BEURLING TYPE THEOREM ON THE BERGMAN SPACE VIA THE HARDY SPACE OF THE BIDISK SHUNHUA SUN AND DECHAO ZHENG A BSTRACT. In this paper, by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk, we give a proof of the Beurling type theorem on the Bergman space of Aleman, Richter and Sundberg [1] via the Hardy space of the bidisk.
1. I NTRODUCTION Let D be the open unit disk in C. Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. The Bergman space L2a is the Hilbert space consisting of the analytic functions on D that are also in the space L2 (D, dA) of square integrable functions on D. The famous Beurling theorem [3] classifies the invariant subspaces of the multiplication operator by the coordinate function z on the Hardy space of the unit disk, the unilateral shift. For an operator T on a Hilbert space H, the subspace H T H is called a wandering subspace of T on H. The Beurling theorem says that all invariant subspaces of the unilateral shift are generated by their wandering subspaces with dimension 1. This result has played an important role in function theory and operator theory. Recently Aleman, Richter and Sundberg have established a remarkable Beurling type theorem [1] for the multiplication operator by the coordinate function z on the Bergman space, the Bergman shift even if the dimension of the wandering subspace ranges from 1 to ∞ [2]. Their result states that all invariant subspaces of the Bergman shift are also generated by their wandering subspaces. This result is a breakthrough in understanding of the invariant subspaces of the Bergman space and becomes a fundamental theorem in the function theory on the Bergman space [6], [10]. Different proofs of the Beurling type theorem were given in [11], [12], [15] later. The goal of this paper is to give a new proof of the Beurling type theorem of Aleman, Richter and Sundberg [1] via the Hardy space of the bidisk. Our main objective is the study of the Bergman space of the unit disk and its operators via the Hardy space of the bidisk. Our project starts in [9], [17], [18] to study multiplication operators on L2a by bounded analytic functions on the unit disk D via the Hardy space of the bidisk. The theme is to use the theory of multivariable operators and functions to study a single operator and the functions of one variable. Our main idea is to lift the Bergman shift up as the compression of a commuting pair of isometries on a subspace of the Hardy space of the bidisk. This idea was used in studying the Hilbert modules by R. Douglas and V. Paulsen [4], The first author was supported in part by the National Natural Science Foundation-10471041 of China. The second author was partially supported by the National Science Foundation. 1
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operator theory in the Hardy space over the bidisk by R. Douglas and R. Yang [5], [19], [20] and [21]; the higher-order Hankel forms by S. Ferguson and R. Rochberg [7] and [8] and the lattice of the invariant subspaces of the Bergman shift by S. Richter [14]. Let T denote the unit circle. The torus T2 is the Cartesian product T × T. The Hardy space H 2 (T2 ) over the bidisk is H 2 (T) ⊗ H 2 (T). For each integer n ≥ 0, let pn (z, w) =
n X
z i wn−i .
i=0
Let H be the subspace of H (T ) spanned by functions {pn }∞ n=0 . Thus every function in H is symmetric with respect to z and w. Let [z −w] denote the closure of (z −w)H 2 (T2 ) in H 2 (T2 ). As every function in [z − w] is orthogonal to each pn , we easily see 2
2
H 2 (T2 ) = H ⊕ [z − w]. Let A be a subspace of L2 (T2 ) and let PA denote the orthogonal projection from L2 (T2 ) onto A. The Toeplitz operator on H 2 (T2 ) with symbol f in L∞ (T2 ) is defined by Tf (h) = PH 2 (T2 ) (f h), for h in H 2 (T2 ). It is not difficult to see that Tz and Tw are a pair of doubly commuting pure isometries on H 2 (T2 ). A little computation gives PH Tz |H = PH Tw |H . We use B to denote the operator above. It was shown explicitly in [16] and implicitly in [4] that B is unitarily equivalent to the Bergman shift, the multiplication operator by the coordinate function z on the Bergman space L2a via the following unitary operator U : L2a (D) → H, pn (z, w) U zn = . n+1 So the Bergman shift is lifted up as the compression of an isometry on a nice subspace H of H 2 (T2 ). In the rest of the paper we identify the Bergman shift with the operator B. This paper is organized as follows. In Section 2, as in [14] we lift each invariant subspace of B as an invariant subspace of the isometry Tz , do the Wold decomposition and identify the the wandering subspace. In Section 3, using the structure of the wandering space and establishing an identity (in Step 8 in the proof of Theorem 3.1) we give a proof of Aleman, Richter and Sundberg theorem [1]. 2. W OLD DECOMPOSITION AND WANDERING SUBSPACES First we introduce notation to lift each invariant subspace of B as an invariant subf to be the direct space of Tz . For an invariant subspace M of B, define the lifting M sum M ⊕ [z − w]. In Theorem 6.8 [14] Richter showed that the mapping f η:M→M
BEURLING TYPE THEOREM VIA HARDY SPACE OF BIDISK
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is a one-to-one correspondence between invariant subspaces of B and invariant subspaces of Tz containing [z − w]. In this section we will get the Wold decomposition of f and identify the wandering subspace of Tz on M. f the isometry Tz on M Although for an invariant subspace M of B, M may not be invariant for Tz∗ , let the ∗ operator BM on M denote the compression of Tz∗ on M, i.e., ∗ q = PM Tz∗ q = PM B ∗ q BM
for q in M. Since the Bergman shift is bounded below, we have the following lemma. ∗ Lemma 2.1. Let M be an invariant subspace of B. Then BM B is invertible on M.
Proof. Since B is unitarily equivalent to the Bergman shift, an easy computation gives 1 kBf k ≥ √ kf k 2 for each f in M. The Cauchy-Schwarz inequality gives ∗ ∗ kBM Bf kkf k ≥ |hBM Bf, f i = kBf k2 1 kf k2 . ≥ 2
Thus we have 1 ∗ kBM Bf k ≥ kf k 2 ∗ ∗ B is invertible on for each f in M. So BM B is bounded below on M. To show that BM ∗ M, we need only show that BM B is onto. If it is not so, then there is a nonzero function f in M such that ∗ hBM Bx, f i = 0 for all x in M. In particular, letting x be equal to f in the above equality, we have kBf k2 = 0, ∗ to get that f is zero since B is injective on M. This shows that BM B is onto on M, to complete the proof. The following lemma will be used in the proof of Theorem 2.3. First we introduce a notation. For two functions x, y in H 2 (T2 ), the symbol x ⊗ y is the operator on H 2 (T2 ) defined by (x ⊗ y)g = [hg, yiH 2 (T2 ) ]x
for g ∈ H 2 (T2 ). Lemma 2.2. On the Hardy space H 2 (T2 ), the identity operator equals X X I = Tz Tz∗ + wl ⊗ wl = Tw Tw∗ + zl ⊗ zl. l≥0
l≥0
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Proof. We will just verify the first equality in the lemma since the same argument leads to the proof of the second equality. P j To do this, let h be in H 2 (T2 ). Writing h(z, w) = ∞ j=0 hj (w)z for some functions hj (w) in H 2 (T), we have ∞ ∞ X X ∗ ∗ j Tz Tz h = hj (w)Tz Tz z = hj (w)z j . j=0
j=1
Using the identity (wl ⊗ wl )h = hh, wl iwl ∞ X = hhj (w)z j , wl iwl = hh0 (w), wl iwl , j=0
we have X X ( wl ⊗ wl )h = hh0 (w), wl iwl = h0 (w), l≥0
l≥0 2
2
to conclude that for each h in H (T ), ∞ X X ∗ l l [Tz Tz + w ⊗ w ]h = hj (w)z j + h0 (w) l≥0
=
j=1 ∞ X
hj (w)z j = h.
j=0
This completes the proof. Let M0 be the wandering space of B on M. Clearly, the wandering subspace LM f of f Tz on M contains M0 . To get understanding LM f, we need to find out the orthogonal complement of M0 in LM f. Let ∗ M00 = {−hg + zPH g − wg(w) : (hg , g) ∈ BM × H 2 (T) and hg = B[BM B]−1 PM g}.
f and shows that the The following theorem gives the Wold decomposition of Tz on M orthogonal complement of M0 in LM f equals M00 . f be the lifting of M. Then Theorem 2.3. Let M be an invariant subspace of B. Let M f is an invariant subspace of the isometry Tz and has the following decomposition: M f = ⊕∞ z n L f M n=0
M
f where LM f is the wandering space of Tz on M given by LM f = M0 ⊕ M00 . f denote the lifting of M. Proof. Let M be an invariant subspace of B and M f is an invariant subspace of Tz . For completeness we It was shown in [14] that M f we write include a proof. To do this, for each f in M, f = f1 + f2
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for f1 in M and f2 in [z − w]. Since [z − w] is an invariant subspace of Tz , Tz f2 = zf2 is in [z − w]. A little computation gives Tz f1 = PH (zf1 ) + P[z−w] (zf1 ) = Bf1 + P[z−w] (zf1 ) f ∈ M ⊕ [z − w] = M. The last equality follows from that M is an invariant subspace of B. Thus we have Tz f = Tz f1 + Tz f2 = Bf1 + Tz f2 f ∈ M ⊕ [z − w] = M, f is an invariant subspace of the isometry Tz . to get that M n f Since for every function f in ∩∞ n=0 Tz M, f and its derivatives vanish at z = 0, we have n f ∩∞ n=0 Tz M = {0}. By the Wold decomposition theorem [13], we have n f = ⊕∞ M f n=0 z LM
f where LM f is the wandering subspace of Tz on M. To finish the proof, we need identify the wandering subspace LM f by showing LM f = M0 ⊕ M00 . First we show that M0 is orthogonal to M00 . To do this, let m be in M0 and −hg + zPH g − wg(w) in M00 . An easy calculation gives hm, −hg + zPH g − wg(w)i = −hm, hg i = 0. The first equality follows from that zPH g − wg(w) is in [z − w] and the last equality follows from that hg is in BM. Thus M0 is orthogonal to M00 . Second we show M0 ⊕ M00 ⊂ LM f. To do this, let m be in M0 and −hg + zPH g − wg(w) in M00 . For each f1 in M and f2 in [z − w], we have that PM Tz f1 = Bf1 and zf2 is contained in [z − w], to get hm, Tz f1 i = hm, Bf1 i = 0, and hm, Tz f2 i = hm, zf2 i = 0. These give hm, Tz (f1 + f2 )i = 0.
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An easy computation gives h−hg + zPH g − wg(w), Tz f1 i = = = =
h−Tz∗ hg + PH g, f1 i − hTz∗ (wg(w)), f1 i h−Tz∗ hg + PH g, PM f1 i ∗ h−BM hg + PM g, f1 i 0.
The second equality follows from that Tz∗ (wg(w)) = 0 and f1 is in M. The third ∗ equality follows from that BM is the compression of Tz∗ on M. Since [z − w] is an ∗ invariant subspace of Tz and Tz (wg(w)) = 0, we also have h−hg + zPH g − wg(w), Tz f2 i = −hhg , Tz f2 i + hzPH g, Tz f2 i − hwg(w), Tz f2 i = hPH g, f2 i − hTz∗ (wg(w)), f2 i = hPH g, f2 i = 0. Thus these give h−hg + zPH g − wg(w), Tz (f1 + f2 )i = 0. So we obtain M0 ⊕ M00 ⊂ LM f. Third we show LM f ⊂ M0 ⊕ M00 . For q in LM f, write q = q1 + q2 for q1 in M and q2 in [z − w]. Noting that q1 is orthogonal to Tz [z − w] and f = Tz M ⊕ Tz [z − w], Tz M we have that q2 is orthogonal to Tz [z − w] and q1 + q2 is orthogonal to Tz M. Using Lemma 2.2, we will derive a special representation of q2 . Since q2 is orthogonal to Tz [z − w], we have that Tz∗ q2 is orthogonal to [z − w]. Letting G = Tz∗ q2 , then G is in H and zG = Tz Tz∗ q2 = q2 − q2 (0, w). The last equality follows from Lemma 2.2: I = Tz Tz∗ +
∞ X
wn ⊗ wn .
n=0
Since q2 is in [z − w], by Lemma 8 in [9], we have that q2 (z, z) = 0 to get zG(z, z) = q2 (z, z) − q2 (0, z) = −q2 (0, z). Letting g(z) = G(z, z), we have that g(z) is in H 2 (T) since q2 (z, w) is in H 2 (T2 ). Rewrite the above equality as q2 (z, w) = zG(z, w) − wg(w).
(2.1)
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Applying the projection PH to both sides of the above equality gives 0 = = = = = =
P H q2 PH [zG − wg(w)] PH Tz G − PH [w(PH g(w) + PH⊥ g(w))] BG − PH (w(PH g(w)) BG − B(PH g(w)) B(G − (PH g(w))).
The first equality follows from that q2 is in [z − w] and the fourth equality follows from that [z − w] is an invariant subspace of Tw . Using the fact that B is injective on H, we have G = PH g. By (2.1), we obtain q2 = zPH g − wg(w).
(2.2)
Since q1 + q2 is orthogonal to Tz M, a simple computation gives 0 = = = =
hq1 + q2 , Tz f i hq1 + zPH g − wg(w), Tz f i hTz∗ q1 + PH g, f i hPM Tz∗ q1 + PM g, f i
for each f in H. This gives ∗ BM q1 + PM g = PM Tz∗ q1 + PM g = 0.
(2.3)
Noting M = M0 ⊕ BM, we write q1 = m0 − Bh1
(2.4)
∗ for some h1 in M and m0 in M0 . Applying BM to both sides of the above equality and using (2.3) give ∗ BM Bh1 = PM g.
Thus ∗ h1 = (BM B)−1 PM g.
Letting hg = Bh1 and using (2.4) and (2.2), we have q = q1 + q2 = m0 − hg + zPH g − wg(w) ∈ M0 ⊕ M00 to complete the proof.
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3. A
PROOF OF
A LEMAN , R ICHTER AND S UNDBERG THEOREM
In this section we will give a proof of the following Beurling type theorem of Aleman, Richter and Sundberg in [1]. Let S be a subspace of H. We use [S] to denote the smallest invariant subspace of B containing S, i.e., [S] = ∨n≥0 B n S, where ∨n≥0 B n S denotes the closed subspace of H spanned by the set {B n S : n ≥ 0}. Theorem 3.1. Let M be an invariant subspace of B. Then M = [M BM]. Proof. Let M0 denote the wandering subspace M BM of B on M. Let N be the orthogonal complement of [M0 ] in M. We will show that N = {0}. The proof is long and will be divided into several steps. Step 1. First we show ∞ X N ⊂{ z n un : un = −hn + zPH gn − wgn (w) ∈ M00 }. n=0
To do this, let q be a function in N . By Theorem 2.3, we have q=
∞ X
k
z m ˜k +
∞ X
z n un ,
n=0
k=0
where m ˜ k is in M0 and un = (−hn + zPH gn − wgn (w)) ∈ M00 . Since q is in N and orthogonal to ∨n≥0 B n M0 , taking inner product of q with B k m ˜ k gives 0 = hq, B k m ˜ ki = hq, PH z k m ˜ ki n = hq, z m ˜ ki = hz k m ˜ k , zk m ˜ k i + hz k uk , z k m ˜ ki 2 = km ˜ k k + huk , m ˜ ki 2 = km ˜ kk . The second equality follows from that B k equals the compression of Tzk on H and the fourth equality follows from that uk is orthogonal to m ˜ k . This gives that m ˜ k = 0 for all k ≥ 0. Thus each function q in N has the following form q=
∞ X
z n un .
(3.1)
n=0
For a function q in N with the above representation, let q1 =
∞ X n=1
z n−1 un .
(3.2)
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Step 2. Next we show that for each q in N , ∗ q1 = B M q
and q1 is still in N . An easy calculation gives Tz∗ q
= PH 2 (T2 ) [¯ z u0 + z¯
∞ X
z n un ]
n=1
= PH 2 (T2 ) [¯ z (−h0 + zPM g0 − wg0 )] +
∞ X
z n−1 un
n=1
Since q and h0 are in M ⊂ H and H is invariant under Tz∗ , we have that both Tz∗ q and f = M ⊕ [z − w] and both M and H are orthogonal Tz∗ h0 are also in H. Noting that M to [z − w], we have that PM f|H = PM |H , to get ∗ PM Tz∗ q = PM fTz q
(3.3)
∗ PM Tz∗ h0 = PM fTz h0 .
(3.4)
and ∗ to q and using (3.3) give Applying the operator BM ∗ ∗ BM q = PM Tz∗ q = PM fTz q
z (−h0 + zPM g0 − wg0 )] + = PM f{PH 2 (T2 ) [¯
∞ X
z n−1 un }
n=1 ∗ = PM f[−Tz h0 + PM g0 ] +
∗ = −PM fTz h0 + PM g0 +
∞ X
z n−1 un
n=1 ∞ X
z n−1 un
n=1
= −PM Tz∗ h0 + PM g0 +
∞ X
z n−1 un
n=1 ∗ = −BM h0 + PM g0 +
∞ X
z n−1 un
n=1
=
∞ X
z n−1 un .
n=1
The sixth equality follows from (3.4) and the last equality follows from that h0 = ∗ B(BM B)−1 PM g0 . This gives ∗ BM q = q1 .
(3.5)
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SUN AND ZHENG
To show that q1 is still in N , let m be in M0 and k nonnegative integers. Easy calculations give ∗ hq1 , B k mi = hBM q, B k mi
= hPM Tz∗ q, B k mi = hTz∗ q, B k mi = hq, Tz B k mi = hq, B k+1 mi = 0. This says that q1 is in N . Step 3. We show ∗ B)−1 q1 . q = B(BM
To do so, using (3.2) we have q = zq1 + [−h0 + zPH g0 − wg0 (w)] = Bq1 − h0 + [(z − B)q1 + zPH g0 − wg0 (w)]. Note that q and Bq1 − h0 are in M and (z − B)q1 + zPH g0 − wg0 (w) is in [z − w] and hence orthogonal to M. Taking the projection at both sides of the above equality onto M and [z − w] respectively gives q = Bq1 − h0
(3.6)
and (z − B)q1 = −zPH g0 + wg0 (w). By the fact that Tz∗ (wg0 (w)) = 0, applying Tz∗ to both sides of the above equality, we have (1 − Tz∗ B)q1 = −PH g0 , to get ∗ PM g0 = −(1 − BM B)q1 . ∗ −1 Since h0 = B(BM B) PM g0 , we have ∗ ∗ h0 = −B(BM B)−1 (1 − BM B)q1 .
(3.7)
So combining (3.6) with (3.7) gives q = Bq1 − h0 ∗ ∗ B)−1 (1 − BM B)q1 = Bq1 + B(BM ∗ −1 = B(BM B) q1 . ∗ Step 4. We show that B(BM B)−1 q is in N for each q in N . By Step 3, hence ∗ ∗ B(BM B)−1 |N is the inverse of BM |N .
Let ∗ q˜ = B(BM B)−1 q.
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Clearly, q˜ is in M. On the other hand, for each m in M0 and n > 0, an easy computation gives ∗ h˜ q , B n mi = hB(BM B)−1 q, B n mi = hq, B n−1 mi = 0,
and ∗ h˜ q , mi = hB(BM B)−1 q, mi ∗ = hq, (BM B)−1 B ∗ mi = 0,
to get that q˜ is in N . Step 5. For each q =
P∞
n=0
∗ k z n un in N as in Step 1, let qk = (BM ) q. Then
kqk−1 k2 + kqk+1 k2 − 2kqk k2 = kuk−1 k2 − kuk k2 . P n−k To prove the above equality, by Step 2 we have that qk = ∞ un , and hence n=k z 2
kqk k =
∞ X
kun k2 ,
n=k
to obtain kqk−1 k2 + kqk+1 k2 − 2kqk k2 ∞ ∞ ∞ X X X 2 2 kun k2 kun k − 2 kun k + = n=k−1 2
n=k+1 2
n=k
= kuk−1 k − kuk k .
∗ k Step 6. For each q in N , let qk = (BM ) q. Then ∗ ∗ kqk−1 k2 + kqk+1 k2 − 2kqk k2 = h(BM B)−1 qk , qk i + h(BBM )qk , qk i − 2hqk , qk i. (3.8)
By Step 4, we have ∗ qk−1 = B(BM B)−1 qk
to obtain ∗ ∗ kqk−1 k2 = hB(BM B)−1 qk , B(BM B)−1 qk i ∗ = h(BM B)−1 qk , qk i.
This gives (3.8). The Dirichlet space D consists of analytic functions on the unit disk whose derivatives are in the Bergman space L2a . We will get a representation of functions in M. Step 7. For each f in M, there is a function g(z) in H 2 (T) ∩ D such that zg(z) − wg(w) f (z, w) = −PH g − . (3.9) z−w For f in M and each h in H, h(B − Tz )f, hi = h(PH Tz − Tz )f, hi = hTz f − Tz f, hi = 0.
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This gives that (B − Tz )f is in [z − w]. On the other hand, for each F in H 2 (T2 ), h(B − Tz )f, z(z − w)F i = h−Tz f, z(z − w)F i = hf, (z − w)F i = 0. Thus (B − Tz )f is in the wandering subspace L{0} g = L[z−w] . By Theorem 2.3, there is 2 a function g in H (T) such that (B − Tz )f = zPH g − wg(w). So Bf = zPH g − wg(w) + zf. Noting that Bf, f, and PH g are in H and hence they are symmetric functions of z and w, we also have Bf = wPH g − zg(z) + wf. Taking the difference of the above equalities gives 0 = (z − w)PH g + zg(z) − wg(w) + (z − w)f. Hence we have zg(z) − wg(w) . z−w Since f is in H, Theorem 9 in [9] gives that g is also in the Dirichlet space D. This completes the proof. f = −PH g −
˜ f˜(w) Clearly, for each function f˜(z) in D, f (z)− is in H. For a function q in the invariant z−w ∗ −1 subspace M of B, let f = (BM B) q. By (3.9) in Step 7, there is a function g in H 2 (T) ∩ D such that
f (z, w) = −PH g −
zg(z) − wg(w) . z−w
Step 8. Let M⊥ denote the orthogonal complement of M in H. Then ∗ ∗ h(BM B)−1 q, qi + h(BBM )q, qi − 2hq, qi g(z) − g(w) 2 g(z) − g(w) 2 1 kPM⊥ BPM⊥ [ ]k − kPM⊥ [ ]k . = 2 z−w z−w
The above identity may be interesting in its own and immediately gives ∗ ∗ B)−1 q, qi + h(BBM )q, qi − 2hq, qi ≤ 0 h(BM
(3.10)
since kPM⊥ BPM⊥ k ≤ 1. (3.10) is the key ingredient in the new proof of the Beurling type theorem of Aleman, Richter and Sundberg in [15].
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∗ ∗ To prove the above identity, we need to identify (BM B)−1 q, q and BM q. Easy calculations give
zg(z) − wg(w) )] z−w z 2 g(z) − zwg(w) −BPH g − PH [ ] z−w z 2 g(z) − w2 g(w) + (w2 − zw)g(w) ] −BPH g − PH [ z−w z 2 g(z) − w2 g(w) z 2 g(z) − w2 g(w) −BPH g − + PH (wg(w)) (by ∈ H) z−w z−w z 2 g(z) − w2 g(w) −BPH g − + BPH g (by wPH⊥ g ∈ [z − w]) z−w z 2 g(z) − w2 g(w) − , z−w
Bf = PH [Tz (−PH g − = = = = =
and hence we have ∗ q = BM Bf
z 2 g(z) − w2 g(w) ] z−w zg(z) − w2 z¯g(w) −PM PH 2 (T2 ) [ ] z−w zg(z) − wg(w) + w(1 − w¯ z )g(w) −PM PH 2 (T2 ) [ ] z−w zg(z) − wg(w) −PM [ ] − PM PH 2 (T2 ) (¯ z wg(w)) z−w zg(z) − wg(w) −PM [ ] z−w zg(z) − wg(w) PM [−PH g − + PH g] z−w PM [f + PH g] (by (3.9)) f + PM g zg(z) − wg(w) . −PM⊥ g − z−w
= PM Tz∗ [− = = = = = = = =
The ninth equality follows from that f is in M. Since M is an invariant subspace of B, M⊥ is an invariant subspace of B ∗ . Thus PM B ∗ (PM⊥ g) = 0.
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So we have ∗ BM q = PM B ∗ q
= PM [−B ∗ PM⊥ g − B ∗ = −PM [
zg(z) − wg(w) ] z−w
g(z) − g(w) ], z−w
to get ∗ ∗ ∗ hBBM q, qi = hBM q, BM qi g(z) − g(w) 2 = kPM [ ]k , z−w
= = = = = =
∗ h(BM B)−1 q, qi = hf, qi zg(z) − wg(w) ]i hf, −PM [ z−w zg(z) − wg(w) hf, −[ ]i (by f ∈ M) z−w zg(z) − wg(w) zg(z) − wg(w) h−PH g − , −[ ]i z−w z−w zg(z) − wg(w) zg(z) − wg(w) 2 ]i + k[ ]k hPH g, [ z−w z−w zg(z) − wg(w) zg(z) − wg(w) 2 zg(z) − wg(w) hg, [ ]i + k[ ]k (by ∈ H) z−w z−w z−w zg(z) − wg(w) 2 kgk2 + k[ ]k , z−w
and zg(z) − wg(w) ]i z−w zg(z) − wg(w) hq, −[ ]i z−w zg(z) − wg(w) zg(z) − wg(w) h−PM⊥ g − , −[ ]i z−w z−w zg(z) − wg(w) zg(z) − wg(w) 2 hPM⊥ g, [ ]i + k[ ]k z−w z−w zg(z) − wg(w) zg(z) − wg(w) zg(z) − wg(w) 2 −hPM⊥ [ ], [ ]i + k[ ]k z−w z−w z−w zg(z) − wg(w) 2 zg(z) − wg(w) 2 −kPM⊥ [ ]k + k[ ]k z−w z−w zg(z) − wg(w) 2 −kPM⊥ gk2 + k[ ]k . z−w
hq, qi = hq, −PM [ = = = = = =
BEURLING TYPE THEOREM VIA HARDY SPACE OF BIDISK
15
The fifth equality and the last equality follow from 0 = PM ⊥ f zg(z) − wg(w) ] z−w zg(z) − wg(w) = −PM⊥ g − PM⊥ [ ]. z−w = PM⊥ [−PH g −
Therefore ∗ ∗ B)−1 q, qi + h(BBM )q, qi − 2hq, qi h(BM zg(z) − wg(w) 2 g(z) − g(w) 2 = kgk2 + k[ ]k + kPM [ ]k z−w z−w zg(z) − wg(w) 2 +2kPM⊥ gk2 − 2k[ ]k z−w g(z) − g(w) 2 g(z) − g(w) 2 = −k[ ]k + kPM [ ]k + 2kPM⊥ gk2 z−w z−w g(z) − g(w) 2 ]k . = 2kPM⊥ gk2 − kPM⊥ [ z−w The second equality follows from
zg(z) − wg(w) 2 g(z) − g(w) 2 ]k = kgk2 + k[ ]k . z−w z−w The above identity comes directly from the computation by using the Fourier series }∞ form an orthonormal basis expansion of the function g(z) and the fact that { p√n (z,w) n+1 n=0 of H. Since g(z) − g(w) zg(z) − wg(w) B[ ] = [ ] − PH g z−w z−w = −f − 2PH g, k[
and f is in M, we have PM⊥ B[
g(z) − g(w) ] = −2PM⊥ g. z−w
Thus 1 g(z) − g(w) 2 kPM⊥ B[ ]k 4 z−w 1 g(z) − g(w) 2 = kPM⊥ BPM⊥ [ ]k . 4 z−w
kPM⊥ gk2 =
The last equality follows from that BPM [ g(z)−g(w) ] is in M. So we obtain z−w ∗ ∗ h(BM B)−1 q, qi + h(BBM )q, qi − 2hq, qi 1 g(z) − g(w) 2 g(z) − g(w) 2 = kPM⊥ BPM⊥ [ ]k − kPM⊥ [ ]k . 2 z−w z−w
16
SUN AND ZHENG
Step 9. Finally we show that N = {0}. P P 2 For each q in N , by Step 1, we write q = n=0 z n un with kqk2 = ∞ n=0 kun k , for ∗ k un in M00 . Let qk = (BM ) q. Step 8 gives that ∗ ∗ )qk , qk i − 2hqk , qk i ≤ 0. B)−1 qk , qk i + h(BBM h(BM
By Steps 5 and 6, we have kuk−1 k2 − kuk k2 ≤ 0, to get that the sequence {kuk k2 } of nonnegative numbers increases, but is summable. Hence kuk k2 = 0 for k ≥ 0. This implies that q = 0, to complete the proof. ACKNOWLEDGMENTS: We thank R. Rochberg for his drawing our attention to his papers [7] and [8] with S. Ferguson. R EFERENCES [1] A. Aleman, S. Richter and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math., 177 (1996), no. 2, 275–310. [2] C. Apostol, H. Bercovici, C. Foias, and C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I, J. Funct. Anal., 63 (1985), no. 3, 369–404. [3] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1948), 239-255. [4] R. Douglas and V. Paulsen, Hilbert Modules over function algebras, Pitman Research Notes in Mathematics, Series 217, Lonman Group UK Limited, 1989. [5] R. Douglas and R. Yang, Operator theory in the Hardy space over the bidisk. I, Integral Equations Operator Theory, 38 (2000), 207–221. [6] P. Duren and A. Schuster, Bergman spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, Providence, RI, 2004. [7] S. Ferguson and R. Rochberg, Higher order Hilbert-Schmidt Hankel forms and tensors of analytic kernels, Math. Scand., 96 (2005), no. 1, 117–146. [8] S. Ferguson and R. Rochberg, Description of certain quotient Hilbert modules, Operator theory, 20, 93–109, Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006. [9] K. Guo, S. Sun, D. Zheng and C. Zhong, Multiplication Operators on the Bergman Space via the Hardy Space of the bidisk, to appear in J. Reine Angew. Math. [10] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000. [11] S. McCullough and S. Richter, Bergman-type reproducing kernels, contractive divisors, and dilations, J. Funct. Anal, 190 (2002), 447–480. [12] A. Olofsson, Wandering subspace theorems, Integral Equations Operator Theory, 51 (2005), 395– 409. [13] B. Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. [14] S. Richter, On invariant subspaces of multiplication operators on Banach spaces of analytic functions, Ph. Dissertation, Univ. of Michigan, 1986. [15] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math., 531 (2001), 147–189. [16] S. Sun and D. Yu, Super-isometrically dilatable operators, Science in China, 32(12), 1989, 14471457. [17] S. Sun, D. Zheng and C. Zhong, Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the Hardy space of the bidisk, to appear in Canadian J. Math.
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[18] S. Sun, D. Zheng and C. Zhong, Multiplication Operators on the Bergman Space and Weighted shifts, to appear in J. Operator Theory. [19] R. Yang, Operator theory in the Hardy space over the bidisk. II, Integral Equations Operator Theory, 42 (2002), 99–124. [20] R. Yang, Operator theory in the Hardy space over the bidisk. III, J. Funct. Anal., 186 (2001), 521– 545. [21] R. Yang, The core operator and congruent submodules, J. Funct. Anal., 228 (2005), 469–489. I NSTITUTE OF M ATHEMATICS , J IAXING U NIVERSITY, J IAXING , Z HEJIANG , 314001, P. R. C HINA E-mail address:
[email protected] D EPARTMENT OF M ATHEMATICS , VANDERBILT U NIVERSITY, NASHVILLE , TN 37240 USA E-mail address:
[email protected]
