Science in China Series A: Mathematics Mar., 2007, Vol. 50, No. 3, 387–411 www.SciChina.com www.springerlink.com

Essentially normal Hilbert modules and Khomology III: Homogenous quotient modules of Hardy modules on the bidisk Kun-yu GUO† & Peng-hui WANG School of Mathematics, Fudan University, Shanghai 200433, China (email: [email protected], [email protected])

Abstract

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers Khomology invariant defined in the case of the homogenous quotient modules on the bidisk. Keywords: MSC(2000):

1

Hardy modules, bidisk, essential normality, K-homology. 47A13, 47A20, 46H25, 46C99

Introduction

Let T = (T1 , . . . , Tn ) be a tuple of commuting operators acting on a Hilbert space H. Using Douglas-Paulsen’s Hilbert module language[1], we endow H with a C[z1 , . . . , zn ]-module structure by p · x = p(T1 , . . . , Tn )x, p ∈ C[z1 , . . . , zn ], x ∈ H. In Arveson’s language[2−4], a Hilbert module is called essentially normal if the self-commutators Tk∗ Tj −Tj Tk∗ of its canonical operators are all compact (one can also refer to such Hilbert modules as “essentially reductive”, see [1, 5, 6]). Much work has been done along this line[1−11] . In the course of Arveson’s studies[2−4] , he considered the essential normality of quotients of standard Hilbert modules. For a d-shift Hilbert module with finite-multiplicity, he established a p-essential normality in case the submodule is generated by monomials[2] . That result on “monomial” submodules was generalized by Douglas to the case in which the d-shift is replaced by more general weighted shifts[5] . In the dimension d = 2, Guo[7] obtained trace estimates, which implies that the homogeneous quotient modules of 2-shift Hilbert module are essentially normal. In the recent work of Guo and Wang[8] , these results were generalized to cases in which the 2-shift is replaced by U-invariant Hilbert modules with finite-multiplicity over 2-dimensional unit ball. In [9], the essential normality and the a K-homology of quasi-homogeneous quotient modules over a 2-dimensional unit ball were characterized by a hard analysis, and the method Received July 3, 2006; accepted November 21, 2006 DOI: 10.1007/s11425-007-0019-2 † Corresponding author This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program

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is different from the case of homogeneous quotient modules. The paper will be devoted to studying the essential normality of quotient modules of the Hardy module over the bidisk. At this point it is completely different from the Hilbert modules on the unit ball. It is well known that neither the Hardy module H 2 (D2 ) nor its nonzero submodules are essentially normal. Let H 2 (D2 ) be the Hardy module on the bidisk with the module action defined by multiplication of coordinate functions. The structure of submodules of H 2 (D2 ) is much more complex than that of submodules of H 2 (D), and for more work on this subject, see [1, 6, 12–19] and references therein. Let M be a submodule of H 2 (D2 ), set N = H 2 (D2 )  M . Then N can be endowed with a C[z, w]-module structure by p · f = p(Sz , Sw )f, p ∈ C[z, w], f ∈ N, where Sz = PN Mz |N and Sw = PN Mw |N . Some efforts have been done to describe the structure of some certain quotient modules over the bidisk, see [20–23]. For any nonzero submodule M of H 2 (D2 ), using the fact that the coordinate functions define a pair of isometries, both of infinite multiplicity, M is not essentially normal. However, in [24], Douglas and Misra showed that some quotient modules of H 2 (D2 ) are essentially normal and some are not. Some other related work has been done in [11, 25]. If a quotient module N of H 2 (D2 ) is essentially normal, then the C ∗ -algebra C ∗ (N ) generated by {Sz , Sw } is essentially commutative. A result of Yang[18,Theorem 4.4] implies that for any quotient module N of H 2 (D2 ), the C ∗ -algebra C ∗ (N ) is irreducible. Hence, we have a C ∗ extension 0 → K → C ∗ (N )→C(σe (Sz , Sw ))→0. This C ∗ -extension gives an element of K1 (σe (Sz , Sw )), which is an invariant for the Hilbert modules[1,26,27] . In this paper, we mainly concern the homogenous quotient modules. Let I be an ideal of C[z, w], and [I] be the submodule of H 2 (D2 ) generated by I. If I is homogeneous, then the submodule [I] also is homogeneous, and in this case, the quotient module H 2 (D2 )/[I] is homogenous. Since the polynomial ring is Noetherian[28] , the ideal I is generated by finitely many polynomials. This implies that I has a greatest common divisor p, and so, I can be uniquely written as I = pL, which is called the Beurling form of I (cf. [29]). In dimension d = 2, by a lemma of Yang[15] , dim C[z, w]/L < ∞ and hence [p]  [I] is of finite dimension, where [p] is the submodule of H 2 (D2 ) generated by p. This means that H 2 (D2 )/[p] and H 2 (D2 )/[I] have the same essential normality. Notice if I is homogeneous, then both p and L are homogeneous. This paper is organized as follows. In sec. 2, we will consider the compactness of the commutators [PM , Mz ] and [PM , Mw ] for homogenous submodules M . For a homogenous submodule M = [p], both [PM , Mz ] and [PM , Mw ] are compact if and only if Z(p)∩∂D2 ⊂ T2 . In sec. 3, we will develop some properties of the asymptotic orthogonality. It plays an important role in the paper. Sec. 4 and sec. 5 are devoted to characterizing the essential normality of homogenous quotient modules of H 2 (D2 ). Let p be a nonzero homogenous polynomial in C[z, w], then p

Homogenous quotient modules of Hardy modules on the bidisk

 can be factorized as p(z, w) = (αi z − βi w)ni . Set  p1 (z, w) = (αi z − βi w)ni , p2 (z, w) = |αi |=|βi |

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(αi z − βi w)ni .

|αi |=|βi |

Then p = p1 p2 ,

(1.1)

with Z(p1 ) ∩ ∂D2 ⊂ T2 and Z(p2 ) ∩ ∂D2 ⊂ (D × T) ∪ (T × D). We state our main theorem as follows: Theorem 1.1. Let p be a nonzero homogenous polynomial in C[z, w], and p = p1 p2 as in (1.1), then the quotient module H 2 (D2 )/[p] is essentially normal if and only if p2 has one of the following forms : (1) p2 = c, with c = 0; (2) p2 = αz + βw, with |α| = |β|; (3) p2 = c(z − αw)(w − βz), with |α| < 1, |β| < 1 and c = 0. In sec. 6, we will describe the essential spectrum of quotient modules. For any quotient module N of H 2 (D2 ), the essential spectrum σe (N ) is defined by σe (Sz , Sw ). It is shown that for a homogenous polynomial p, σe (H 2 (D2 )/[p]) = Z(p) ∩ ∂D2 . Therefore, if a homogenous quotient module H 2 (D2 )/[p] is essentially normal, then we have a C ∗ -extension 0 → K → C ∗ (Sz , Sw )→C(Z(p) ∩ ∂D2 ) → 0. It is shown that this extension yields a nontrivial K-homology element in K1 (Z(p) ∩ ∂D2 ). Moreover, we also describe the essential spectrum of a general quotient module. Using the result in this paper, one can construct a quotient module N of H 2 (D2 ), such that σe (N ) = ∂D2 . 2

Compactness of [PM , Mz ] and [PM , Mw ]

Let M be a submodule of H 2 (D2 ), and N = H 2 (D2 )M be the corresponding quotient module. ∗ The quotient module N is called essentially normal, if both [Sz , Sz∗ ] and [Sw , Sw ] are compact, where for two operators A and B, the operator [A, B] = AB − BA is the commutator of A and B. If both Sz and Sw are essentially normal, then by Fuglede-Putnam Theorem, the ∗ ] is compact. Since N is an invariant subspace of Mz∗ and Mw∗ , the quotient commutator [Sz , Sw module N is essentially normal if and only if both Mz∗ |N and Mw∗ |N are essentially normal. In [24], Douglas and Misra established by a direct calculation that the quotient module 2 H (D2 ) [(z − w)2 ] is essentially normal and H 2 (D2 )  [z 2 ] is not. They also showed that H 2 (D2 )  [z n − wm ] is essentially normal, and this result was generalized by Clark in [25]. Recently, Izuchi and Yang[11] showed that if ϕ(w) ∈ H 2 (D) is an inner function, then H 2 (D2 )  [z − ϕ(w)] is essentially normal if and only if ϕ is a finite Blaschke product. In this section, the homogenous quotient modules H 2 (D2 )/[p] with Z(p) ∩ ∂D2 ⊂ T2 will be concerned. Let p be a homogenous polynomial, and M = [p] be the submodule of H 2 (D2 ) generated by p. We will prove that both [PM , Mz ] and [PM , Mw ] are compact if and only if Z(p) ∩ ∂D2 ⊂ T2 , where PM denotes the orthogonal projection from H 2 (D2 ) onto M .

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As a corollary, we will prove that if Z(p) ∩ ∂D2 ⊂ T2 , then the quotient module H 2 (D2 )/[p] is essentially normal. At first, we need some results on the evaluation operator and the difference quotient operator, which are studied by Yang[16,17,19] . The evaluation operators L(0) and R(0) at z = 0 and w = 0 respectively are defined by L(0)f (z, w) = f (0, w), R(0)f (z, w) = f (z, 0), f ∈ H 2 (D2 ). In fact, L(0) = 1 − Mz Mz∗ and R(0) = 1 − Mw Mw∗ are projections. Let M be a submodule of H 2 (D2 ), and N = H 2 (D2 )  M be the corresponding quotient module. Write Dz = PM Mz |N , Dw = PM Mw |N , then we can rewrite Dz = PM Mz (1 − PM ) = [PM , Mz ],

Dw = PM Mw (1 − PM ) = [PM , Mw ].

∗ are given by the difference quotient operators in [19], The operators Dz∗ and Dw

f (z, w) − f (0, w) f (z, w) − f (z, 0) ∗ , Dw . f (z, w) = z w The following proposition comes from [16], which gives the relationship between the evaluation operators and the difference quotient operators. Dz∗ f (z, w) =

Proposition 2.1 If M is a submodule of H 2 (D2 ), let N = H 2 (D2 )  M be the corresponding quotient module, then ∗ ∗ (1) Sz∗ Sz + Dz∗ Dz = IN , Sw Sw + D w Dw = IN ; ∗ ∗ (2) Sz Sz + PN L(0)PN |N = IN , Sw Sw + PN R(0)PN |N = IN . Let M be a submodule of H 2 (D2 ), and N = H 2 (D2 )  M . Proposition 2.1 implies that, if both [PM , Mz ] and L(0)PN are compact, then Sz is essentially normal. Similarly, if both [PM , Mw ] and R(0)PN are compact, then Sw is essentially normal. We have the following Lemma 2.2

Let p be a homogeneous polynomial with the factorization p(z, w) = λ

n 

(w − αi z),

with αi = 0.

i=1

Set M = [p] and N = H 2 (D2 )  M , then [PM , Mz ] is compact if and only if L(0)PN is compact. Similarly, the commutator [PM , Mw ] is compact if and only if R(0)PN is compact. Proof. p as

Firstly, we will prove that both ker Sz and ker Sz∗ are of finite dimension. We rewrite p(z, w) = λ

n  (w − αi z) = λwn + zp1 (z, w). i=1

Notice that ker Sz∗ = ker Mz∗ ∩ [p]⊥ = Hw2 ∩ [p]⊥ , ∞ where Hw2 = span{wn , n = 0, 1, 2, . . .}. If f ∈ ker Sz∗ , then f = k=0 ak wk and f ∈ [p]⊥ . For any m  0, ¯ m+n + M ∗ f, wm p1  = f, λwm+n  + f, zwm p1  = f, wm p = 0. ¯ m+n = λa λa z

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It follows that dim(ker Sz∗ )  n + 1. Now we will show that ker Sz = {0}. Notice that ker Sz = {f | f ∈ [p]⊥ , zf ∈ [p]}.

(2.1)

∞ For any f ∈ ker Sz , let f = i=0 Fi (z, w) be the homogeneous expansion of f . Since p is homogeneous and zf ∈ [p], zFi ∈ [p]. By the fact that Fi is a polynomial, it follows that there is a polynomial q such that zFi = pq. Noting that GCD(z, p) = 1, this means that q/z is a polynomial, and hence Fi = pq/z ∈ [p]. So f ∈ [p]. By (2.1), f = 0. Therefore ker Sz = 0. By Proposition 2.1, if one of [PM , Mz ] and L(0)PN is compact, then Sz is Fredholm, and hence Sz∗ is the essential inverse of Sz . This insures that the other of [PM , Mz ] and L(0)PN is compact. The same reason gives the case of [PM , Mw ] and R(0)PN . We state the main theorem in this section as follows. Theorem 2.3. Let p be a homogenous polynomial in C[z, w]. Set M = [p], then both [PM , Mz ] and [PM , Mw ] are compact if and only if Z(p) ∩ ∂D2 ⊂ T2 . Combining Theorem 2.3, Lemma 2.2 with Proposition 2.1, we have the next corollary. Corollary 2.4. Let p be a homogenous polynomial in C[z, w] satisfying Z(p) ∩ ∂D2 ⊂ T2 , then the quotient module H 2 (D2 )  [p] is essentially normal. Proof of the necessity of Theorem 2.3. We will prove that if Z(p) ∩ ∂D2  T2 , then at least one of [PM , Mz ] and [PM , Mw ] is not compact. As mentioned in the introduction, for any homogenous polynomial p, p can be factorized as p = λz k wl

n 

(w − αi z),

with αi = 0.

i=1

Below, the necessity of Theorem 2.3 will be proved in two cases. Case 1. Both k and l are zero. Since Z(p) ∩ ∂D2  T2 , there is an i0 such that |αi0 | = 1. Set β = αi0 . Suppose now that 0 < |β| < 1. We rewrite p as p = (z − βw)p , where p is a homogenous polynomial. Set N1 = [z − βw]⊥ and N2 = [z − βw]  [p], then M ⊥ = N1 ⊕ N2 . ¯ j βz) ¯ j−1 w+···+w j ∞ √ +( Since {1} ∪ {ej = (βz) }j=1 is an orthonormal basis of N1 , 2j 2j−2 |β|

+|β|

+···+1

1 L(0)PN1 ej =  wj . 2j 2j−2 |β| + |β| + ··· + 1 Then limj→∞ L(0)PN1 ej  = 1 − |β|2 = 0. It follows that L(0)PN1 is not compact, and hence L(0)PM ⊥ is not compact. Lemma 2.2 implies that [PM , Mz ] is compact if and only if L(0)PM ⊥ is compact. Thus [PM , Mz ] is not compact. The same reason implies that if |β| > 1, then [PM , Mw ] is not compact. Case 2. At least one of k and l is not zero. We suppose now k = 0, then p = zp , where p is a homogenous polynomial. It follows that M ⊥ = [z]⊥ ⊕ ([z]  M ), ⊥ and hence {wj }∞ j=1 ⊂ M . Let m = deg p, then p = claim that PM Mzi0 |M ⊥ is not compact.

m

i=0

ai z i wm−i . Now taking ai0 = 0, we

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Since p is homogenous, {fj = wj p/p}∞ j=1 is an orthonormal set of M . By Bessel’s inequality, for sufficiently large j, PM Mzi0 |M ⊥ wj 2 = PM z i0 wj 2 ∞   2   

z i0 wj , fi fi  i=1

= | z i0 wj , fj−m+i0 |2 = |ai0 |2 /p2 = 0. Since wj converges to zero weakly, [PM , Mzi0 ] = PM Mzi0 |M ⊥ is not compact. Since {PM }e = {A ∈ B(H 2 (D2 )) | [PM , A] is compact} is a C ∗ -algebra, the compactness of [PM , Mz ] implies that [PM , Mzi0 ] is compact. And hence [PM , Mz ] is not compact. The same reason implies that if l = 0, then [PM , Mw ] is not compact, thus completing the proof. The proof of the sufficiency of Theorem 2.3 is long. The remaining part of this section will be devoted to the proof of the sufficiency of Theorem 2.3. Let Hn = {fn ∈ C[z, w] | fn is homogenous with degree n}, then H 2 (D2 ) can be decomposed as H 2 (D2 ) = ⊕∞ n=0 Hn . It is easy to see that dim Hn = n + 1. Given a homogenous polynomial p of degree m, then for sufficiently large n, dim([p] ∩ Hn ) = n − m + 1, and hence dim([p]⊥ ∩ Hn ) = m. To prove the sufficiency of Theorem 2.3, we need the next lemma. In the case n = 1, the next lemma is considered in [16]. Lemma 2.5. compact. Proof.

For n  1, let M0 = [(z − w)n ], then both [PM0 , Mz ] and [PM0 , Mw ] are

Set N0 = H 2 (D2 )  M0 . For i = 1, 2, . . . , n, set Ni = [(z − w)i−1 ]  [(z − w)i ],

then N0 = N1 ⊕ · · · ⊕ Nn . Let Nik = Ni ∩ Hk , k = 0, 1, 2, . . ., then Ni has a homogeneous decomposition Ni =

∞

Nik .

k=0

It is easy to see that for sufficiently large k, Nik is of dimension 1. Let eki (z, w) be in Nik with eki  = 1. Define a unitary operator V : H 2 (D2 ) → H 2 (D2 ) by V f (z, w) = f (w, z), f ∈ H 2 (D2 ). Obviously, every Ni is a reduced subspace of V , and hence every Nik is a reduced subspace of (k) (k) V . It follows that there are complex numbers λi with |λi | = 1, such that (k)

eki (w, z) = V eki (z, w) = λi eki (z, w).

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This implies that (k)

| z k , eki | = | V z k , V eki | = | wk , λi eki | = | wk , eki |.

(2.2)

Below, we will prove that L(0)PNi L(0) is compact for i = 1, . . . , n, by induction. Since k k−1 w+···+w k {1, z +z √k+1 , k = 1, 2, . . .} is an orthonormal basis of N1 , L(0)PN1 L(0)wk =

wk , k+1

and this insures that L(0)PN1 L(0) is compact. Suppose now that L(0)PNi L(0) is compact for all i < m. Since for sufficiently large k, L(0)PNi L(0)wk = L(0)PNik L(0)wk = wk , eki  eki , wk wk , i = 1, . . . , m − 1, we have limk→∞ | wk , eki |2 = 0 for i = 1, . . . , m − 1, and hence by (2.2)  m−1  m−1   k k k k  lim 

z , ei  ei , w   lim | wk , eki |2 = 0. k→∞ k→∞ i=1

(2.3)

i=1

By [16, Lemma 4.1.2], for any quotient module N , the operator L(0)PN R(0) is Hilbert-Schmidt. m Now take N = H 2 (D2 )  [(z − w)m ], then N = i=1 Ni . Since PNi z k = PNik z k = z k , eki eki , L(0)PN R(0)z k =

m 

L(0)PNi R(0)z k =

i=1

m 

z k , eki  eki , wk wk .

i=1

Let L(0)PN R(0)H.S denote the Hilbert-Schmidt norm of L(0)PN R(0), then  ∞  ∞    m k k k k 2  

z , ei  ei , w  = L(0)PN R(0)z k 2  L(0)PN R(0)2H.s < +∞.  k=m

i=1

k=m

Thus lim

k→∞

m 

z k , eki  eki , wk  = 0.

(2.4)

i=1

By (2.3) and (2.4), we have lim z k , ekm  ekm , wk  = 0.

k→∞

It follows that L(0)PNm L(0)wk  = | wk , ekm  ekm , wk | = | z k , ekm  ekm , wk | → 0. So L(0)PNm L(0) is compact, and hence for 1  i  n, L(0)PNi L(0) is compact. This insures that

  n L(0)PN0 L(0) = L(0) PNi L(0) i=1

is compact, and hence L(0)PN0 is compact. By Lemma 2.2, [PM0 , Mz ] is compact. The same reason implies that [PM0 , Mw ] is compact. Let pi ∈ C[z, w], i = 1, 2, setting r = GCD(p1 , p2 ), the greatest common divisor of p1 and p2 , then p1 = rp1 and p2 = rp2 . If Z(r) ∩ D2 = ∅, then by [12, Proposition 2.2.13], [r] = H 2 (D2 ), and hence [pi ] = [pi ]. To prove the sufficiency of Theorem 2.3, we still need several lemmas.

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Lemma 2.6.

Given p1 , p2 ∈ C[z, w], if Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅, then [p1 ]⊥ + [p2 ]⊥ = [p1 p2 ]⊥ .

Proof.

At first, we claim that [p1 ]⊥ + [p2 ]⊥ is closed. Since Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅,

as mentioned above, we may assume GCD(p1 , p2 ) = 1. By [15, Lemma 6.1], it follows that [p1 ]⊥ ∩ [p2 ]⊥ = ([p1 ] + [p2 ])⊥ is of finite dimension. Let M = [p1 ]⊥  ([p1 ]⊥ ∩ [p2 ]⊥ ) and N = [p2 ]⊥  ([p1 ]⊥ ∩ [p2 ]⊥ ), then [p1 ]⊥ + [p2 ]⊥ is closed if and only if M + N is closed. Now suppose that M + N is not ∞ closed, then we can take {fn | fn  = 1}∞ n=1 ⊂ M and {gn | gn  = 1}n=1 ⊂ N , such that both fn and gn converge weakly to 0, and fn + gn  tends to 0 as n → ∞. Since both Mp∗1 fn = 0 and Mp∗2 gn = 0, we have lim ( (Mp1 Mp∗1 + Mp2 Mp∗2 )fn , fn  + (Mp1 Mp∗1 + Mp2 Mp∗2 )gn , gn )

n→∞

= lim ( Mp2 Mp∗2 fn , fn  + Mp1 Mp∗1 gn , gn  + Mp1 Mp∗1 fn , fn + gn  n→∞

+ Mp1 Mp∗1 gn , fn  + Mp2 Mp∗2 gn , fn + gn  + Mp2 Mp∗2 fn , gn )

= lim (Mp1 Mp∗1 + Mp2 Mp∗2 )(fn + gn ), fn + gn  n→∞

 lim Mp1 Mp∗1 + Mp2 Mp∗2 fn + gn 2 n→∞

= 0. Since Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅, ker(Mp1 Mp∗1 + Mp2 M2∗ ) = [p1 ]⊥ ∩ [p2 ]⊥ is of finite dimension. By [30, Theorem 4.3] and [30, Lemma 4.5], the operator Mp1 Mp∗1 + Mp2 Mp∗2 has a closed range, and hence it is a positive Fredholm operator. This implies that there exist a positive invertible operator A and a compact operator K, such that Mp1 Mp∗1 + Mp2 Mp∗2 = A + K. Since both fn and gn converge weakly to zero as n → ∞, we have lim Kfn  = 0

n→∞

and

lim Kgn = 0.

n→∞

Since A is positive and invertible, there are some ci > 0, i = 1, 2, such that lim (Mp1 Mp∗1 + Mp2 Mp∗2 )fn , fn  = lim (A + K)fn , fn   lim c1 fn 2 = c1 ,

n→∞

n→∞

n→∞

and lim (Mp1 Mp∗1 + Mp2 Mp∗2 )gn , gn  = lim (A + K)gn , gn   lim c2 gn 2 = c2 .

n→∞

n→∞

n→∞

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This contradiction implies that M + N is closed, and hence [p1 ]⊥ + [p2 ]⊥ is closed, the claim is proved. Using the characteristic space theory for polynomials in [12, Chapter 2], we have [p1 ] ∩ [p2 ] = [p1 p2 ], and hence [p1 ]⊥ + [p2 ]⊥ = ([p1 ] ∩ [p2 ])⊥ = [p1 p2 ]⊥ , thus completing the proof. Lemma 2.7. Let pi ∈ C[z, w], i = 1, 2, such that Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅. Write Pi = P[pi ] and set M = [p1 p2 ]. If both [Pi , Mz ] and [Pi , Mw ] are compact for i = 1, 2, then so are [PM , Mz ] and [PM , Mw ]. Proof. Since both [Pi , Mz ] and [Pi , Mw ] are compact for i = 1, 2, a simple reason shows that for any polynomial p, the commutator [Pi , Mp ] are compact for i = 1, 2. Since Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅, by Lemma 2.6, M ⊥ = [p1 ]⊥ + [p2 ]⊥ . It is easy to check that ∗ Mp1 M ⊥ ⊂ [p2 ]⊥ , and hence Mp∗1 PM ⊥ = P2⊥ Mp∗1 PM ⊥ , where P2⊥ = I − P2 . Moreover, notice that M = [p1 p2 ] ⊂ [p2 ], then for any polynomial p, we have PM Mp Mp1 Mp∗1 PM ⊥ = PM (P2 Mp Mp1 P2⊥ )Mp∗1 PM ⊥ = PM [P2 , Mpp1 ]Mp∗1 PM ⊥ . This implies that the operator PM Mp Mp1 Mp∗1 PM ⊥ is compact. Similarly, the operator PM Mp Mp2 Mp∗2 PM ⊥ is compact. Therefore, the operator PM Mp (Mp1 Mp∗1 + Mp2 Mp∗2 )PM ⊥ is compact. Since Z(p1 ) ∩ Z(p2 ) ∩ ∂D2 = ∅, the same argument as in the proof of Lemma 2.6 implies that the operator Mp1 Mp∗1 + Mp2 Mp∗2 is a positive Fredholm operator, and hence there exist a finite rank operator F and a positive invertible operator X, such that Mp1 Mp∗1 + Mp2 Mp∗2 = X + F . Hence PM Mp XPM ⊥ = PM Mp (Mp1 Mp∗1 + Mp2 Mp∗2 )PM ⊥ − PM Mp F PM ⊥ (2.5) is compact. Considering the matrix representation of X, ⎞ ⎛ M X11 X12 ⎠ , X=⎝ ∗ X12 X22 M⊥ and taking p = 1 in the left side of (2.5), we have that X12 = PM XPM ⊥ is compact. Hence X22 is Fredholm. Noting that M is an invariant subspace, the operator Mp has a matrix representation  M , and hence Sp D p M⊥ 0 Rp ⎞ ⎛ ∗ Sp X12 + Dp X22 M Sp X11 + Dp X12 ⎠ Mp X = ⎝ . ∗ Rp X12 Rp X22 M⊥

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By (2.5) again, Sp X12 + Dp X22 is compact. Since X12 is compact, the operator Dp X22 is compact. The Fredholmness of X22 implies that Dp is compact. Now taking p = z and p = w, we have that both Dz and Dw are compact, that is, both [PM , Mz ] and [PM , Mw ] are compact, as desired. Proof of the sufficiency of Theorem 2.3. The hypothesis Z(p) ∩ ∂D2 ⊂ T2 implies that p can be factorized as m  p(z, w) = λ (z − αi w)ni , with |αi | = 1 and λ = 0. i=1

Without loss of generality, we may assume λ = 1. Below, we will prove the theorem by induction. When m = 1 and |α| = 1, define a unitary operator U : H 2 (D2 ) → H 2 (D2 ) by U f (z, w) = f (z, αw), for f ∈ H 2 (D2 ), then Mz U = U Mz ,

αMw U = U Mw .

Let M0 = [(z − w)n ] and M = [(z − αw)n ], then PM0 = U ∗ PM U,

PM0⊥ = U ∗ PM ⊥ U.

Notice that PM Mz PM ⊥ = U PM0 U ∗ Mz U PM0⊥ U ∗ = U PM0 Mz PM0⊥ U ∗ = U [PM0 , Mz ]U ∗ . Lemma 2.5 implies that [PM0 , Mz ] is compact, and hence [PM , Mz ] = PM Mz PM ⊥ is compact. Similarly, [PM , Mw ] is compact. Now we suppose that the theorem holds for m = k, that is, for pk (z, w) =

k 

(z − αi w)ni ,

with |αi | = 1

i=1

both [Pk , Mz ] and [Pk , Mw ] are compact, where Pk = P[pk ] . For any |β| = 1 and β = αi , i = 1, . . . , k, setting pk+1 = (z − βw)nk+1 pk , write Pk+1 = P[pk+1 ] . It suffices to show that both [Pk+1 , Mz ] and [Pk+1 , Mw ] are compact. Since β = αi , Z(z − βw) ∩ Z(pk ) = {0}, and hence by Lemma 2.7, both [Pk+1 , Mz ] and [Pk+1 , Mw ] are compact. This completes the proof. 3

Asymptotic orthogonality

In this section, we will develop some properties of the asymptotic orthogonality for closed subspaces. It plays an important role in this paper.

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Let H be a Hilbert space, N1 and N2 be two closed subspaces of H. The subspaces N1 and N2 are called asymptotically orthogonal, denoted by N1  N2 if PN1 PN2 is compact. Proposition 3.1. Let H be a Hilbert space, N1 and N2 be two closed subspaces of H. If N1  N2 , then N1 + N2 is closed. Proof. Suppose N1 + N2 is not closed. Then there are {fn | fn  = 1}∞ n=1 ⊂ N1 and w w ∞ {gn | gn  = 1}n=1 ⊂ N2 satisfying fn → 0 and gn → 0, such that fn + gn  tends to zero as n → ∞. Since PN2 PN1 is compact, and since fn converges weakly to zero, lim PN2 fn  = lim PN2 PN1 fn  = 0.

n→∞

n→∞

Similarly, lim PN1 gn  = 0. Now, on the one hand, n→∞

lim | (PN2 + PN1 )(fn + gn ), (fn + gn )|  lim PN2 + PN1 fn + gn 2 = 0,

n→∞

n→∞

on the other hand, lim | (PN2 + PN1 )(fn + gn ), (fn + gn )|

n→∞

= lim [ PN1 fn , fn  + PN2 gn , gn  + PN1 gn , fn + gn  n→∞

+ PN2 fn , fn + gn  + PN1 fn , gn  + PN2 gn , fn ]

= lim (fn 2 + gn 2 ) = 2. n→∞

This contradiction shows that N1 + N2 is closed. The closeness of N1 + N2 is not enough to insure N1  N2 . The following proposition helps understanding the asymptotic orthogonality. Proposition 3.2. Let H be a Hilbert space, N1 and N2 be two closed subspaces of H. Then N1 and N2 are asymptotically orthogonal if and only if N1 + N2 is closed and PN1 +N2 − (PN1 + PN2 ) is compact. Proof.

Suppose first that N1 + N2 is closed and PN1 +N2 − (PN1 + PN2 ) is compact. Since PN1 PN2 + PN2 PN1 = (PN1 + PN2 )2 − (PN1 + PN2 ) = (PN1 + PN2 )2 − PN1 +N2 + PN1 +N2 − (PN1 + PN2 ) = [(PN1 + PN2 − PN1 +N2 )(PN1 + PN2 + PN1 +N2 )] +[PN1 +N2 − (PN1 + PN2 )],

we have that PN1 PN2 + PN2 PN1 is compact. It follows that PN1 PN2 PN1 PN2 =

1 PN (PN1 PN2 + PN2 PN1 )PN1 PN2 2 1

is compact. Since (PN2 PN1 PN2 )∗ (PN2 PN1 PN2 ) = PN2 PN1 PN2 PN1 PN2 , and hence PN2 PN1 PN2 is compact. From the equality PN2 PN1 PN2 = (PN1 PN2 )∗ (PN1 PN2 ),

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we see that PN1 PN2 is compact. Suppose now that PN1 PN2 is compact. By Proposition 3.1, N1 + N2 is closed. Notice that the compactness of PN1 PN2 implies N1 ∩ N2 being of finite dimension, and hence we can assume without loss of generality that N1 ∩ N2 = {0}. Now, we claim that Q = PN1 + PN2 , viewed as an operator on N1 + N2 , is Fredholm. At first, for f1 ∈ N1 , f2 ∈ N2 , if Q(f1 + f2 ) = 0, then by N1 ∩ N2 = {0}, f1 + PN1 f2 = 0 and f2 + PN2 f1 = 0, and hence f2 = PN2 PN1 f2 . Since N1 ∩ N2 = {0}, we have f2 = 0. Thus f1 = 0. Therefore ker Q = {0}. To prove that Q is Fredholm, it suffices to show that Q has a closed range. Let N1  N2 = {(f1 , f2 ) | fi ∈ Ni , (f1 , f2 ) = f1  + f2 }, then N1 N2 is a Banach space. Define an operator T : N1 N2 → N1 +N2 by T (f1 , f2 ) = f1 +f2 , then T is bounded and invertible. It follows that there are c0 > 0 and c1 > 0 such that c0 f1 + f2   f1  + f2   c1 f1 + f2 , for f1 ∈ N1 , f2 ∈ N2 . Notice that PN1 PN2 PN1 can be regarded as an operator on N1 , and it is compact. By the Fredholm Alternative Theorem[31,Theorem 5.22] , for any ε > 0, there is a finitely codimensional subspace N1ε of N1 such that, for any f ∈ N1ε , 1

PN2 f  = (PN1 PN2 PN1 ) 2 f  < εf . Similarly, there is a finitely codimensional subspace N2ε of N2 , such that for any f ∈ N2ε , 1

PN1 f  = (PN2 PN1 PN2 ) 2 f  < εf . Hence for f1 ∈ N1ε and f2 ∈ N2ε , (PN1 + PN2 )(f1 + f2 )  f1  + f2  − (PN1 f2  + PN2 f1 ) 1

1

= f1  + f2  − ((PN2 PN1 PN2 ) 2 f2  + (PN1 PN2 PN1 ) 2 f1 )  f1  + f2  − ε(f1  + f2 )  c0 f1 + f2  − εc1 f1 + f2  = (c0 − εc1 )f1 + f2 . We can take ε to be enough small, such that c0 − εc1 > 0. Now, fix ε, since Niε is a finitely codimensional subspace of Ni , the space N ε = N1ε + N2ε is a finitely codimensional subspace of N1 + N2 . The above reason implies that Q|N ε is bounded below, and hence Q has a closed range. Therefore, the operator Q is Fredholm, and this means that 0 ∈ / σe (Q). Now, since PN1 PN2 is compact, Q2 − Q = (PN1 + PN2 )2 − (PN1 + PN2 ) = PN1 PN2 + PN2 PN1 is compact, and hence σe (Q) ⊂ {0, 1}. It follows that σe (Q) = {1}. Since Q is self-adjoint, there is an exact sequence, π 0 → K → C ∗ (Q) + K → C → 0,

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where K is the ideal of all compact operators on N1 +N2 and C ∗ (Q) is the C ∗ -algebra generated by Q and the identity operator I on N1 + N2 . Since Q is Fredholm, π(Q) = 0. Notice that π(Q)2 = π(Q), and we have π(Q) = 1. It follows that I − Q is compact. This insures that PN1 +N2 − (PN1 + PN2 ) is compact, as desired. Let H be an infinitely dimensional Hilbert space, T ∈ B(H). Suppose N1 and N2 are invariant subspaces of T satisfying N1  N2 , by Proposition 3.1, N = N1 + N2 is closed. It is easy to see that N is an invariant subspace of T . Let T1 = T |N1 and T2 = T |N2 be the restrictions of T to N1 and N2 respectively. The following theorem maybe is familiar to some readers, we sketch it here for convenience. Theorem 3.3 Under the above assumption, if both T1 and T2 are essentially normal, then T |N , the restriction of T to N , is essentially normal. Proof. At first, we consider the matrix of T |N relative to the decomposition N = N1 ⊕(N N1) to obtain ⎛ ⎞ PN1 T PN1 PN1 T PN N1 ⎠. T |N = ⎝ PN N1 T PN1 PN N1 T PN N1 Since N1 is an invariant subspace of T , we have T PN1 = PN1 T PN1 , and hence PN N1 T PN1 = PN N1 PN1 T PN1 = 0. Since N2 is also an invariant subspace of T , we have T PN2 = PN2 T PN2 , and hence PN1 T PN N1 = PN1 T (PN N1 − PN2 ) + PN1 T PN2 = PN1 T (PN N1 − PN2 ) + PN1 PN2 T PN2 . Since N1  N2 , by Proposition 3.2, both PN N1 − PN2 and PN1 PN2 are compact, and hence PN1 T PN N1 is compact. A simple reason implies that T |N is essentially normal if and only if both PN1 T PN1 and PN N1 T PN N1 are essentially normal. Since T1 is essentially normal, PN1 T PN1 = T1 is essentially normal. Now noticing that PN N1 − PN2 is compact, we have PN N1 T PN N1 = PN2 T PN2 + a compact operator. Since PN2 T PN2 = T2 is essentially normal, this implies that PN N1 T PN N1 is essentially normal, and hence T |N is essentially normal, thus completing the proof. Now, let us turn to quotient modules of H 2 (D2 ). Let M1 , M2 be two submodules of H 2 (D2 ), and N1 = H 2 (D2 )M1 , N2 = H 2 (D2 )M2 be the corresponding quotient modules. If N1 N2 , then by Proposition 3.1, N = N1 + N2 is closed, and hence N = H 2 (D2 )  (M1 ∩ M2 ). As an immediate application of Theorem 3.3, we have Corollary 3.4. Under the above assumption, if both N1 and N2 are essentially normal, then N is essentially normal. 4

The sufficiency of Theorem 1.1

In this section, we use some results of the asymptotic orthogonality to prove the sufficiency of Theorem 1.1. To this end, we still need several facts. Proposition 4.1. Let p1 and p2 be two homogenous polynomials in C[z, w], setting N1 = 2 2 H (D )  [p1 ] and N2 = H 2 (D2 )  [p2 ], then the followings are equivalent :

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(i) N1  N2 , (ii) for any fn ∈ N1 ∩ Hn , gn ∈ N2 ∩ Hn satisfying fn  = 1 and gn  = 1, lim fn , gn  = 0,

n→∞

(iii) for fn ∈ N1 ∩ Hn with fn  = 1, there exist qn ∈ [p2 ] ∩ Hn such that lim fn − qn  = 0.

n→∞

(1)

(2)

Proof. (i)=⇒(iii): Since H 2 (D2 ) = N2 ⊕ [p2 ], there are fn ∈ N2 ∩ Hn and fn ∈ [p2 ] ∩ Hn (1) (2) such that fn = fn + fn . Since PN2 PN1 is compact and fn converges weakly to zero, lim fn − fn(2)  = lim fn(1)  = lim PN2 PN1 fn  = 0.

n→∞

n→∞

n→∞

(iii)=⇒(ii): By (iii), for any fn ∈ N1 ∩ Hn with fn  = 1, there is qn ∈ [p2 ] ∩ Hn such that lim fn − qn  = 0.

n→∞

It follows that for any gn ∈ N2 ∩ Hn with gn  = 1, we have lim | fn , gn |  lim (| fn − qn , gn | + | qn , gn |)  lim fn − qn gn  = 0.

n→∞

n→∞

n→∞

(ii)=⇒(i): Set k1 = deg p1 and k2 = deg p2 . Then, as mentioned in sec. 2, there is a positive integer N , such that for any n  N , dim(N1 ∩ Hn ) = k1 and dim(N2 ∩ Hn ) = k2 . 1 2 and {fnj }kj=1 be orthonormal bases of N1 ∩ Hn and N2 ∩ Hn respectively, then there Let {ein }ki=1 is a finite rank operator F , so that

PN1 PN2 = F +

 k1 ∞ 

ein

⊗

ein

n=N i=1

=F+

k2 k1  ∞   n=N

Since have

k1 k2 i=1

j i i j=1 fn , en en

Since for any i, j,

j fm

⊗

j fm



m=N j=1

fnj , ein ein

⊗

fnj

 .

i=1 j=1

⊗ fnj can be viewed as an operator from N2 ∩ Hn to N1 ∩ Hn , we

k1  k2 ∞   n=N

  k2 ∞ 

fnj , ein ein ⊗ fnj

i=1 j=1



k1  k2 ∞ 

=

n=N

limn→∞ fnj , ein 



fnj , ein ein ⊗ fnj .

i=1 j=1

= 0, this gives

 k1 k2   j i i  j  lim 

fn , en en ⊗ fn  = 0. n→∞ i=1 j=1

Therefore, PN1 PN2 is compact. From Proposition 4.1, the next corollary follows easily. Corollary 4.2.

Let |α1 | < 1 and |α2 | < 1, then [z − α1 w]⊥  [w − α2 z]⊥ .

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Proof.

Let

401

α1 z)k−1 w · · · + wk (¯ α1 z)k + (¯ , ek =  |α1 |2k + |α1 |2k−2 + · · · + 1

and

α2 w)k−1 z · · · + z k (¯ α2 w)k + (¯ fk =  , |α2 |2k + |α2 |2k−2 + · · · + 1

then ek ∈ [z − α1 w]⊥ ∩ Hk and fk ∈ [w − α2 z]⊥ ∩ Hk , with ek  = 1 and fk  = 1. Note that dim([z −α1 w]⊥ ∩Hk ) = 1 and dim([w−α2 z]⊥ ∩Hk ) = 1. It is easy to verify limn→∞ ek , fk  = 0, by Proposition 4.1, [z − α1 w]⊥  [w − α2 z]⊥ . Proposition 4.3. Let pi , i = 1, 2, 3, be the homogenous polynomials in C[z, w] satisfying GCD(p2 , p3 ) = 1. If both [p1 ]⊥  [p2 ]⊥ and [p1 ]⊥  [p3 ]⊥ , then [p1 ]⊥  [p2 p3 ]⊥ . Proof.

Since p2 and p3 are homogenous and GCD(p2 , p3 ) = 1, Z(p2 ) ∩ Z(p3 ) = {0}.

By Lemma 2.6, [p2 p3 ]⊥ = [p2 ]⊥ + [p3 ]⊥ . Below, we will prove [p1 ]⊥  ([p2 ]⊥ + [p3 ]⊥ ). By [15, Lemma 6.1], [p2 ]⊥ ∩ [p3 ]⊥ is of finite dimension, and hence for sufficiently large n, ([p2 ]⊥ ∩ Hn ) ∩ ([p3 ]⊥ ∩ Hn ) = {0}. Using the argument in the proof of Lemma 2.6, there are constants ci > 0, i = 1, 2, such that for sufficiently large n and fn ∈ [p2 ]⊥ ∩Hn , gn ∈ [p3 ]⊥ ∩Hn , fn   c1 fn + gn  and gn   c2 fn + gn . Suppose now fn +gn  = 1, then fn   c1 and gn   c2 . Since [p1 ]⊥ [p2 ]⊥ and [p1 ]⊥ [p3 ]⊥ , by Proposition 4.1, for any hn ∈ [p1 ]⊥ ∩ Hn satisfying hn  = 1, lim (fn + gn ), hn  = lim fn , hn  + lim gn , hn  = 0.

n→∞

n→∞

n→∞

By Proposition 4.1 again, [p1 ]⊥  ([p2 ]⊥ + [p3 ]⊥ ), thus completing the proof. Lemma 4.4. For |α| = 1 and |β| = 1, the quotient modules [z − αw]⊥  [(z − βw)n ]⊥ . Proof.

¯ we claim that Without loss of generality, assume |α| < 1. Setting α = αβ, [z − α w]⊥  [(z − w)n ]⊥ .

Since |α | < 1, it follows from Corollary 4.2 that [z]⊥  [w − α ¯  z]⊥ . Now, let Ni = [(z − w)i−1 ]  [(z − w)i ], and eki ∈ Ni ∩ Hn with eki  = 1. The argument in the proof of Lemma 2.5 implies that for i = 1, 2, . . . , n, limk→∞ wk , eki  = 0. By Proposition 4.1, P[z]⊥ PNik is compact, and hence P[z]⊥ P[(z−w)n ]⊥ is compact, which means [z]⊥  [(z − w)n ]⊥ . Since |α | < 1, GCD((¯ α z − w), (z − w)n ) = 1, and hence by Proposition 4.3, the quotient modules [z]⊥ 

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[(¯ α z − w)(z − w)n ]⊥ . By Proposition 4.1, it follows that there are polynomials pk ∈ Hk−n , such that lim wk+1 − (¯ α z − w)(z − w)n pk  = 0. Now, setting fk−1 =

k−1 i=0

k→∞

(¯ α z)i wk−1−i , we have (¯ α z)k − wk = (¯ α z − w)fk−1 , and hence

α z − w)fk − (¯ α z − w)(z − w)n pk  0  lim (¯ n→∞

α z)k+1  + lim wk+1 − (¯ α z − w)(z − w)n pk   lim (¯ n→∞

n→∞

= 0. Since for any f ∈ H 2 (D2 ), (¯ α z − w)f (z, w)2 =

 T2

|(¯ α z − w)f (z, w)|2 dm2  (1 − |α |)2 f 2 ,

we have 1 α z − w)fk − (¯ α z − w)(z − w)n pk  = 0. lim (¯ 1 − |α| k→∞

0  lim fk − (z − w)n pk   k→∞

1

Noticing that limk→∞ fk  = (1 − |α |2 )− 2 , we have lim fk − (z − w)n pk /fk  = 0.

k→∞

Taking qk = fpkk , then qk ∈ [(z − w)n ]. Since ffkk ∈ [z − α w]⊥ ∩ Hk , by Proposition 4.1, the quotient modules [z − α w]⊥  [(z − w)n ]⊥ . The claim is proved. For |β| = 1, define a unitary operator U : H 2 (D2 ) → H 2 (D2 ) by U f (z, w) = f (z, βw) for f ∈ H 2 (D2 ). Then it is easy to see that [z − αw]⊥ = U [z − α w]⊥ ,

[(z − βw)n ]⊥ = U [(z − w)n ]⊥ .

This implies that [z − αw]⊥  [(z − βw)n ]⊥ , thus completing the proof. For a homogenous polynomial p satisfying Z(p) ∩ ∂D2 ⊂ T2 , p can be factorized as p(z, w) = λ

m 

(z − αi w)ni , with |αi | = 1 and λ = 0.

i=1

Combining Lemma 4.4 with Proposition 4.3, we have the following Lemma 4.5.

Let p be a homogenous polynomial in C[z, w] satisfying Z(p) ∩ ∂D2 ⊂ T2 ,

then for |α| = 1, [z − αw]⊥  [p]⊥ . With the above preparations, we are able to prove the sufficiency of Theorem 1.1. As mentioned in the introduction, let p be a nonzero homogenous polynomial in C[z, w], then p can be factorized as p = p1 p2 ,

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with Z(p1 ) ∩ ∂D2 ⊂ T2 and Z(p2 ) ∩ ∂D2 ⊂ (D × T) ∪ (T × D). We restate the sufficiency of Theorem 1.1 as follows. Theorem 4.6. Let p be a nonzero homogenous polynomial in C[z, w], and p = p1 p2 as in (1.1), if p2 has one of the following forms: (1) p2 = c, with c = 0; (2) p2 = αz + βw, with |α| = |β|; (3) p2 = c(z − αw)(w − βz), with |α| < 1, |β| < 1 and c = 0, then the quotient module H 2 (D2 )/[p] is essentially normal. Proof. (1) Suppose p2 = c, then [p] = [p1 ]. Corollary 2.4 implies that the quotient module H 2 (D2 )/[p] is essentially normal. (2) Suppose p2 = αz + βw, with |α| = |β|. Without loss of generality, we may assume that α = 0. Notice that Z(p2 ) ∩ Z(p1 ) ∩ ∂D2 = ∅. By Lemma 2.6, [p]⊥ = [p1 ]⊥ + [p2 ]⊥ . Since |α| = 0 and |α| = |β|, β   β ⊥   [p2 ]⊥ = z − w , with   = 1. α α 2 2 ⊥ Notice that Z(p1 ) ∩ ∂D ⊂ T . Lemma 4.5 insures [p2 ]  [p1 ]⊥ . To show that [p]⊥ is essentially normal, by Corollary 3.4, it suffices to show that both [p1 ]⊥ and [p2 ]⊥ are essentially normal. Since Z(p1 ) ∩ ∂D2 ⊂ T2 , the essential normality of [p1 ]⊥ comes from Corollary 2.4. Now, set α = αβ , then [p2 ]⊥ = [z − α w]⊥ . It follows that α z)k−1 w + · · · + wk (¯ α z)k + (¯ ek =  |α |2k + |α |2k−2 + · · · + 1 is an orthonormal basis of [p2 ]⊥ . So Sz ek = zek , ek+1 ek+1 , and hence it is easy to verify that Sz is essentially normal. Similarly, Sw is essentially normal. This implies that [p2 ]⊥ is essentially normal. (3) Suppose p2 = c(z − αw)(w − βz), with |α| < 1, |β| < 1 and c = 0, then [p]⊥ = [(z − αw)(w − βz)p1 ]⊥ . Since Z(z − αw) ∩ Z((w − βz)p1 ) = {0}, by Lemma 2.6, we have [p]⊥ = [z − αw]⊥ + [(w − βz)p1 ]⊥ . As done in proof (2), the same reason shows that both [z−αw]⊥ and [(w−βz)p1 ]⊥ are essentially normal. To show that [p]⊥ is essentially normal, by Corollary 3.4, it suffices to show [z − αw]⊥  [(w − βz)p1 ]⊥ . Since |α| < 1, |β| < 1, Corollary 4.2 insures [z − αw]⊥  [w − βz]⊥ . Since |α| < 1 and Z(p1 ) ∩ ∂D2 ⊂ T2 , Lemma 4.5 implies [z − αw]⊥  [p1 ]⊥ .

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Notice that Z(w − βz) ∩ Z(p1 ) ∩ ∂D2 = ∅, then by Proposition 4.3, [z − αw]⊥  [(w − βz)p1 ]⊥ , as desired, thus completing the proof. 5

The necessity of Theorem 1.1

In this section, we will prove the necessity of Theorem 1.1. In fact, we will prove the following result, from which the necessity of Theorem 1.1 easily follows. Theorem 5.1. Let M  be a submodule of H 2 (D2 ), and |αi | < 1, i = 1, 2, then the quotient module [(z − α1 w)(z − α2 w)M  ]⊥ is not essentially normal. Similarly, the quotient module [(w − α1 z)(w − α2 z)M  ]⊥ is not essentially normal. Let p be a homogenous polynomial. Factorizing p = p1 p2 as in Theorem 1.1, if p2 has not one of the forms in Theorem 1.1, then p can be factorized as p = (z − α1 w)(z − α2 w)p ,

or

p = (w − α1 z)(w − α2 z)p ,

with |αi | < 1, i = 1, 2. Now taking M  = [p ] in Theorem 5.1, the necessity of Theorem 1.1 immediately comes from Theorem 5.1. Let H be an infinitely dimensional Hilbert space, and T be a bounded linear operator on H. For an invariant space M of T ∗ , T has a matrix representation, ⎛ ⎞ PM T |M M ⎠ T =⎝ . M⊥ PM ⊥ T |M PM ⊥ T |M ⊥ To prove Theorem 5.1, the following lemma is needed, whose proof is similar to the proofs of [3, Theorem 1] and [5, Theorem 4.3]. Lemma 5.2. Under the above assumption, if PM T |M is essentially normal, then the followings are equivalent: (1) T is essentially normal, (2) PM ⊥ T |M is compact, and PM ⊥ T |M ⊥ is essentially normal. To continue, the following lemma is needed. Lemma 5.3. Given |αi | < 1, i = 1, 2, write M = [(z −α1 w)(z −α2 w)] and N = H 2 (D2 )M . Then the quotient module N is not essentially normal. Proof.

We will show that Sz is not essentially normal.

Let V1 = [z − α1 w]⊥ and V2 = [z − α1 w]  M , then N = V1 ⊕ V2 . For any f ∈ V1 and g ∈ V2 , it is easy to see

Sz∗ f, g = f, zg = 0, that is, V1 is an invariant subspace of Sz∗ . It follows that Sz has the matrix representation, ⎞ ⎛ V Az ⎠ 1 . Sz = ⎝ Bz Cz V2

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Since V1 = [z − αw]⊥ , Theorem 4.6 implies that Az = PV1 Mz |V1 is essentially normal. To show that Sz is not essentially normal, by Lemma 5.2, it suffices to show that Bz is not compact. Notice that for n  2, dim Vi ∩ Hn = 1, i = 1, 2. Let α1 z)n−1 w + · · · + wn (¯ α1 z)n + (¯ , en =  |α1 |2n + |α1 |2(n−1) + · · · + 1 then en ∈ V1 ∩ Hn and en  = 1. For gn ∈ V2 ∩ Hn satisfying gn  = 1, we have Bz en = zen , gn+1 gn+1 . Hence Bz is compact if and only if zen , gn+1  → 0 as n → ∞. We will show that Bz is not compact in the following three cases. Case 1. α1 = α2 . Since Z(z − α1 w) ∩ Z(z − α2 w) = {0}, by Lemma 2.4, N = M ⊥ = [z − α1 w]⊥ + [z − α2 w]⊥ . Set

α1 z)n−1 w + · · · + wn (¯ α1 z)n + (¯ en =  , |α1 |2n + |α1 |2(n−1) + · · · + 1

and

α2 z)n−1 w + · · · + wn (¯ α2 z)n + (¯ fn =  . |α2 |2n + |α2 |2(n−1) + · · · + 1

Since α1 = α2 , by the Cauchy inequality, | en , fn | 

n 

|α1 α2 |j

j=0

n  

|α1 |2j

j=0

n 

|α2 |2j

 12 < 1.

j=0

It follows that for n  2, {en , fn } is a linear basis of N ∩ Hn . Using the Gram-Schmidt orthogonalization,  1  gn = fn − fn , en en 1 − | en , fn |2 2 . Since |αi | < 1, i = 1, 2, by direct calculations, we have  (1 − |α1 |2 )(1 − |α2 |2 ) , lim en+1 , fn+1  = n→∞ 1−α ¯ 1 α2  α2 (1 − |α1 |2 )(1 − |α2 |2 ) lim zen , fn+1  = , n→∞ 1−α ¯ 1 α2 and lim zen , en+1  = α1 .

n→∞

√

Set c0 =

|1−α ¯ 1 α2 | (1−|α1 |2 )(1−|α2 |2 ) , (1−α ¯ 1 α2 )|α1 −α2 |

then we have

lim zen , gn+1  = lim

n→∞

zen , fn+1  − en+1 , fn+1  zen , en+1 

n→∞

= c0 (α2 − α1 ).

1

(1 − | en , fn |2 ) 2

Kun-yu GUO & Peng-hui WANG

406

In the case α1 = α2 , we have c0 = 0, and hence limn→∞ zen , gn+1  = 0. This implies that in this case, Bz is not compact. Case 2.

α1 = α2 = α = 0. Let αz)n−1 w + · · · + wn (¯ αz)n + (¯ en =  . |α|2n + |α|2n−2 + · · · + 1

Now it is easy to verify that for any nonnegative integers j1 , j2 ,

(z − αw)2 z j1 wj2 , (n + 1)(¯ αz)n + n(¯ αz)n−1 w + · · · + wn  = 0, for n  2. It follows that (n + 1)(¯ αz)n + n(¯ αz)n−1 w + · · · + wn ∈ N ∩ Hn . Write αz)n−1 w + · · · + wn (n + 1)(¯ αz)n + n(¯ hn =  , (n + 1)2 |α|2n + n2 |α|2n−2 + · · · + 1 then hn  = 1. For n  2, by the Cauchy inequality,

    12 n n n  2j 2 2j 2j

en , hn  = (j + 1)|α| (j + 1) |α| |α| < 1. j=0

j=0

j=0

Since dim N ∩ Hn = 2 for n  2, {en , hn } is a linear basis of N ∩ Hn . Using the Gram-Schmidt orthogonalization again,  1 gn = (hn − hn , en en ) (1 − | en , hn |2 ) 2 ∈ V2 ∩ Hn , and gn  = 1. Now, we will show that lim zen , gn+1  = 0.

(5.1)

n→∞

Since en+1 , gn+1  = 0 and α = 0, (5.1) holds if and only if lim ¯ αzen − ben+1 , gn+1  = 0

n→∞

for any b ∈ R. Taking b=

n+1 

2j

|α|

 n

j=1

2j

|α|

 12 ,

j=1

we have αze ¯ n − ben+1 = −wn+1

  n

|α|2j

 12 .

j=1

Therefore, it suffices to show that  1 − |α|2 lim wn+1 , gn+1  = lim n→∞



n→∞

w

n+1

  n j=1

Since 0 < |α| < 1, by direct calculations, we have lim wn , en  =

n→∞

 1 − |α|2 ,

2j

|α|

 12

 0. , gn+1 =

Homogenous quotient modules of Hardy modules on the bidisk

407

 3 1 lim wn , hn  = (1 − |α|2 ) 2 (1 + |α|2 ) 2 ,

n→∞

and

1 lim en , hn  =  . 1 + |α|2

n→∞

Therefore,   lim wn , hn  − en , hn  wn , en  3 1 1   (1 − |α|2 ) 2 − (1 − |α|2 ) 2 = 0. = lim wn , gn  = n→∞ |α| lim 1 − | en , hn |2 n→∞

n→∞

Therefore, in this case, Bz is not compact. Case 3.

α1 = α2 = 0. This case is considered in [24], we sketch the proof here for convenience. M ⊥ = N1 ⊕ N2 ,

where N1 = H 2 (D2 )  zH 2 (D2 ) and N2 = zH 2 (D2 )  z 2 H 2 (D2 ). It is easy to see that N1 = span{wn | n = 0, 1, . . .}, N2 = span{zwn | n = 0, 1, . . .}. Since Sz wn = zwn and Sz zwn = 0, Sz can be decomposed as

⎛ Sz = ⎝

⎞ 0

0

U

0

⎠

N1

,

N2

where U wn = zwn . That is, Bz = U is not compact. The above reason shows that Sz is not essentially normal, thus completing the proof. The proof of Theorem 5.1. We will only prove that, if |αi | < 1 for i = 1, 2, then for any submodule M  of H 2 (D2 ), the quotient module N  = H 2 (D2 )  [(z − α1 w)(z − α2 w)M  ] is not essentially normal. Let M1 = [z − α1 w], M2 = [(z − α1 w)(z − α2 w)] and M3 = [(z − α1 w)(z − α2 w)M  ], then N  can be decomposed as N  = M1⊥ ⊕ (M1  M2 ) ⊕ (M2  M3 ). Relative to this decomposition, Sz has the matrix representation, ⎛

⎞

Az

⎜ Sz = ⎜ ⎝ Bz Ez

Cz Fz

M1⊥ ⎟ ⎟ M M . 1 2 ⎠ Gz M2  M3

Kun-yu GUO & Peng-hui WANG

408

Since M1 = [z − α1 w], Theorem 4.6 implies that Az = PM1⊥ Mz |M1⊥ is essentially normal. The  argument in the proof of Lemma 5.3 implies that Bz is not compact, and hence Bz is not Ez compact. By Lemma 5.2, Sz is not essentially normal, and hence Theorem 5.1 is proved. 6

Essential spectrum of the quotient modules and K-homology

In this section, we will describe the essential spectrum of quotient modules of H 2 (D2 ). Some similar techniques can be seen in [10]. The K-homology will also been considered. Let M be submodule of H 2 (D2 ), Z(M ) = {λ ∈ D2 | f (λ) = 0, ∀f ∈ M }, and Z∂ (M ) is defined by

! Z∂ (M ) = λ ∈ ∂D2 | there are λn ∈ Z(M ), such that lim λn = λ . n→∞

Theorem 6.1. σe (N ).

For any submodule M of H 2 (D2 ), set N = H 2 (D2 )  M . Then Z∂ (M ) ⊆

Proof. The proof is routine. Some similar techniques appear in [10]. For the reader’s convenience, we give the detail of the proof. Given λ ∈ Z∂ (M ), by the definition of Z∂ (M ), there is μn ∈ Z(M ) such that limn→∞ μn = λ. If the pair (λ1 − Sz , λ2 − Sw ) is Fredholm, by a result of Curto [32, Corollary 3.11], the operator ⎛ ⎞ λ1 − Sz λ2 − Sw ⎠:N ⊕N →N ⊕N A=⎝ −(λ2 − Sw )∗ (λ1 − Sz )∗ is Fredholm. Set T1 = λ1 − Sz and T2 = λ2 − Sw . It follows that ⎞ ⎛ ∗ ∗ T T + T T 0 1 1 2 2 ⎠ AA∗ = ⎝ 0 T1∗ T1 + T2∗ T2 is Fredholm. Hence (λ1 − Sz )(λ1 − Sz )∗ + (λ2 − Sw )(λ2 − Sw )∗ = T1 T1∗ + T2 T2∗ is Fredholm. This implies that there exist a positive invertible operator B and a compact operator K, such that (λ1 − Sz )(λ1 − Sz )∗ + (λ2 − Sw )(λ2 − Sw )∗ = B + K. Now, let kμn be the normalized reproducing kernel of H 2 (D2 ) at μn . Since kμn converges weakly to zero, and since B is positive and invertible, there is a positive constant c such that lim [(λ1 − Sz )(λ1 − Sz )∗ + (λ2 − Sw )(λ2 − Sw )∗ ]kμn , kμn 

n→∞

= lim (B + K)kμn , kμn  n→∞

= lim Bkμn , kμn   c. n→∞

However, since μn = follows that

(1) (2) (μn , μn )

∈ Z(M ), for any f ∈ M , f, kμn  = 0, and then kμn ∈ N . It

lim [(λ1 − Sz )(λ1 − Sz )∗ + (λ2 − Sw )(λ2 − Sw )∗ ]kμn , kμn 

n→∞

2 (2) 2 = lim (|λ1 − μ(1) n | + |λ2 − μn | ) = 0. n→∞

Homogenous quotient modules of Hardy modules on the bidisk

409

This contradiction implies Z∂ (M ) ⊆ σe (N ), as desired. Example.

Let ∞  α ¯ n αn − z φ(z) = , |α ¯n z n| 1 − α n=1

∞  β¯n βn − w ϕ(w) = |βn | 1 − β¯n w n=1

be two infinite Blaschke products. It is well-known that there exist infinite Blaschke products φ and ϕ, such that T ⊂ Z(φ) and T ⊂ Z(ϕ). For detailed information of this kind of inner functions, one can see [33, sec. 3]. Now, let M = [φϕ], then Z∂ (M ) = ∂D2 . By Theorem 6.1, ∂D2 ⊂ σe (Sz , Sw ). By [19, Theorem 4.3], σe (Sz , Sw ) = ∂D2 . Theorem 6.2. Z(p) ∩ ∂D2 . Proof.

Given a homogenous polynomial p. Set N = H 2 (D2 )  [p], then σe (N ) =

On the one hand, since p is a homogenous polynomial, Z∂ ([p]) = Z([p]) ∩ ∂D2 = Z(p) ∩ ∂D2 .

By Theorem 6.1, Z(p) ∩ ∂D2 ⊂ σe (N ). On the other hand, by [19, Theorem 4.3], σe (N ) ⊂ ∂D2 . Since p(Sz , Sw ) = 0, by the Spectral Mapping Theorem[34,Theorem 4.8] , σe (N ) ⊂ Z(p) ∩ ∂D2 . This completes the proof. Let p be a homogenous polynomial. If the quotient module [p]⊥ is essentially normal, then we get an extension 0 → K → C ∗ ([p]⊥ )→C(Z(p) ∩ ∂D2 ) → 0. This extension yields a K-homology element in K1 (Z(p) ∩ ∂D2 ), which is denoted by ep . By Theorem 1.1, the polynomial p can be factorized as p = (w − α1 z)n1 (z − α2 w)n2

m 

(z − αi w)ni ,

i=3

with |αi | < 1, ni  1 for i = 1, 2, and |αi | = 1 for i  3. Set p1 = (w − α1 z)n1 , and pi = (z − αi w)ni for i  2. Without loss of generality, assume that Z(pi ) ∩ Z(pj ) = {0} for i = j. The same argument as [8, Proposition 4.2] implies that ep = ep1 ⊕ ep2 ⊕ · · · ⊕ epm .

(6.1)

Below, we will show that if deg pi = 0, then epi are nontrivial K-homology elements in K1 (Z(pi ) ∩ ∂D2 ) for i = 1, . . . , m. In fact, we will show that the corresponding extension is not split. Proposition 6.3.

Let α be a complex number with |α| < 1, then the extension 0 → K → C ∗ ([z − αw]⊥ )→C(Z(z − αw) ∩ ∂D2 ) → 0

is not split.

Kun-yu GUO & Peng-hui WANG

410

Proof.

By Theorem 6.2, σe ([z − αw]⊥ ) = Z(z − αw) ∩ ∂D2 = {(αw, w)| |w| = 1}.

By the Spectral Mapping Theorem[34,Theorem 4.8] , σe (Sw ) = T. This implies that Sw is Fredholm. It is easy to see that the Fredholm index Ind(Sw ) = −1. By [8, Proposition 4.6], the extension 0 → K → C ∗ ([z − αw]⊥ )→C(Z(z − αw) ∩ ∂D2 ) → 0 is not split, as desired. Proposition 6.4.

Let α be a complex number with |α| = 1, then the extension 0 → K → C ∗ ([(z − αw)n ]⊥ )→C(Z(z − αw) ∩ ∂D2 ) → 0

is not split. Proof. The same reason as in the proof of Proposition 6.3 shows that Sz is Fredholm. The argument in the proof of Lemma 2.2 implies that ker Sz = {0}. Now since 1 ∈ ker Sz∗ , the Fredholm index Ind(Sz ) = 0. By [8, Proposition 4.6] again, the extension 0 → K → C ∗ ([(z − αw)n ]⊥ )→C(Z(z − αw) ∩ ∂D2 ) → 0 is not split, thus completing the proof. Combining Proposition 6.3, Proposition 6.4 with (6.1), we have the following theorem. Theorem 6.5. Let p be a homogenous polynomial. If [p]⊥ is essentially normal, then the short exact sequence 0 → K → C ∗ ([p]⊥ )→C(Z(p) ∩ ∂D2 ) → 0 is not split. References [1] Douglas R, Paulsen V. Hilbert modules over function algebras. Pitman Research Notes in Mathematics Series, 217, 1989 [2] Arveson W. p-Summable commutators in dimension d. J Oper Theory, 54: 101–117 (2005) [3] Arveson W. Quotients of standard Hilbert modules. Trans AMS, to appear [4] Arveson W. The dirac operator of a commuting d-tuple. J Funct Anal, 189: 53–79 (2002) [5] Douglas R. Essentially reductive Hilbert modules. J Oper Theory, 55: 117–133 (2006) [6] Douglas R. Invariants for Hilbert Modules. Proceedings of Symposia in Pure Mathematics, Vol. 51, Part 1, 1990, 179–196 [7] Guo K. Defect operator for submodules of Hd2 . J Reine Angew Math, 573: 181–209 (2004) [8] Guo K, Wang K. Essentially normal Hilbert modules and K-homology. Preprint [9] Guo K, Wang K. Essentially normal Hilbert modules and K-homology II: Quasi-homogeneous Hilbert modules over two dimensional unit ball. Preprint [10] Guo K, Duan Y. Spectrum property of the submodule of the Hardy space over Bd . Studia Math, to appear [11] Izuchi K, Yang R. Nϕ -type quotient modules on the torus. Preprint [12] Chen X, Guo K. Analytic Hilbert modules. π-Chapman & Hall/CRC Research Notes in Math, 433, 2003 [13] Curto R, Muhly P, Yan K. The C ∗ -algebra of an homogeneous ideal in two variables is type I. Current Topics in Operator Algebras (Nara, 1990). River Edge, NJ: World Sci. Publishing, 1991. 130–136

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