Toeplitz operators are either nor-mal or analytic? In 1976, M.B. Abrahamse showed that ifφ=g+f∈L∞(f, g∈H2) is such thatφorφis of bounded type, ifTφis hyponormal, and if ker[Tφ∗, Tφ]is invariant underTφthen
Tφis normal or analytic. The purpose of this paper is to establish an extension of Abrahamse’s
theorem for block Toeplitz operators. In Section 2 we make a brief sketch on Halmos’s Problem 5 and Abrahamse’s theorem. Section 3 is devoted to the proof of the main result. In Section 4 we consider the scalar Toeplitz operators with finite rank self-commutators.
2. Halmos’s Problem 5 and Abrahamse’s theorem
In 1970, P.R. Halmos posed the following problem, listed as Problem 5 in his lectures “Ten problems in Hilbert space”[19,20]:
Is every subnormal Toeplitz operator either normal or analytic?
A Toeplitz operatorTφis calledanalyticifφ∈H∞. Any analytic Toeplitz operator is easily seen
to be subnormal: indeed,Tφh=P (φh)=φh=Mφhforh∈H2, whereMφ is the normal
op-erator of multiplication byφonL2. The question is natural because the two classes, the normal and analytic Toeplitz operators, are fairly well understood and are subnormal. Halmos’s Prob-lem 5 has been partially answered in the affirmative by many authors (cf.[1,2,8,9,27], and etc.). In 1984, Halmos’s Problem 5 was answered in the negative by C. Cowen and J. Long[7]: they found an analytic functionψfor whichTψ+αψ (0< α <1) is subnormal – in fact, this Toeplitz operator is unitarily equivalent to a subnormal weighted shiftWβwith weight sequenceβ≡ {βn},
whereβn=(1−α2n+2) 1
2 for n=0,1,2, . . . .Unfortunately, Cowen and Long’s construction
does not provide an intrinsic connection between subnormality and the theory of Toeplitz op-erators. Until now researchers have been unable to characterize subnormal Toeplitz operators in terms of their symbols. On the other hand, surprisingly, as C. Cowen notes in[4,5], some analytic Toeplitz operators are unitarily equivalent to non-analytic Toeplitz operators; i.e., the analytic-ity of Toeplitz operators is not invariant under unitary equivalence. In this sense, we might ask whether Cowen and Long’s non-analytic subnormal Toeplitz operator is unitarily equivalent to an analytic Toeplitz operator. To this end, we have:
Observation.Cowen and Long’s non-analytic subnormal Toeplitz operatorTφ is not unitarily
equivalent to any analytic Toeplitz operator.
Proof. Assume to the contrary thatTφis unitarily equivalent to an analytic Toeplitz operatorTf.
Then by the above remark,Tf is unitarily equivalent to the subnormal weighted shiftWβ with
weight sequenceβ≡ {βn}, whereβn=(1−α2n+2) 1
2 (0< α <1) forn=0,1,2, . . .;i.e., there
exists a unitary operatorV such that
V∗TfV =Wβ.
Thus if{en}is the canonical orthonormal basis for2then
V∗TfV ej=Wβej=βjej+1 forj=0,1,2, . . . . We thus have V∗T|f|2V ej=Wβ∗Wβej=βj2ej,
and hence,
T|f|2−β2
j(V ej)=0 forj=0,1,2, . . . .
Fixj0 and observe thatV ej ∈ker(T|f|2−β2
j). By Coburn’s theorem, if|f| 2−β2 j is nonzero then eitherT|f|2−β2 j or T ∗ |f|2−β2 j
is one-one. It follows that|f|2=βj2 forj =0,1,2, . . . .This readily implies thatβ0=β1=β2= · · ·,a contradiction. 2
Consequently, even if we interpret “is” in Halmos Problem 5 as “is up to unitary equivalence”, the answer to Halmos Problem 5 is still negative.
We would like to reformulate Halmos’s Problem 5 as follows:
Halmos’s Problem 5 reformulated.Which Toeplitz operators are subnormal?
The most interesting partial answer to Halmos’s Problem 5 was given by M.B. Abrahamse[1]. M.B. Abrahamse gave a general sufficient condition for the answer to Halmos’s Problem 5 to be affirmative.
Abrahamse’s theorem can be then stated as:
Abrahamse’s theorem.(See[1, Theorem].) Letφ=g+f ∈L∞(f, g∈H2)be such thatφ
or φis of bounded type. IfTφ is hyponormal andker[Tφ∗, Tφ]is invariant underTφthenTφis
normal or analytic.
Consequently, ifφ=g+f ∈L∞(f, g∈H2) is such thatφorφ is of bounded type, then every subnormal Toeplitz operator must be normal or analytic.
We say that a block Toeplitz operatorTΦisanalyticifΦ∈HM∞n. Evidently, any analytic block
Toeplitz operator with a normal symbol is subnormal because the multiplication operatorMΦis
a normal extension ofTΦ. As a first inquiry in the above reformulation of Halmos’s Problem 5
the following question can be raised:
Is Abrahamse’s theorem valid for block Toeplitz operators?
In this paper we answer this question in the affirmative (Theorem3.5).
3. Abrahamse’s theorem for matrix-valued symbols Recall the representation (5), and forΨ ∈L∞M
n such thatΨ
∗is of bounded type, writeΨ =
Θ2B∗=B∗Θ2. LetΩ be the greatest common left inner divisor ofBandΘ2. ThenB=ΩB
andΘ2=ΩΩ2for someB∈HM2nand some inner matrixΩ2. Therefore we can write
Ψ =B∗Ω2, whereBandΩ2are left coprime; in this case,B∗Ω2is called aleft coprime factorizationofΨ. Similarly,
Ψ = 2Br∗, whereBrand 2are right coprime; in this case, 2Br∗is called aright coprime factorizationofΨ.
To prove our main result (Theorem3.5), we need several auxiliary lemmas. We begin with:
Lemma 3.1.SupposeΦ=Φ−∗ +Φ+∈L∞M
n is such thatΦ andΦ
∗ are of bounded type of the
form
Φ+=A∗Θ1 and Φ−=B∗Θ2,
whereΘi :=θiIn with an inner functionθi (i=1,2). IfTΦ is hyponormal, thenΘ2is a right
inner divisor ofΘ1.
Proof. SupposeTΦis hyponormal. Then there exists a matrix functionK∈HM∞nsuch thatΦ
∗ −−
KΦ+∗ ∈HM2
n. ThusBΘ
∗
2−KAΘ1∗=F for someF ∈H 2
Mn, which implies thatBΘ
∗
2Θ1∈H 2
Mn.
Now we writeΦ−=[fij]n×n. SinceΦis of bounded type we can writefij =θijcij, whereθij
is an inner function,cijis inH2, andθij andcijare coprime. WriteB=[bij]n×n. We thus have
fij=θijcij=θ2bj i for eachi, j=1, . . . , n,
which implies thatbj i=θijθ2cij. But sinceBΘ2∗Θ1= [θ1θ2bij] ∈HM2n, we haveθ1θj icj i∈H2.
Sinceθj iandcj iare coprime for eachi, j=1, . . . , n, it follows thatθj iθ1∈H2, which implies that θ2θ1∈H2 and therefore, Θ2 divides Θ1, i.e.,Θ1=Θ0Θ2 for some inner matrix func-tionΘ0. 2
In the sequel, when we consider the symbolΦ=Φ−∗+Φ+∈L∞M
n, which is such thatΦand
Φ∗are of bounded type and for whichTΦis hyponormal, we will, in view of Lemma 3.1, assume
that
Φ+=A∗Ω1Ω2 and Φ−=B∗Ω2 (left coprime factorization),
whereΩ1Ω2=Θ =θ In. We also note that Ω2Ω1=Θ: indeed, if Ω1Ω2=Θ=θ In, then
(θ InΩ1)Ω2=In, so that Ω1(θ InΩ2)=In, which implies that (θ InΩ2)Ω1=In, and hence
Ω2Ω1=θ In=Θ.
Lemma 3.2.SupposeΦ=Φ−∗ +Φ+∈L∞M
n is such thatΦ andΦ
∗ are of bounded type of the
form
Φ+= 1A∗r (right coprime factorization)
and
Φ−= 2Br∗ (right coprime factorization).
Proof. SupposeTΦ is hyponormal. Then there exists K∈HM∞n such that Φ−KΦ∗∈HM∞n.
Thus HΦ=HKΦ∗=TK∗HΦ∗, which implies that kerHΦ+∗ ⊆kerHΦ∗−, so that by Lemma1.1,
1HC2n⊆ 2HC2n. It follows (cf.[13, Corollary IX.2.2]) that 2is a left inner divisor of 1. 2 On the other hand, the condition “(left/right) coprime factorization” is not so easy to check in general. For example, consider a simple case:Φ−:=z zz z. One is tempted to write
Φ−:= z 0 0 z 1 1 1 1 ∗ .
Butz00zand1 11 1are not right coprime as we have seen in the Introduction. On the other hand, observe that 1 1 1 1 ≡ B∗, where :=√1 2 1 z −1 z is inner and B:=√1 2 0 2z 0 2z .
Again, andBare not right coprime because kerH1 1 1 1
=HC22. Thus we might choose
Φ−=(zI2 )·B∗ or Φ−= ·(zI2B)∗.
A straightforward calculation shows that kerHΦ−∗ = HC22. Hence the latter of the above
factor-izations is the desired factorization: i.e., andzI2Bare right coprime.
However, ifΘis given in a formΘ=θ Inwith a finite Blaschke productθ, then we can obtain
a more tractable criterion on the coprime-ness ofΘandB∈HM2
n. To see this, recall that ann×n
matrix-valued functionDis calleda finite Blaschke–Potapov productifDis of the form
D(z)=ν M m=1 bm(z)Pm+(I−Pm) ,
whereνis ann×nunitary constant matrix,bmis aBlaschke factor, which is of the form
bm(z):=
z−αm
1−αmz
(αm∈D),
andPmis an orthogonal projection inCn. In particular, a scalar-valued functionDreduces to a
finite Blaschke productD(z)=νMm=1bm(z), whereν=eiω. It was known[29]that ann×n
matrix-valued function D is rational and inner if and only if it can be represented as a finite Blaschke–Potapov product.
We writeZ(θ )for the set of zeros of an inner functionθ. We then have: Lemma 3.3.Let B∈HM2
n andΘ:=θ In with a finite Blaschke productθ. Then the following
(a) B(α)is invertible for eachα∈Z(θ ); (b) BandΘare right coprime;
(c) BandΘare left coprime.
Proof. We first write
θ (z)=eiξ N i=1 z−αi 1−αiz mi N i=1 mi=:d .
(a)⇔(b): SupposeB(α)is invertible for eachα∈Z(θ ). Assume to the contrary thatBand
Θare not right coprime. Thus there exists a finite Blaschke–Potapov productDof the form
D(z)=ν M m=1 bm(z)Pm+(I−Pm) satisfying that
B=B1D and Θ=Θ0D for some inner functionΘ0.
Thus ifα∈Z(bm0)for some 1m0M, thenΘ(α)=Θ0(α)D(α)is not invertible. But since
Θ=θ In, it follows thatΘ(α)=0 and henceα∈Z(θ ). Moreover,
detB(α)=detB1(α)detD(α)=det(ν)detB1(α)
M m=1 detbm(α)Pm+(I−Pm) =0,
giving a contradiction. ThereforeBandΘare right coprime.
For the converse we assume that B(αi0)is not invertible for some i0. Then the following
matrix is not invertible:
B:= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ B0 0 0 0 · · · 0 B1 B0 0 0 · · · 0 B2 B1 B0 0 · · · 0 .. . . .. . .. ... ... ... Bmi0−2 Bmi0−3 . .. ... B0 0 Bmi0−1 Bmi0−2 . . . B2 B1 B0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Bj:= B(j )(αi0) j! .
Thus there exists a nonzeron×mi0 matrixG=(G0G1· · ·Gmi0−1)
t such thatBG=0.We now
want to show that there existsh=( h1h2· · ·hn)t∈HC2nsatisfying the following property:
h(j )(αi) j! = G j (i=i0), 0 (i=i0). (7)
This is exactly the classical Hermite–Fejér interpolation problem (cf.[13]), so that we use an argument of a solution for the interpolation of this type. Thus we can construct a function (in
fact, a polynomial)h(z)≡P (z)satisfying (7) (see[13, p. 299]). ThenP (z)belongs to kerHBΘ∗.
Since
G=[G0G1· · ·Gmi0−1]
t=
0,
it follows thatP (z) /∈ΘHC2n. Therefore we have kerHBΘ∗=ΘHC2n, which implies thatB and
Θare not right coprime.
(b)⇔(c): SupposeBandΘare right coprime. IfBandΘare not left coprime, there exists a nonconstant inner matrix ∈HM2
n such thatB= B1 andΘ= Ω.We thus have that for
eachi=1,2, . . . , N ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ i,0 0 0 0 · · · 0 i,1 i,0 0 0 · · · 0 i,2 i,1 i,0 0 · · · 0 .. . . .. . .. . .. . .. ... i,mi−2 i,mi−3 . .. . .. i,0 0 i,mi−1 i,mi−2 . . . i,2 i,1 i,0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ωi,0 Ωi,1 Ωi,2 .. . Ωi,mi−2 Ωi,mi−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ =0, where i,j:= (j )(α i) j! and Ωi,j := Ω(j )(αi) j! .
But since B(α) is invertible for each α∈Z(θ ), we have that i,0 is invertible for each i= 1,2, . . . , N. Thus
Ωi,j =0 (i=1,2, . . . , N, j=0,1,2, . . . , mi−1),
which implies thatΩ=ΘΩ1for someΩ1∈HM2n. ThusΘ= Ω= ΘΩ1, so thatI = Ω1 and hence ∗=Ω1, which implies that is a constant matrix, a contradiction. ThusBandΘ
are left coprime. The converse follows from the same argument. This completes the proof. 2 Lemma 3.4.Letθ0be a nonconstant inner function. ThenHθ0 contains an outer function that is
invertible inH∞.
Proof. Ifθ0has at least one Blaschke factor, say 1z−−αzα (|α|<1), then 1−1αz is an outer function and 1−1αz∈Hθ0 because 1−1αz is the reproducing kernel forα, so that for anyf∈H2,
θ0f, 1 1−αz =θ0(α)f (α)=0. Now supposeθ0is a nonconstant singular inner function of the form
θ0(z):=exp − π −π eiθ+z eiθ−zdμ(θ ) ,
whereμ is a finite positive Borel measure on T which is singular with respect to Lebesgue measure. We put ω(z):=exp − π −π eiθ+z eiθ−zd μ 2(θ ) .
Then ω2=θ0. If α:=ω(0) then evidently, 0<|α|<1 since ω is not constant. Note that
θ0(ω−α1)=ω−α1θ0∈(H2)⊥, since(ω−α1θ0)(0)=α−α1α2=0. We thus haveω−1α ∈Hθ0.
Also a straightforward calculation shows that 1
ω−α1 is bounded and analytic inD, which says
thatω−1α is invertible inH∞. Henceω−α1 is an outer function inHθ0. This completes the
proof. 2
Before proving the main result, we recall the inner–outer factorization of vector-valued func-tions. IfD andE are Hilbert spaces and ifF is a function with values inB(E, D) such that
F (·)e∈HD2(T)for eache∈E, thenF is called a strongH2-function. The strongH2-function
F is called aninner functionifF (·)is an isometric operator fromDintoE. WritePE for the
set of all polynomials with values inE, i.e.,p(ζ )=nk=0p(k)ζk,p(k)∈E. Then the function
Fp=nk=0Fp(k)zkbelongs toHD2(T). The strongH2-functionF is calledouterif clF·PE=HD2(T).
Note that if dimD=dimE=n <∞, then evidently, everyF ∈HM2
n is a strongH
2-function. We then have an analogue of the scalar inner–outer factorization theorem.
Inner–outer factorization.(Cf.[28].) Every strongH2-functionF with values inB(E, D)can be expressed in the form
F =FiFe,
whereFeis an outer function with values inB(E, D)andFi is an inner function with values inB(D, D)for some Hilbert spaceD.
We are now ready to prove the main result of this paper.
Theorem 3.5(Abrahamse’s theorem for matrix-valued symbols). SupposeΦ:=Φ−∗+Φ+∈L∞M
n
is such thatΦandΦ∗are of bounded type. In view of Lemma3.1, we may write
Φ+=A∗Θ0Θ2 and Φ−=B∗Θ2,
whereΘi=θiInwith an inner functionθi (i=0,2)andA, B∈HM2n. Assume thatA, BandΘ2
are left coprime. If
(i) TΦis hyponormal;and
then TΦ is normal or analytic. Hence, in particular, if TΦ is subnormal then it is normal or
analytic.
Remark 3.6.We note that ifn=1 (i.e.,TΦis a scalar Toeplitz operator) thenΦ+=aθ0θ2and
Φ−=bθ2 witha, b∈H2. Thus, we can always arrange that a,b andθ are coprime. Conse-quently, ifn=1 then our matrix version reduces to the original Abrahamse’s theorem.
Proof of Theorem 3.5. IfΘ2is constant thenΦ−=0, so thatTΦis analytic. Suppose thatΘ2 is nonconstant.
We split the proof into three steps.
STEP1: We first claim that
Θ0HC2n⊆ker
TΦ∗, TΦ
. (8)
To see this, we observe that
TΦ∗, TΦ
=HΦ∗∗
+HΦ+∗ −HΦ∗−∗HΦ−∗ =HAΘ∗ 2∗Θ0∗HAΘ2∗Θ0∗−HΘ∗2∗BHΘ2∗B, (9)
which implies that
Θ0Θ2HC2n⊆ker
TΦ∗, TΦ
. (10)
On the other hand, sinceΘ0Θ2is diagonal, we have that for allg∈PCn,
TΦ(Θ0Θ2g)=Pn Θ2∗BΘ0Θ2g+Φ+Θ0Θ2g =Θ0Bg+Θ0Θ2Φ+g =PHΘ 0Θ2(Θ0Bg)+PΘ0Θ2HC2n(Θ0Bg)+Θ0Θ2Φ+g.
SinceHΘ0Θ2=HΘ0⊕Θ0HΘ2, it follows that
PHΘ 0Θ2(Θ0Bg)=PΘ0HΘ2(Θ0Bg). We thus have TΦ(Θ0Θ2g)=PΘ0HΘ2(Θ0Bg)+PΘ0Θ2HC2n(Θ0Bg)+Θ0Θ2Φ+g. (11) We claim that HΘ2=cl PHΘ 2(Bg): g∈PCn . (12)
In view of the above mentioned inner–outer factorization, letB=BiBebe the inner–outer fac-torization ofB (as a strongH2-function), whereBi∈HM∞
n×r andB
e∈H2
Mr×n. SinceBandΘ2
are left coprime,Bi andΘ2are left coprime. Thus it follows from the Beurling–Lax–Halmos theorem that
Θ2HC2n∨clBPCn=Θ2HC2n∨Bi
clBePCn=Θ2HC2n∨BiHC2r =HC2n,
giving (12). Thus we have
Θ0HΘ2=clΘ0 PHΘ 2(Bg): g∈PCn =clPΘ0HΘ 2(Θ0Bg): g∈PCn . (13)
If ker[TΦ∗, TΦ]is invariant underTΦthen since ker[TΦ∗, TΦ]is a closed subspace it follows from
(10)–(13) that Θ0HΘ2⊆ker TΦ∗, TΦ . We thus have Θ0HC2n=Θ0HΘ2⊕Θ0Θ2HC2n⊆ker TΦ∗, TΦ , which proves (8).
STEP2: We next claim that
E(Φ)contains an inner functionK. (14) To see this, we first observe that ifK∈E(Φ)then by (4),
TΦ∗, TΦ =HΦ∗∗ +HΦ∗+−HKΦ∗ +∗HKΦ+∗ =HΦ∗+∗ I−TKT∗ K HΦ∗ +, (15) so that kerTΦ∗, TΦ =kerI−TKT∗ K HΦ+∗. Thus by (8), {0} =I−TKT∗ K HAΘ∗ 2Θ0∗ Θ0HC2n =I−TKT∗ K HAΘ∗ 2 HC2n , which implies cl ranHAΘ∗ 2 ⊆ker I−TKT∗ K . (16)
Since by assumption, A andΘ2 are left coprime, and hence AandΘ2 are right coprime, it follows from Lemma1.1that
cl ranHAΘ∗2 =(kerHAΘ2∗)⊥= Θ2HC2n
⊥
=HΘ2 , (17)
which together with (16) implies
HΘ2 ⊆ker I−TKT∗ K . (18)
We thus have
F =TKT∗
KF for eachF∈HΘ2. (19)
But since K ∞= K ∞1, we have
Pn K∗F2K∗F2 F 2=TKTK∗F2=KPn K∗F2Pn K∗F2, which gives
Pn K∗F2=K∗F2,
which impliesK∗F ∈HC2n. Therefore by (19), we have
F=KK∗F for eachF ∈HΘ2.
In view of Lemma3.4, we can choose an outer functionf∈Hθ
2, which is invertible inH ∞. For
eachj=1,2, . . . , n, we define
Fj:=(0, . . . ,0, f,0, . . . ,0)t (wheref is thej-th component).
Then Fj ∈HΘ2 for each j =1,2, . . . , n, so that (I−KK∗)Fj =0 for eachj =1,2, . . . , n.
If we writeI −KK∗≡ [qij]1i,jn∈L∞Mn, thenqijf =0 for eachi, j =1,2, . . . , n, so that
qij =0 for each i, j =1,2, . . . , nbecause f is invertible. Therefore we have K∗K=I =I,
which implies thatKis an inner function. This proves (14).
STEP3: Now sinceKis inner it follows from (3) that
I−TKT∗ K=TKK∗−TKTK∗=H ∗ K∗HK∗. Thus by (18), we have HΘ2⊆kerH ∗ K∗HK∗=kerHK∗. (20)
WriteK:= [kij]1i,jn∈HM∞n. Since, by Lemma3.4,Hθ2 contains an outer functionhthat is
invertible inH∞, it follows from (20) that
kij(z)h∈H2 for eachi, j=1,2, . . . , n,
so thatki,j(z)∈1hH2⊆H2for eachi, j=1,2, . . . , n. Therefore eachkij is constant and hence,
Remark 3.7.Theorem3.5may fail if the condition “BandΘ2are left coprime” is dropped even thoughAandΘ2are left coprime. To see this, letθ be a nonconstant finite Blaschke product. Consider the matrix-valued function
Φ= 2θ+θ θ θ 2θ+θ . Write Θ:= θ 0 0 θ . Then Φ+=2Θ and Φ−= θ θ θ θ = 1 1 1 1 ∗ Θ.
A direct calculation shows that Φ is normal. Put K:=121 1
1 1
. Remember that for a matrix-valued functionA, we define A ∞:=supt∈T A(t ) (where · means the operator norm).
Then K ∞=1 andΦ−KΦ∗∈HM∞
2, so thatTΦis hyponormal. Observe that
[TΦ∗, TΦ] =HΦ∗∗ +HΦ+∗ −H ∗ Φ−∗HΦ−∗ =4 H∗ θHθ 0 0 H∗ θHθ −2 H∗ θHθ H ∗ θHθ H∗ θHθ H ∗ θHθ =2 H∗ θHθ −H ∗ θHθ −H∗ θHθ H ∗ θHθ =2 PHθ −PHθ −PHθ PHθ , which gives ker[TΦ∗, TΦ] =ker PHθ −PHθ −PHθ PHθ = f g : PHθf =PHθg =ΘHC22⊕ {f⊕f:f ∈Hθ}.
We now claim that ker[TΦ∗, TΦ]is invariant underTΦ. To show this we suppose
F = f g ∈kerTΦ∗, TΦ . Then TΦF = 2Tθ+Tθ Tθ Tθ 2Tθ+Tθ f g = 2Tθf +Tθ(f+g) 2Tθg+Tθ(f+g) . Observe that
PHθ2Tθf+Tθ(f+g)
=PHθTθ(f+g)=PHθ2Tθg+Tθ(f+g)
,
which implies that ker[TΦ∗, TΦ]is invariant underTΦ. But since
{f⊕f: f ∈Hθ}HΘ, and hence, ker
TΦ∗, TΦ
=HC22,
we can see thatTΦis not normal. Note that by Lemma3.3,
1 1 1 1
andΘare not left coprime.
If, in the left coprime factorization Φ− =B∗Θ2 (Θ2=θ2In) of Theorem 3.5, θ2 has a Blaschke factor, then the assumption of the “left coprime factorization” for the analytic part
Φ+ofΦcan be dropped in Theorem3.5.
Corollary 3.8.SupposeΦ=Φ−∗ +Φ+∈L∞M
n is such thatΦ andΦ
∗are of bounded type. In
view of (5), we may write
Φ−=B∗Θ,
whereΘ:=θ Inwith an inner functionθ. Assume thatB andΘare left coprime. Assume also
thatθcontains a Blaschke factor. If
(i) TΦ is hyponormal;and
(ii) ker[TΦ∗, TΦ]is invariant underTΦ;
then TΦ is normal or analytic. Hence, in particular, if TΦ is subnormal then it is normal or
analytic.
Proof. For notational convenience, we letΘ2:=Θandθ2:=θ. Now supposeθ2has a Blaschke factorbαand writeBα:=bαIn. By assumption,BandBαare left coprime, so that by Lemma3.3,
B andBα are right coprime. Thus, in view of Lemmas3.1and3.2, we can write
Φ+=A∗Θ0Θ2=Bα 1A∗r and Φ−=B∗Θ2,
whereΘi =θiIn with an inner functionθi (i=0,2), Ar andBα 1are right coprime, andB andΘ2are left coprime. In particular, we note thatAandBα are right coprime, so that again by
Lemma3.3,AandBα are left coprime. On the other hand, an analysis for the proof of STEP 1
of Theorem3.5shows that
Θ0HC2n⊆ker
TΦ∗, TΦ
. (21)
(Note that we didn’t employ the assumption “AandΘ2are left coprime” to get (21) in the proof of STEP 1 of Theorem3.5.) Thus ifK∈E(Φ)then by the same argument as (16), we have
cl ranHAΘ2∗⊆ker
I−TKT∗
K
. (22)
SinceAandBα are left coprime it follows thatAandBαare right coprime. Thus we can write
for some inner functionΩ andA1∈HM2n. It thus follows from Lemma1.1that cl ranHAΘ2∗=(kerHAΘ2∗)⊥=BαΩH C2n
⊥ ⊇BαHC2n ⊥ =HBα. Thus by (22), HBα ⊆ker I−TKT∗ K .
Then the exactly same argument as the argument from (18) to the end of the proof of Theorem3.5
withBα in place ofΘ2 shows that TΦ is normal. (We again note that we didn’t employ the
assumption “AandΘ2are left coprime” there.) This completes the proof. 2 We thus have:
Corollary 3.9.SupposeΦ=Φ−∗ +Φ+∈L∞M
n is a matrix-valued rational function. In view of
(5)and(2), we may write
Φ−=B∗Θ,
where Θ :=θ In with a finite Blaschke product θ. Assume that B(α) is invertible for each
α∈Z(θ ). IfTΦis subnormal thenTΦis normal or analytic.
Proof. This follows at once from Corollary3.8together with Lemma3.3. 2 4. Scalar Toeplitz operators with finite rank self-commutators
IfΦ is normal and analytic then[TΦ∗, TΦ] =HΦ∗∗HΦ∗, so that by the Kronecker’s lemma,
TΦ has a finite rank self-commutator if and only ifΦ is rational. Therefore Corollary3.9
il-lustrates the case of subnormal Toeplitz operators with finite rank self-commutators. But it is still open whether subnormal (even scalar-valued) Toeplitz operators with finite rank self-commutators are either normal or analytic. We would like to state:
Conjecture 4.1.IfTφis a subnormal Toeplitz operator with finite rank self-commutator, thenTφ
is normal or analytic.
We need not expect that if Tφ is a hyponormal Toeplitz operator with finite rank
self-commutator thenφ is of bounded type. Indeed, if ψ∈H∞ is such that ψ¯ is not of bounded type andφ=ψ+zψ (and henceφis not of bounded type) then a straightforward calculation shows thatTφis hyponormal and rank[Tφ∗, Tφ] =1.
We would like to take this opportunity to give a positive evidence for Conjecture4.1. First of all, we recall a theorem of Nakazi and Takahashi[27, Theorem 10]which states that ifTφis
hyponormal then[Tφ∗, Tφ]is of finite rank if and only if there exists a finite Blaschke product
binE(φ)such that the degree ofbequals the rank of[Tφ∗, Tφ]. In what follows we letbM:=
Theorem 4.2.SupposeTφ is a hyponormal Toeplitz operator with finite rank self-commutator.
Ifker[Tφ∗, Tφ]andbker[Tφ∗, Tφ](someb∈E(φ))are invariant underTφ, thenTφis normal or
analytic.
Proof. Write K:=ker[Tφ∗, Tφ] and R:=ran[Tφ∗, Tφ]. If φ or φ is of bounded type then by
Abrahamse’s theorem,Tφis either normal or analytic. Suppose bothφandφare not of bounded
type. We first claim that
clHφ
kerTφ∗, Tφ
=H2. (23)
To see this we observe that by the Nakazi–Takahashi theorem, there exists a finite Blaschke productb∈E(φ)such that deg(b)=dimR. Since
Tφ∗Tφ−TφTφ∗=Hφ∗Hφ−Hφ∗Hφ=Hφ∗Hφ−Hbφ∗ Hbφ=Hφ∗HbHb∗Hφ, we have kerTφ∗, Tφ =kerH∗ bHφ=ker(TφTb−TbTφ),
which shows that H˜
bHφ(K)=0, and henceHφ(K)⊆ ˜bH
2, so that clH
φK⊆ ˜bH2. But since
dimR <∞and by (1),Hφis one-one and has dense range, we have
H2=clHφ(K+R)=cl(HφK+HφR)=clHφK+HφR.
We therefore have clHφK= ˜bH2since dimHφR=deg(b). Hence clHφK=clHbφK=clTb˜∗HφK=Tb˜bH˜
2=H2,
which proves (23). On the other hand, we note thatE(φ)is a singleton set: otherwise, φis of bounded type. Thus E(φ)consists of only a finite Blaschke product b. We next argue that if
Tφ(bK)⊆bH2then
Tφ(bk)=bTφk for eachk∈K. (24)
To see this, letk∈Kand writek1:=Tφk. Thusφk=k1+k2for somek2∈H02=zH2. Then
Tφ(bk)=P (bφk)=P (bk2+bk1)=P (bk2)+bk1=P (bk2)+bTφk.
Since, by assumption, Tφ(bK)⊆bH2, it follows that P (bk2)∈bH2. But since P (bk2)∈
(bH2)⊥, we haveP (bk2)=0, which proves (24). SinceTφTb−TbTφ=Hb˜Hφit follows from
(23) and (24) that H˜ bH 2=H ˜ b(clHφK)=clHb˜HφK=cl(TφTb−TbTφ)K=0,
which implies thatbH˜ 2⊆H2, so thatb=eiθ for someθ∈ [0,2π ). Thereforeφis of the form
φ=f +eiθf for somef ∈H∞andθ∈ [0,2π )which implies thatTφis normal. 2
We thus have:
Corollary 4.3.SupposeTφis a subnormal Toeplitz operator with finite rank self-commutator. If
bker[Tφ∗, Tφ]is invariant underTφ(someb∈E(φ)), thenTφis normal or analytic.
Proof. Since ker[T∗, T]is invariant underT for every subnormal operatorT, the result follows at once from Theorem4.2. 2
We were not unable to decide whether the condition “bker[Tφ∗, Tφ](someb∈E(φ)) is
invari-ant underTφ” can be dropped from Corollary4.3: in other words, ifTφis a subnormal operator
with finite rank self-commutator andb∈E(φ), isbker[Tφ∗, Tφ]invariant underTφ? If the answer
to this question is affirmative we can conclude that Conjecture4.1is true.
Acknowledgment
The authors are deeply indebted to the referee for many helpful comments that helped im-proved the presentation and mathematical content of the paper.
References
[1] M.B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976) 597–604. [2] I. Amemiya, T. Ito, T.K. Wong, On quasinormal Toeplitz operators, Proc. Amer. Math. Soc. 50 (1975) 254–258. [3] A. Brown, P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964) 89–102. [4] C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367 (1986) 215–219.
[5] C. Cowen, Hyponormal and subnormal Toeplitz operators, in: J.B. Conway, B.B. Morrel (Eds.), Survey of Some Recent Results in Operator Theory, I, in: Pitman Res. Notes Math., vol. 171, Longman, 1988, pp. 155–167. [6] C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988) 809–812.
[7] C. Cowen, J. Long, Some subnormal Toeplitz operators, J. Reine Angew. Math. 351 (1984) 216–220.
[8] R.E. Curto, W.Y. Lee, Joint Hyponormality of Toeplitz Pairs, Mem. Amer. Math. Soc., vol. 712, Amer. Math. Soc., Providence, RI, 2001.
[9] R.E. Curto, W.Y. Lee, Subnormality andk-hyponormality of Toeplitz operators: A brief survey and open question, in: Operator Theory and Banach Algebras, Theta, Bucharest, 2003, pp. 73–81.
[10] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
[11] R.G. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators, CBMS Reg. Conf. Ser. Math., vol. 15, Amer. Math. Soc., Providence, RI, 1973.
[12] D.R. Farenick, W.Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996) 4153–4174.
[13] C. Foia¸s, A. Frazho, The Commutant Lifting Approach to Interpolation Problems, Oper. Theory Adv. Appl., vol. 44, Birkhäuser, Boston, 1993.
[14] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. II, Birkhäuser, Basel, 1993.
[15] C. Gu, A generalization of Cowen’s characterization of hyponormal Toeplitz operators, J. Funct. Anal. 124 (1994) 135–148.
[16] C. Gu, On a class of jointly hyponormal Toeplitz operators, Trans. Amer. Math. Soc. 354 (2002) 3275–3298. [17] C. Gu, J.E. Shapiro, Kernels of Hankel operators and hyponormality of Toeplitz operators, Math. Ann. 319 (2001)
553–572.
[18] C. Gu, J. Hendricks, D. Rutherford, Hyponormality of block Toeplitz operators, Pacific J. Math. 223 (2006) 95–111. [19] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970) 887–933.
[21] I.S. Hwang, W.Y. Lee, Hyponormality of trigonometric Toeplitz operators, Trans. Amer. Math. Soc. 354 (2002) 2461–2474.
[22] I.S. Hwang, W.Y. Lee, Hyponormality of Toeplitz operators with rational symbols, Math. Ann. 335 (2006) 405–414. [23] I.S. Hwang, W.Y. Lee, Hyponormal Toeplitz operators with rational symbols, J. Operator Theory 56 (2006) 47–58. [24] I.S. Hwang, I.H. Kim, W.Y. Lee, Hyponormality of Toeplitz operators with polynomial symbols, Math. Ann. 313 (2)
(1999) 247–261.
[25] I.S. Hwang, I.H. Kim, W.Y. Lee, Hyponormality of Toeplitz operators with polynomial symbols: An extremal case, Math. Nachr. 231 (2001) 25–38.
[26] W.Y. Lee, Cowen sets for Toeplitz operators with finite rank selfcommutators, J. Operator Theory 54 (2) (2005) 301–307.
[27] T. Nakazi, K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993) 753–769.
[28] N.K. Nikolskii, Treatise on the Shift Operator, Springer, New York, 1986.
[29] V.P. Potapov, On the multiplicative structure of J-nonexpansive matrix functions, Tr. Mosk. Mat. Obs. (1955) 125– 236 (in Russian); English trasl.: Amer. Math. Soc. Transl. Ser. 2 15 (1966) 131–243.
[30] D.V. Yakubovich, Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory, J. Funct. Anal. 236 (2006) 25–58.
[31] K. Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equations Operator Theory 21 (1996) 376–381.
