Integr. Equ. Oper. Theory (2018) 90:13 https://doi.org/10.1007/s00020-018-2435-9 Published online March 1, 2018 c Springer International Publishing AG,  part of Springer Nature 2018

Integral Equations and Operator Theory

Essential Normality of Homogenous Quotient Modules Over the Polydisc: Distinguished Variety Case Penghui Wang and Chong Zhao Abstract. In the present paper, we investigate the essential normality of quotient modules over the polydisc. Let I be a homogeneous ideal in C[z1 , . . . , zd ], we show that if the homogeneous quotient module [I]⊥ of H 2 (Dd ) is essentially normal, then dimC Z(I) ≤ 1. It is shown that if Z(I) is distinguished, then [I]⊥ is (1, ∞)-essentially normal, i.e. the [Sz∗i , Szj ]’s are not necessarily trace class operators but indeed belong to the interpolation ideal L(1,∞) , see the monograph “Noncommutative Geometry” of Connes. This result leads to the answer to the polydisc version of Arveson–Douglas problem. Moreover, we study the boundary representation of [I]⊥ . Mathematics Subject Classification. Primary 47A13; Secondary 46H25. Keywords. Essential normality, Hardy space over polydisc, Quotient module, Boundary representation.

1. Introduction Let Dd = {(z1 , . . . , zd ) : |zi | < 1, i = 1, . . . , d} represent the unit polydisc in Cd . The present paper is devoted to investigate the essential normality of homogenous quotient modules of H 2 (Dd ) when d ≥ 3, and the associated boundary representations. Hilbert modules [13] are the natural framework to study multi-variable operator theory. Given a tuple T = (T1 , . . . , Td ) of commuting operators on a Hilbert space H, one can naturally make H into a Hilbert module over the polynomial ring C[z1 , . . . , zd ], with the module action defined by p · x = p(T1 , . . . , Td )x,

p ∈ C[z1 , . . . , zd ],

x ∈ H.

One of the fundamental problems in the Hilbert module theory is to study the essential commutativity of the C ∗ -algebra C ∗ (T ) generated by Id, T1 , . . . , Td , This work was partially supported by NSFC (No. 11471189), NSFC (No. 11501329), Shandong Province Natural Science Foundation ZR2014AQ009 and The Fundamental Research Funds of Shandong University 2015GN017.

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where Id is the identity on H, which arises from the well-known BDF-theory [6]. When C ∗ (T ) is essentially commutative, H is said to be essentially normal. A long-standing open problem, due to Arveson [4], states that Conjecture: All graded submodules of the d-shift module over the unit ball are essentially normal. In this paper, we focus on essential normality of analytic Hilbert modules over the polydisc. As the Arveson conjecture over the unit ball has been extensively studied by many mathematicians such as [4,5,11,12,15,20–22,26], the polydisc case is quite different as well as interesting as will be seen later. Let H 2 (Dd ) be the Hardy space over the polydisc, it is not difficult to see that nontrivial submodules are not essentially normal provided d ≥ 2. Hence on the unit polydisc, essential normality of quotient modules rather than that of submodules is worth of investigation. Let M be a submodule of H 2 (Dd ). Denote by N = M⊥ and let Szi = PN Mzi |N the compression of the multiplication operator Mzi on N . Then N is naturally equipped with a C[z1 , . . . , zd ]-module structure by the tuple (Sz1 , . . . , Szd ), and called a quotient module of H 2 (Dd ). The first result on essential normality in the polydisc case was due to Douglas and Misra [14], who exhibited both essentially normal and nonessentially normal quotient modules. By restricting the Hardy space to the variety, Clark [7] identified the quotient module generated by {Bi (zi ) − Bj (zj ); i, j = 1, . . . , d} for finite Blaschke products Bi (zi ) with a kind of Bergman space on its variety, which therefore is essentially normal. The essential normality of (quasi-) homogenous quotient modules of H 2 (D2 ) was completely characterized by Guo and the first author [23,24], and the p-essential normality was concerned in [25]. To sum up, the answer to the polydisc version of Arveson’s conjecture is totally different from that on the unit ball. In this paper, we focus on quotient modules of H 2 (Dd ) with homogeneous distinguished variety, which was introduced by Alger and McCarthy in the bidisc. According to [1], we call an algebraic variety V distinguished if V ∩ ∂Dd ⊂ Td . Previous work [23,24,27,28] suggests that essential normality of quotient modules of H 2 (Dd ) is intimately related to distinguished varieties. To introduce our main result we need the concept of interpolation ideals L(p,∞) [8] as mentioned in the abstract. Let T be a compact operator and |T | = (T ∗ T )1/2 be its absolute value. Let μ0 (T ) ≥ μ1 (T ) ≥ · · · be the sequence of eigenvalues of |T | repeated as many times as their multiplicity. Set σN (T ) =

N −1  n=0

μn (T ), N = 1, 2, . . . .

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For 1 < p < ∞ we say T ∈ L(p,∞) if it holds σN (T ) = O(N 1−1/p ), and T ∈ L(1,∞) if σN (T ) = O(log N ). The following theorem is our main result. Theorem 1.1. Suppose I ⊂ C[z1 , . . . , zd ] is a homogeneous ideal such that Z(I) ∩ ∂Dd ⊂ Td , then the quotient module M = [I]⊥ is (1, ∞)-essentially normal. In fact, in a subsequent paper [29] we give a complete criteria for the essential normality of homogenous quotient modules over H 2 (Dd ), which shows that the distinguished components are the “main part” of essentially normal quotient modules. Let J ⊂ C[z1 , . . . , zd ] denote the prime ideal with the variety Z(J) = {(λ, . . . , λ) : λ ∈ C} . We begin our proof of Theorem 1.1 by showing that [J N ]⊥ is essentially normal, where the key tool is the restriction map r : H 2 (Dd ) → Hol(D) which is defined as r(f )(z) = f (z, . . . z) and generated from [18]. It is worth to point out that our proof of essential normality of [J N ]⊥ is also valid on weighted Bergman quotient modules. Moveover, we give the following trace formula on [J N ]⊥ . Theorem 1.2. Let N = [J N ]⊥ , then for f1 , f2 ∈ C[z1 , . . . , zd ],   d+N −2 ∗ (rf2 ) , (rf1 ) L2a (D) . tr [Sf1 , Sf2 ] = d−1 In the bidisk case, such a trace formula was given in [25]. However, the high dimensional case is much more complicated. The theory of boundary representation of C ∗ -algebra was developed by Arveson [2,3]. Recently, Kennedy and Shalit [26] showed that for the dshift module on the unit ball, the essential normality is closely related to the boundary representation. In this paper we also make some discussion on the boundary representation of the distinguished homogenous quotient modules of H 2 (Dd ). The present paper is organized as follows. In Sect. 2, we prove the essential normality of [I]⊥ provided that the zero variety is simple. In Sect. 3 the main result is proved. In Sect. 4, we study the boundary representation.

2. Restriction of H 2 (Dd ) to the Simple Homogenous Distinguished Variety Let Vθ be the simple homogenous distinguished variety in Dd through (θ1 , . . . , θd ), namely Vθ = {(θ1 z, . . . , θd z) : z ∈ D}

(2.1)

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for some fixed θ = (θ1 , . . . , θd ) ∈ Td . Denote by Jθ = I(Vθ ) the prime ideal of C[z1 , . . . , zd ] with variety Vθ . This section is devoted to prove the 1-essential normality of the quotient modules Nθ = [JθN ]⊥ of H 2 (Dd ), where N ≥ 1 is an arbitrary integer. Without loss of generality, we assume θ = (1, . . . , 1) throughout this section. As in [19], we define the restriction map r : H 2 (Dd ) → Hol (D) such that (rf )(z) = f (z, . . . , z) for z ∈ D, then [J] = ker r. √ 2π Denote by ω = e d −1 the d-th primitive root of unit, and define linear polynomials wi (z) :=

d 

ω (i−1)(j−1) zj , i = 1, . . . , d.

j=1

Set w = (w1 , . . . , wd ). Obviously wi , wl = 0 for i = l, and ||wi ||2 = d for i = 1, . . . , d. Evidently r maps w1 to d · z and wi to 0 for i = 2, . . . , d. Simple division shows that J is precisely the ideal generated by w2 , . . . , wd , and J n is generated by Jn := span {wα : |α| = n, α1 = 0} . Write ∂ = ( ∂z∂ 1 , . . . , ∂z∂d ). Differentiation by parts shows r∂ α f = 0 for all f ∈ J n and |α| < n. Direct computation gives r∂ α Mwi = r[wi (∂)z α ](∂), i = 2, . . . , d, and r∂ α Mw1 = r[w1 (∂)z α ](∂) + r(w1 )r∂ α . Then for p ∈ C[z1 , . . . , zd ] we have rp(∂)Mwβ = r[wβ (∂)p](∂)

(2.2)

and rp(∂)Mh Mωβ = r[h(∂)wβ (∂)p](∂) + r(h)[wβ (∂)p](∂) (2.3) when β1 = 0 and h ∈ C[z1 , . . . , zd ] is linear. It is routine to check that the operator ∂ α∗ maps constant 1 to α!z α . Therefore given any g ∈ C[z1 , . . . , zd ], we can find  a unique pg ∈ C[z1 , . . . , zd ] making pg (∂)∗ 1 = g. Set rg = rpg (∂). For g = |γ|=n cγ z γ one can verify  zγ (2.4) pg = c¯γ . γ! |γ|=n  For f = |γ|=n dγ z γ we have  dγ c¯γ  ∂γ zγ = f (∂)pg = dγ c¯γ = f, g . (2.5) γ! |γ|=n

|γ|=|α|

Then by (2.2) rg Mf = rpg (∂)Mf = r[f (∂)pg ](∂) = f, g r, ∀f, g ∈ Jn .

(2.6)

The following proposition generalizes Proposition 1 of [19] to the multivariable case.

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Proposition 2.1. Let h ∈ C[z1 , . . . , zd ], then h ∈ J N if and only if rwα h = 0 whenever |α| < N and α1 = 0. Consequently   N = span (ker rwα )⊥ : |α| < N, α1 = 0 . Proof. By previous discussion we have rwα [J N ] = 0 for |α| < N . Conversely assume h ∈ C[z1 , . . . , zd ] and rwα h = 0 whenever |α| < N and α1 = 0, and we shall prove h ∈ J N by induction. If N = 1, then since rh = 0 we have h ∈ [w2 , . . . , wd ] = [J], and the proposition is proved in this case. Assume the proposition is proved for N = 1, . . . , n, and suppose rwα h = 0 whenever |α| < n + 1 and α1 = 0. By the assumption for induction, h ∈ J n . Denote Jn = {α ∈ Zd+ : |α| = n and α1 = 0}, and then we can write  h = β∈Jn wβ hβ where each hβ is a polynomial. For α ∈ Jn we have   rwα Mwβ hβ = wβ , wα rhβ = 0, (2.7) rwα h = β∈Jn

β∈Jn

where the second equality is deduced by (2.3). Since each homogeneous polynomial of degree n is a linear combination of {wα : |α| = n}, the later forms a linearly independent system. This implies the invertibility of the matrix (wβ , wα )α,β∈Jn , and then each rhβ should be zero by (2.7). Therefore hβ ∈ J and h ∈ J n+1 . By induction, the proposition holds for all natural numbers N .  Remark 2.2. By linearity, Proposition 2.1 actually states that h ∈ J n if and only if rg h = 0 for every g ∈ J0 ∪ J1 . . . ∪ Jn−1 . Similar to [19] for λ ∈ D we write Kλ⊗ (z) =

d 

Kλ (zi ), z ∈ Dd

i=1

and kλ⊗ (z) =

d 

kλ (zi ), z ∈ Dd ,

i=1

where kλ =

Kλ ||Kλ || 2

is the normalized reproducing kernel for H 2 (D).

For f ∈ H (Dd ) and g ∈ C[z1 , . . . , zd ], we have rg f (λ) = r[pg (∂)f ](λ) = pg (∂)f (λ, . . . , λ) = pg (∂)f, Kλ⊗ = f, pg (∂)∗ Kλ⊗ Therefore (ker rg )⊥ = span {pg (∂)∗ Kλ⊗ : λ ∈ D}.  By equality (2.4), for wα = |γ|=n cγ z γ ∈ Jn we have

(2.8)

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pwα (∂)∗ Kλ⊗ (z) =

|γ|=n



=

cγ

1 γ∗ ⊗ ∂ Kλ (z) γ!



|γ|=n

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cγ

d 

γi

γi +1

z Kλ (zi )

i=1

= wα (ψλ (z))Kλ⊗ (z), zd 1 where ψλ (z) := 1−zλz , λ ∈ D. Obviously ¯ 1 , . . . , 1−λz ¯ 

(2.9)

d

r(wi ◦ ψλ )(μ) = 0, ∀μ ∈ D, i = 2, . . . , d. Therefore wi ◦ ψλ ∈ [J] and wα ◦ ψλ ∈ [J n ]. By linearity, g ◦ ψλ ∈ [J n ] whenever g ∈ Jn . Combining this fact with (2.8) and (2.9) we obtain the following lemma. Lemma 2.3. (ker rg )⊥ = span {g ◦ ψλ · Kλ⊗ : λ ∈ D} ⊂ [J n ] for g ∈ Jn . Denote Hn = span {(ker rg )⊥ : g ∈ Jn }, then by Proposition 2.1 and its remark, N = span {Hn : n = 0, 1, . . . , N − 1} . Moreover, we have the following proposition on the structure of N . Proposition 2.4. Hm ⊥Hn whenever m = n. As a consequence, N =

N −1 n=0 Hn . Proof. By the previous lemma Hn ⊂ [J n ], and since [J n ]⊥ = span {Hm : 0 ≤ m ≤ n − 1} for each n, we have Hn ⊥Hm provided m < n.



For λ ∈ D, denote the Mobius transform on D by   λ − z1 λ − zd d ϕλ : z → ¯ 1 , . . . , 1 − λz ¯ d ,z ∈ D . 1 − λz d

It is well-known that the linear map V : f → f ◦ ϕλ · kλ⊗ is a unitary on H 2 (Dd ), such that V 2 = Id. Suppose wα , wβ ∈ Jn , λ ∈ D, then by (2.9) we have pwα (∂)∗ kλ⊗ , pwβ (∂)∗ kλ⊗ = wα ◦ ψλ · kλ⊗ , wβ ◦ ψλ · kλ⊗ = wα ◦ ψλ ◦ ϕλ , wβ ◦ ψλ ◦ ϕλ      λ − z1 λ − z1 λ − zd λ − zd α β = w ,..., ,..., ,w 1 − |λ|2 1 − |λ|2 1 − |λ|2 1 − |λ|2   2 −2n α β w (λ − z1 , . . . , λ − zd ), w (λ − z1 , . . . , λ − zd ) = (1 − |λ| )   = (1 − |λ|2 )−2n wα (−z1 , . . . , −zd ), wβ (−z1 , . . . , −zd ) = (1 − |λ|2 )−2n wα , wβ . This induces   pf (∂)∗ Kλ⊗ , pg (∂)∗ Kλ⊗ = (1 − |λ|2 )−d−2n f, g , ∀λ ∈ D

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whenever f, g ∈ Jn . Notice that the function   F (λ, μ) = pf (∂)∗ Kλ⊗ , pg (∂)∗ Kμ⊗ is analytic on μ and co-analytic on λ. By Proposition 1 of [16],   pf (∂)∗ Kλ⊗ , pg (∂)∗ Kμ⊗ = (1 − λ, μ )−d−2n f, g , ∀λ, μ ∈ D.

(2.10)

Next, let Ad+2n−2 denote the weighted Bergman space on the unit disc (d+2n−2) with the reproducing kernel Kλ . For homogeneous g ∈ Jn of ||g|| = 1, by (2.10) we can define an isometry Rg : (ker rg )⊥ → Ad+2n−2 , satisfying (d+2n−2) Rg (pg (∂)∗ Kλ⊗ ) = Kλ . Then it holds for λ, μ ∈ D that   ∗ ⊗ rg [pg (∂) Kλ ](μ) = pg (∂)pg (∂)∗ Kλ⊗ , Kμ⊗   = pg (∂)∗ Kλ⊗ , pg (∂)∗ Kμ⊗ ¯ −d−2n (by 2.10) = (1 − λμ)   (d+2n−2) = Kλ , Kμ(d+2n−2) = Rg [pg (∂)∗ Kλ⊗ ](μ), which implies that Rg is actually the restriction of rg on (ker rg )⊥ . Namely we have (d+2n−2) rg [pg (∂)∗ Kλ⊗ ] = Kλ , ∀ λ ∈ D, (2.11) ⊥ and that rg is an isometry on (ker rg ) . For each subspace Jn ⊂ H 2 (Dd ) we choose an orthonormal basis    d+n−2 (n) Bn = fj : j = 1, . . . , , d−2 and denote    N −1 N +d−2 B := Bn = gj : j = 1, . . . , . d−1 n=0 By (2.10), (ker rf (n) )⊥ is orthogonal to (ker rf (n) )⊥ whenever i = j, j

i ⊥ and so Hn = g∈Bn (ker rg ) . Then by Proposition 2.4 we have N =

N −1

⊥ g∈B (ker rg ) . Define a linear mapping U : N → g∈Bn Ad+2n−2... n=0 by U h = (rg h)g∈B . (2.12) ⊥ Since each rg is an isometry on (ker rg ) , U is unitary. Proposition 2.5. For g ∈ Bn and f ∈ C[z1 , . . . , zd ], it holds that rg (f h) = r(f )rg (h), ∀h ∈ (ker rg )⊥ . Proof. For λ ∈ D we have   g ◦ ψλ · f kλ⊗ , g ◦ ψλ · kλ⊗ = f ◦ ϕλ · g ◦ ψλ ◦ ϕλ , g ◦ ψλ ◦ ϕλ      λ − z1 λ − z1 λ − zd λ − zd = f ◦ ϕλ · g , . . . , , . . . , , g 1 − |λ|2 1 − |λ|2 1 − |λ|2 1 − |λ|2 = (1 − |λ|2 )−2n f ◦ ϕλ · g(λ − z1 , . . . , λ − zd ), g(λ − z1 , . . . , λ − zd )

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= (1 − |λ|2 )−2n f ◦ ϕλ · g(−z1 , . . . , −zd ), g(−z1 , . . . , −zd ) = (1 − |λ|2 )−2n f ◦ ϕλ · g, g = (1 − |λ|2 )−2n f ◦ ϕλ (0)g, g = (1 − |λ|2 )−2n r(f )(λ), and then by equality (2.11)   (d+2n−2) (d+2n−2) rg (f rg∗ Kλ ), Kλ   = rg (g ◦ ψλ · f Kλ⊗ ), rg (g ◦ ψλ · Kλ⊗ )   = g ◦ ψλ · f Kλ⊗ , g ◦ ψλ · Kλ⊗ = (1 − |λ|2 )−d−2n r(f )(λ)   (d+2n−2) (d+2n−2) . = r(f )Kλ , Kλ Again by Proposition 1 in [16],    (d+2n−2) (d+2n−2) rg f rg∗ Kλ , Kμ(d+2n−2) = r(f )Kλ , Kμ(d+2n−2) whenever λ, μ ∈ D, and hence  (d+2n−2) (d+2n−2) = r(f )rg rg∗ Kλ , ∀ λ ∈ D. rg f rg∗ Kλ Since {rg∗ Kλ proved.

(d+2n−2)

: λ ∈ D} is dense in (ker rg )⊥ , the proposition is 

Denote by Pg (g ∈ B) the orthogonal projection onto (ker rg )⊥ , and  (f,g) := Pf Szi Pg . Then we have Szi = f,g∈B Ai . By Proposition 2.5,

(f,g) Ai (g,g) Ai

is unitarily equivalent to Mz acting on Ad+2n−2 , which is 1-essentially (g,g)∗ (g,g) normal with tr [Mz∗ , Mz ] = 1. Therefore [Ai , Ai ] belongs to the trace (f,g) class and is of trace 1. Next proposition concerns the compactness of Ai where f = g. Proposition 2.6. Given f ∈ Bm and g ∈ Bn , then (f,g)

(a) Ai = 0 if m ≤ n and f = g; (f,g) (b) Ai ∈ L2 if m > n. Proof. For λ ∈ D we have   g ◦ ψλ · zi kλ⊗ , f ◦ ψλ · kλ⊗   λ − zi = · g ◦ ψ ◦ ϕ , f ◦ ψ ◦ ϕ λ λ λ λ ¯ i 1 − λz      λ − zi λ − z1 λ − z1 λ − zd λ − zd = g , . . . , , . . . , , f ¯ i 1 − |λ|2 1 − |λ|2 1 − |λ|2 1 − |λ|2 1 − λz   λ − zi = (1 − |λ|2 )−n−m g(λ − z , . . . , λ − z ), f (λ − z , . . . , λ − z ) 1 d 1 d ¯ i 1 − λz   λ − zi = (1 − |λ|2 )−n−m ¯ i g(−z1 , . . . , −zd ), f (−z1 , . . . , −zd ) 1 − λz

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2 −n−m

= (1 − |λ| )



λ − zi ¯ i g, f 1 − λz

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 .

(2.13)

In the cases m ≤ n and f ⊥g and deg(f ) ≤ deg(g). Notice that ∞  λ − zi ¯ i )n g. g = (λ − z ) (λz i ¯ i 1 − λz n=0   λ−zi Since zin g ⊥ f for any n ≥ 0, we have 1− g, f = 0 and therefore ¯ λzi   (d+2n−2) ∗ (d+2m−2) = 0, zi rg∗ kλ , rf kλ

inducing

  (d+2n−2) ∗ (d+2m−2) zi rg∗ kλ = 0, ∀λ, μ ∈ D , rf kμ

by Proposition 1 of [16]. This equality together with Lemma 2.3 shows (f,g)

Ai

= Pf Mzi Pg = 0

in these cases. In the case m > n, by the expansion ∞  λ − zi 2 ¯ k−1 z k = λ − (1 − |λ| ) λ i ¯ i 1 − λz k=1

we find by equality (2.13)   (d+2n−2) ∗ (d+2m−2) ¯ m−n−1 (1 − |λ|2 )1−n−m z m−n g, f = −λ zi rg∗ kλ , rf kλ i and therefore     (d+2n−2) ∗ (d+2m−2) ¯ m−n−1 (1 − |λ|2 )1−d−n−m z m−n g, f . = −λ zi rg∗ Kλ , rf Kλ i An application of Proposition 1 of [16] shows     (d+2n−2) ∗ (d+2m−2) ¯ m−n−1 (1 − μλ) ¯ 1−d−n−m z m−n g, f . zi rg∗ Kλ = −λ , rf Kμ i (f,g) (f,g) Denote by A˜i := rf Ai rg∗ : Ad+2n−2 → Ad+2m−2 , then we have   (f,g) (d+2n−2) ¯ m−n−1 (1 − z λ) ¯ 1−d−n−m z m−n g, f . A˜i Kλ = −λ (2.14) i

¯ k (k ≥ m − n − 1) we find By comparing the coefficients of λ      d + 2n + k − 1 k−m+n+1  m−n (f,g) d + 2n + k − 1 A˜i zi z g, f , zk = − d+m+n−2 d + 2n − 1 and therefore

  −1  m−n  d + 2n + k − 1 d + 2n + k − 1 (f,g) A˜i z k = − zi g, f z k−m+n+1 . d+m+n−2 d + 2n − 1

Then from ||z n−m+k+1 ||2d+2m−2 = ||z k ||2d+2n−2



 −1 d + 2n + k − 1 d + n + m + k d + 2n − 1 d + 2m − 1

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we obtain (f,g) A˜i

 m−n  zk z k−m+n+1 = z g, f a (k) , m,n i ||z k ||d+2n−2 ||z k−m+n+1 ||d+2m−2

(2.15)

where

  −1/2  −1/2 d + 2n + k − 1 d + 2n + k − 1 d+n+m+k am,n (k) = − . d+m+n−2 d + 2n − 1 d + 2m − 1 (f,g) Since am,n (k) = O(k −1 ) as k → +∞, we find A˜i ∈ L2 .



Corollary 2.7. is unitarily equivalent  There is an isometry S ∈ B(N ) which  +d−2 -shift and compact operators Ki ∈ L(1,∞) such that Szi = S−Ki . to a Nd−1 Moreover, there is a constant c such that if h ∈ N is homogeneous and of degree k, then ||Ki h|| ≤ c(k + 1)−1 ||h||. As a result, Sz∗i − Sz∗j is compact for 1 ≤ i, j ≤ d. Proof. By Proposition 2.5, for each f ∈ Bm (f,f ) A˜i

zk ||z k ||d+2m−2

z k+1 ||z k+1 ||d+2m−2 k k+1 ||z ||d+2m−2 ||z ||d+2m−2  1/2 k+1 z k+1 = , k+1 d + 2m + k ||z ||d+2m−2 =

and then zk z k+1 (f,f ) A˜i − ||z k ||d+2m−2 ||z k+1 ||d+2m−2   1/2 k+1 z k+1 = −1 k+1 d + 2m + k ||z ||d+2m−2 =−

z k+1 d + 2m − 1  1/2  ||z k+1 ||  d+2m−2 k+1 (d + 2m + k) 1 + d+2m+k

(2.16)

N −1 Denote by S˜ the isometry on m=0 f ∈Bm rf N that maps each monomial zk ||z k ||d+2m−2

∈ rf N to

z k+1 , ||z k+1 ||d+2m−2

then by equalities (2.15) and (2.16),

˜ i = S˜ − U Sz U ∗ ∈ L(1,∞) . K i ∗˜ ˜ iU . The corollary holds for S = U SU and Ki = U ∗ K



Remark 2.8. In the case θ = (1, . . . , 1), define linear transform Lθ : Cd → Cd , (z1 , . . . , zd ) → (θ1 z1 , . . . , θd zd ). Then Uθ : H 2 (Dd ) → H 2 (Dd ), f → f ◦ Lθ maps Nθ = [JθN ]⊥ unitarily onto N = [J N ]⊥ . By the corollary, the is an isometry S ∈ B(N ) and compact operators Ki such that Szi = ui S − Ki for 1 ≤ i ≤ d. Therefore Szi ◦ Lθ = ui S ◦ Lθ − Ki ◦ Lθ .

(2.17)

Denote by Sθ = S ◦ Lθ , then Sθ is an isometry on Nθ , and Ki ◦ Lθ is compact. Then equality (2.17) actually shows that the compression of Mzi on Nθ is the sum of the isometry ui Sθ and some compact operator.

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⊥ Corollary 2.9. PN Mzi PN ∈ L(2,∞) .

Proof. By Corollary 2.7, Szi = S − Ki where S is an isom etry and there is a constant c such that ||Ki h|| ≤ c(deg h)−1 ||h|| for homogeneous h ∈ N . Suppose h ∈ N is homogeneous, then  ⊥  ⊥ Mzi PN h||2 = PN Mzi h, Mzi h ||PN = ||Mzi h||2 − Szi h, Szi h = ||h||2 − (S − Ki )h, S − Ki h = Sh, Ki h + Ki h, Sh − Ki h, Ki h ≤ 2||Ki h|| ≤ 2c(deg h)−1 ||h||. ⊥ Therefore we conclude PN Mzi PN ∈ L(2,∞) as desired.



Theorem 2.10. N is 1-essentially normal. Proof. Let U be the unitary defined in (2.12). By definition we need to prove for i, j = 1, . . . , d that  (h,f )∗ ˜(h,g) (f,h) (g,h)∗ A˜i Aj − A˜j A˜i ∈ L1 . (2.18) U [Sz∗i , Szj ]U ∗ = f,h,g∈B

By Proposition 2.6, the terms for which h = f and h = g must belong to L1 . Then it is sufficient to prove for f, g ∈ B that (f,f )∗ ˜(f,g) (f,g) (g,g)∗ A˜j Ai − A˜i A˜j ∈ L1

(2.19)

and (g,f )∗ ˜(g,g) (f,f ) (g,f )∗ A˜i Aj − A˜j A˜i ∈ L1 .

Suppose f ∈ Bm and g ∈ Bn . By symmetry we can assume m ≥ n, and then by Proposition 2.6 (g,f )∗ ˜(g,g) (f,f ) (g,f )∗ A˜i Aj − A˜j A˜i =0

if f = g, and (f,f )∗ ˜(f,g) (f,g) (g,g)∗ A˜j Ai − A˜i A˜j =0 (f,f )

if m = n and f = g. When f = g then A˜i is the shift Mz acting on (f,f )∗ ˜(f,f ) 1 ˜ Ad+2n−2 , and therefore [Ai , Aj ] ∈ L . Then it suffices to prove (2.19) for m > n. It is routine to check  k z k−1 z k (f,f )∗ A˜i = , ∀k ∈ Z+ k ||z || d + 2m + k − 1 ||z k−1 || and

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(g,g)∗ A˜j

zk = ||z k ||



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z k−1 k , ∀k ∈ Z+ . d + 2n + k − 1 ||z k−1 ||

Then by (2.15) zk (f,g) (f,g) ˜(g,g)∗ − A˜i ) k A˜i Aj ||z ||d+2n−2

    k−m+n+1 k m−n = zi g, f am,n (k) − am,n (k − 1) d+m+n+k d + 2n + k − 1 (f,f )∗

(A˜j

z k−m+n ||z k−m+n ||d+2m−2   z k−m+n = zim−n g, f bm,n (k) k−m+n ||z ||d+2m−2 ×

where bm,n (k) = O(k −2 ) as k → +∞. Therefore (2.19) holds for m > n, completing the proof of the theorem.  To find the traces of commutators of multiplication operators, we need the following lemma. Lemma 2.11. tr [Sf∗1 , Sf2 ] = 0 for f1 ∈ J and f2 ∈ C[z1 , . . . , zd ]. Proof. [Sf∗1 , Sf2 ] belongs to the trace class by Theorem 2.10. We have  [Sf∗1 , Sf2 ] = Pg Sf∗1 Pl Sf2 Ph − Pg Sf2 Pl Sf∗1 Ph . (2.20) g,h,l∈B

If g, h, l are different from each other, Proposition 2.6 shows that both Pg Sf∗1 Pl Sf2 Ph and Pg Sf2 Pl Sf∗1 Ph belong to the trace class. Then the argument before Proposition 2.5 shows Pg ⊥Ph and we have tr Pg Sf∗1 Pl Sf2 Ph = tr Pg Sf2 Pl Sf∗1 Ph = 0. When g = h, the proof of Theorem 2.10 shows Pg Sf∗1 Pg Sf2 Ph − Pg Sf2 Ph Sf∗1 Ph ∈ L1 and Pg Sf∗1 Pg Sf2 Ph − Pg Sf2 Ph Sf∗1 Ph ∈ L1 , then since Pg ⊥Ph we find tr (Pg Sf∗1 Pg Sf2 Ph − Pg Sf2 Ph Sf∗1 Ph )

= tr (Pg Sf∗1 Pg Sf2 Ph − Pg Sf2 Ph Sf∗1 Ph ) = 0.

This equality together with (2.21) shows  tr (Pg Sf∗1 Pl Sf2 Ph − Pg Sf2 Pl Sf∗1 Ph ) = 0. g,h,l∈B,g =h

Since f1 ∈ J, Sf1 maps each Jn into J n+1 , and then we have

(2.21)

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tr [Sf∗1 , Sf2 ] = tr

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Pg Sf∗1 Pl Sf2 Pg − Pg Sf2 Pl Sf∗1 Pg

g,l∈B



= tr

Pg Sf∗1 Pl Sf2 Pg − Pl Sf2 Pg Sf∗1 Pl

g,l∈B



=

tr (Pg Sf∗1 Pl Sf2 Pg − Pl Sf2 Pg Sf∗1 Pl )

g,l∈B,deg l>deg g

= 0, where the last equality comes from the fact that both Pg Sf∗1 Pl and Pl Sf2 Pg belong to L2 .  Then we can give the trace formula. Theorem 2.12. tr [Sf∗1 , Sf2 ] = C[z1 , . . . , zd ].

d+N −2 d−1

(rf2 ) , (rf1 ) L2a (D) for f1 , f2 ∈

Proof. Denote by gi (z) = rfi (z1 ), i = 1, 2. The previous lemma shows tr [Sf∗1 , Sf2 ] = tr [Sg∗1 , Sg2 ]. By the proof of Lemma 2.11 we have  tr (Pg Sg∗1 Pl Sg2 Ph − Pg Sg2 Pl Sg∗1 Ph ) = 0. g,h,l∈B,g =h

When g = h and l = h, Proposition 2.6 shows Pg Sg∗1 Pl , Pl Sg2 Ph ∈ L2 , and then  tr (Ph Sg∗1 Pl Sg2 Ph − Ph Sg2 Pl Sg∗1 Ph ) = 0. h,l∈B,l =h

Therefore 

tr [Sg∗1 , Sg2 ] = tr

Pg Sg∗1 Pl Sg2 Ph − Pg Sg2 Pl Sg∗1 Ph

g,h,l∈B

= tr



Ph Sg∗1 Ph Sg2 Ph − Ph Sg2 Ph Sg∗1 Ph

h∈B

=

N −1 



  (d+2n−2)∗ (d+2n−2) tr Mrf1 , Mrf2

n=0 h∈Bn

N −1  

 d+n−2 = (rf2 ) , (rf1 ) L2a (D) (by [17]) d − 2 n=0   d+N −2 = (rf2 ) , (rf1 ) L2a (D) . d−1



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To prove the essential normality of general homogeneous quotient modules, we need the following lemma.  +d−2 -multicyclic. Lemma 2.13. For each 1 ≤ i ≤ d, Szi is Nd−1 Proof. Suppose f ∈ N is a homogeneous polynomial of degree n. Then we can write f as  n−|α| f (z) = cα wα zi , ∀z ∈ Dd , α∈Zd + ,α1 =0 n−|α|

∈ [J N ] whenever |α| ≥ N , we where the cα ’s are coefficients. Since wα zi find  cα Szn−|α| wα . f= i  α∈B

Theorem 2.14. Suppose I ⊂ C[z1 , . . . , zd ] is a homogeneous ideal with variety Z(I) = V . Then the quotient module M = [I]⊥ is 1-essentially normal. Proof. Let I = I1 ∩ I2 be the primary decomposition, where Z(I1 ) = V and Z(I2 ) = {0}. Let √ {f1 , . . . , fn } be a set of generators of I1 . By Hilbert’s Nullstellensatz, J = I1 and therefore for each i = 2, . . . , d there is an d positive integer ni such that wini ∈ I2 . Let N1 = i=2 ni − d + 2 then we have J N2 ⊂ I1 . Similarly we can find N2 ∈ N such that mN2 ⊂ I2 where m ⊂ C[z1 , . . . , zd ] is the maximal ideal generated by w1 , . . . , wd . Set N = max{N1 , N2 } then J N ⊂ I1 ∩ I2 = I. Denote by N = [J N ]⊥ then M ⊂ N. For i = 1, . . . , d we decomposite Szi with respect to PN = PN M ⊕ PM as follows,

(1) Si Ci Szi = . (2) 0 Si  +d−2 (1) By the previous lemma, Szi is Nd−1 -multicyclic, and therefore Si is at N +d−2 most d−1 -multicyclic on PN  PM . Since (1)∗

[Si

, Si ] = [PN M Sz∗i , Szi PN M ] = PN M [Sz∗i , Szi ]PN M + PN M Szi PM Sz∗i PN M (1)

being the sum of a positive operator and a trace class operator, a generalization of the Berger-Shaw Theorem by Voiculescu [9, pp. 155, Theorem 2.7 ] (1)∗ (1) implies [Si , Si ] ∈ L1 and   N +d−2 (1)∗ (1) tr [Si , Si ] ≤ . d−1 Since

[Sz∗i , Szi ]

=

[Si , Si ] − Ci Ci∗ (1)∗ (2)∗ Ci∗ Si − Si Ci (1)∗

(1)

(1)∗

(2)∗

Si Ci − Ci Si (2)∗ (2) [Si , Si ] + Ci∗ Ci

∈ L1 ,

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both [Si , Si ] − Ci Ci∗ and [Si , Si ] + Ci∗ Ci are trace class operators. (2)∗ (2) Therefore Ci ∈ L2 and [Si , Si ] ∈ L1 , which together with Corollory 2.7 (2)∗ (2) shows [Sj , Si ] ∈ L1 for 1 ≤ i, j ≤ d.  (1)∗

(1)

(2)∗

(2)

Remark 2.15. The approach to Theorem 2.14 is also valid to the homogeneous weighted Bergman quotient module case. Corollary 2.16. Suppose I ⊂ C[z1 , . . . , zd ] is a homogeneous ideal with vari⊥ ety Z(I) = V and denote M = [I]⊥ . Then PM Mzi PM ∈ L(2,∞) and ∗ (2,∞) [Mzi , Mzj ]PM ∈ L for 1 ≤ i, j ≤ d. Proof. Let N be as in the proof of Theorem 2.14, then C = PN M Mzi PM ∈ L2 . By Theorem 2.14 and Corollary 2.9 we have [Mz∗i PM , PM Mzj ] ∈ L1 and ⊥ Mzi PN ∈ L(2,∞) . Then we get PN ⊥ ⊥ PM Mzi PM = PN M Mzi PM + PN Mzi PM ∈ L(2,∞)

and ⊥ Mzj PM ∈ L(1,∞) . PM [Mz∗i , Mzj ]PM = [Mz∗i PM , PM Mzj ] + PM Mz∗i PM

Since [Mz∗i , Mzj ] is an projection, we have [Mz∗i , Mzj ]PM ∈ L(2,∞) .



Similarly we can prove the following theorem. Theorem 2.17. Let Vθ be a simple homogenous distinguished variety of Dd , and suppose I ⊂ C[z1 , . . . , zd ] is an ideal with Z(I) = Vθ , then the quotient ⊥ module M = [I]⊥ is 1-essentially normal. Moveover, PM Mzi PM ∈ L(2,∞) ∗ (2,∞) and [Mzi , Mzj ]PM ∈ L for 1 ≤ i, j ≤ d.

3. Essential Normality of Homogenous Distinguished Quotient Modules Before proving our main result, we need to make some observation on homogenous distinguished varieties. Proposition 3.1. If I ⊂ C[z1 , . . . , zd ] is a homogeneous ideal that satisfies Z(I) ∩ ∂Dd ⊂ Td , then Z(I) is of complex dimension 1. Proof. Define the map



Φ : C \{0} → C d

d−1

\{0}, z →

z1 zd−1 ,..., zd zd

 .

Then Φ is holomorphic, and maps Z(I) into Td−1 . Suppose homogeneous polynomials f1 , . . . , fn generate I, and define polynomials gi ∈ C[z1 , . . . , zd−1 ] by g(z1 , . . . , zd−1 ) = f (z1 , . . . , zd−1 , 1). Obviously for z ∈ Cd \{0}, fi (z) = 0 if and only if gi (Φ(z)) = fi (Φ(z), 1) = 0. Conversely if gi (0) = 0 for each i, then fi (0, . . . , 0, 1) = 0, and Z(I) must contain {(0, . . . , 0, z) ∈ Cd : z ∈ C}, contracting to the assumption. By now we see that Φ(Z(I)) = i Z(gi ) is an analytic variety contained in Td−1 .

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Suppose V is an irreducible component of Φ(Z(I)), and n = dimC V . Then there are d−n polynomials g1 , . . . , gd−n such  that V = Z(g1 , . . . , gd−n ).

Choose a regular point z0 ∈ V , where the matrix

∂gi ∂zj (z0 )

i,j

is of rank d−n.

If n > 0, by the implicit function theorem, there is an analytic embedding ϕ : U → V on some neighbourhood U of 0 in Cn . Thus ϕ is an analytic embedding from U to Φ(Z(I)) ⊂ Td−1 , which is impossible. Hence n must be zero, and each component of Z(I) must be of complex dimension 1.  Theorem 3.2. If I ⊂ C[z1 , . . . , zd ] is a homogeneous ideal satisfying Z(I) ∩ ∂Dd ⊂ Td , then the quotient module M = [I]⊥ is (1, ∞)-essentially normal. !n Proof. By Proposition 3.1, dimC Z(I)=1. Suppose Z(I)∩Dd = k=1 Vk , then dimC Vk = 1 and therefore Vk is a simple distinguished variety for each k. Denote by Jk the prime ideal with zero variety Vk for k = 1, . . . , n, and m n the maximal ideal at {0}. Let I = k=0 Ik be the primary decomposition, with Z(I0 ) = {0} and Z(Ik ) = Vk . Since I0 is of finite codimension, we may assume I = I1 ∩ . . . ∩ In . There is a positive integer N such that JkN ⊂ Ik for each k. When 1 ≤ j, k ≤ n and j = k, we can choose a linear polynomial gk,j ∈ Jj , such that gk,j (z) = 0 whenever z ∈ Vk . Since gk,j is homogeneous and Vk is a simple distinguished variety, |gk,j (z)| is constant on Vk ∩ ∂Dd . Denote by N , then ϕk ∈ j =k Jj and |ϕk (z)| is a nonzero constant on ϕk = Πj =k gk,j d Vk ∩ ∂D . Denote by Mk = [Ik ]⊥ , Nk = [JkN ]⊥ , and Sϕ k = PNk Mϕk |Nk be the contraction of Mϕk on the quotient module Nk . We want to represent PM by the PMk ’s, and then induce the essential normality of M from that of the Mk ’s. First we prove that Sϕ k is a Fredholm operator. Without loss of generality we may assume Vk = V(1,...,1) . By corollary 2.7, there is an isometry S ∈ B(Nk ) which is unitarily equivalent to a multi-shift, and some operator Kk ∈ L(1,∞) such that Sϕ k = ϕ(Sk , . . . , Sk ) + Kk . Therefore σe (Sϕ k ) = {rϕk (z) : z ∈ ∂D} = ϕk (Vk ∩ ∂Dd ), which do not contain {0}. This implies that Sϕ k is Fredholm, and so is Sϕ∗k . Since Sϕ∗k is homogeneous and ker Sϕ∗k is of finite dimension, there is a constant ρk > 0 and positive integer Nk such that ||Sϕ∗k h|| ≥ ρk ||h|| whenever h ∈ Nk and deg h > Nk . Let ρ = min{ρk : k = 1, . . . , n} and N  = max{Nk : k = 1, . . . , n}. Denote by M the closed subspace of M spanned by homogeneous polynomials of degree greater than N  . n Denote by Sϕk = PM Mϕk |M and T = k=1 Sϕk Sϕ∗ k . For homogeneous f ∈ M , denote by " # L = g ∈ H 2 (Dd ) : g is homogeneous, deg g = deg f .

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We have



f ∈M ∩L⊂M∩L = =

n $

⊥

∩L=L

Ik

k=1 n %

L  (Ik ∩ L) = L ∩

k=1

n

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n $

13

Ik ∩ L

k=1 n %

[Ik ]⊥ .

k=1 ⊥

Then we can write f = k=1 fk where each fk ∈ [Ik ] = Mk is homogeneous and of the same degree as f . Then ||Sϕ∗ k fk || = ||Sϕ∗k fk || ≥ ρ||fk ||. When j = k, since ϕk ∈ j =k Jj we have Mϕ∗k PMj = 0, and therefore Sϕ∗ k fj = 0. From this we obtain Sϕ∗ k f = Sϕ∗ k fk ∈ Mk . Then & n ' n    ∗  ∗ T f, f = Sϕk f, Sϕ∗ k f Sϕk Sϕk f, f = k=1

=

n  



Sϕ∗ k fk , Sϕ∗ k fk ≥

k=1

k=1 n 

ρ2 ||fk ||2 ≥

k=1

ρ2 ||f ||2 . n

Therefore T is bounded below on M , and must have closed range. Since T keeps degree of polynomials, T maps M into a dense subspace of itself. Fredholm. Hence M ⊂ T M and T is (n n ∗  Denote by T  = k=1 PMk Sϕk Sϕk PMk , then T M ⊂ k=1 Mk . By Theorem 2.17 for each k we have PMk Sϕ∗ k − Sϕ∗ k PMk = PMk Mϕ∗k (PM − PMk ) ∈ L(2,∞) , and therefore T − T =

n 

Sϕk PMk Sϕ∗ k −

k=1

n 

PMk Sϕk Sϕ∗ k PMk ∈ L(2,∞) ,

k=1

(n  which(implies T is Fredholm since (n T is. By k=1 Mk ⊃ T M we conclude n that k=1 Mk is closed. Since k=1 Mk is dense in M, it is just M. Define an operator 

S:

n )

Mk → M, (h1 , . . . , hn ) →

k=1

n 

hk ,

k=1

then S is surjective hence invertible on (ker S)⊥ . Moreover, S∗ : M →

n )

Mk , h → (PM1 h, . . . , PMn h)

k=1

is bounded below. Choose a constant c > 0 such that ||h|| < c||S ∗ h|| whenever h ∈ M. Then for each h ∈ H 2 (Dd ) we have PM h, h = ||PM h||2 < c2 S ∗ PM h, S ∗ PM h

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= c2 2

=c

n 

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PMk h, PMk h

k=1 & n 

' PMk h, h ,

k=1

n which implies PM < c2 k=1 PMk . Therefore all the eigenvalues of  n −2 . Let{ej : j = 1, 2, . . .} be an complete k=1 PMk are greater than c n system of normalized eigenvectors of k=1 PMk , and {λj : j = 1, 2, . . .} the associated eigenvalues, then n ∞   PMk = λj ej ⊗ ej . j=1

k=1

Let A=

∞ 

λ−1 j ej ⊗ ej

j=1

n

n and we have PM = A k=1 PMk = k=1 PMk A. According to Theorem ⊥ (2,∞) ∗ (2,∞) 2.17, PM M P ∈ L and [M , M for 1 ≤ i, j ≤ d z M zi ]PMk ∈ L z i k j k and 1 ≤ k ≤ n. Therefore n n   ⊥ ⊥ ⊥ ⊥ PM Mzi PM = PM Mzi PMk A = PM PMk Mzi PMk A ∈ L(2,∞) , k=1

k=1

and [Mz∗j , Mzi ]PM

=

n 

[Mz∗j , Mzi ]PMk A ∈ L(2,∞) .

k=1

Finally ⊥ [Mz∗j PM , PM Mzi ] = PM [Mz∗j , Mzi ]PM − PM Mz∗j PM Mzi PM ∈ L(1,∞) ,

for 1 ≤ i, j ≤ d.



At the end of this section, we give the following proposition, which reveals the structure of semisimple homogenous quotient module. Following [23], two subspaces N1 and N2 of a Hilbert space H are said to be asymptotically orthogonal if PN1 PN2 is compact. Proposition 3.3. Let I be a homogenous ideal such that Z(I) ∩ ∂Dd ⊂ Td , and I = I1 . . . In be its primary decomposition, with dimC Z(Ii ) = 1 for 1 ≤ i ≤ n. Suppose {Ii : 1 ≤ i ≤ n} are prime. Then {[Ii ]⊥ : 1 ≤ i ≤ n} are asymptotically orthogonal to each other. Proof. Assume Vi = Vθi for θi ∈ Td . Since [Ii ]⊥ is prime, we have " # [Ii ]⊥ = span Kλθi ∈ H 2 (Dd ) : λ ∈ D . Direct computation shows Kλθi =

∞  k=1

¯ k gk ◦ L−1 λ θi

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where Lθi is defined as in Remark 2.8, and gk (z) = {

gk ◦L−1 θi ||gk ◦L−1 θi ||

 |α|=k

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13

z α . We see that

: k ∈ Z+ } forms an orthonormal basis for [Ii ]⊥ . One can compute

for λ ∈ D that

    −1 λk gk ◦ L−1 , g ◦ L = gk ◦ L−1 k θi θj θi , Kλθj = gk ◦ L−1 θi (λθj ) = λk gk (L−1 θi θj ),

−1 −1 which induces gk ◦ L−1 θi , gk ◦ Lθj = gk (Lθi θj ). When i = j, at least two −1 coordinates of L−1 θi θj are not equal. Denote by ρ = Lθi θj , and without loss of generality we assume ρ1 = ρ2 . k (ρ)| Claim: limk→∞ |g ||gk ||2 = 0. We prove the claim by induction on the number of variables. To do this, set

gm,k (z) = gk (z1 , . . . , zm , 0, . . . , 0), 1 ≤ m ≤ d, k ∈ Z+ . When m = 2, direct computation shows

* * k * * |g2,k (ρ)| * −1 l k−l * lim = lim (k + 1) ρ ρ * * 1 2 * * k→∞ ||g2,k ||2 k→∞ l=0 * * * * ¯k+1 ρk+1 * −1 * k 1 − ρ 1 2 = lim (k + 1) *ρ1 * * k→∞ 1 − ρ¯1 ρ2 * ≤ lim (k + 1)−1 k→∞

2 = 0. |1 − ρ¯1 ρ2 |

Suppose the claim is proved for some m(2 ≤ m ≤ d − 1), then from the observation gm+1,k =

k 

l zm+1 gm,k−l

l=0

we obtain

 −1  k m+k |gm+1,k (ρ)| ≤ lim |gm,k−l (ρ)| k→∞ ||gm+1,k ||2 k→∞ m l=0  −1 m+k−1 = lim |gm,k (ρ)| (by Stolz’s Theorem) k→∞ m−1 |gm,k (ρ)| =0 = lim k→∞ ||gm,k ||2 lim

by the assumption. Then the claim holds for m+1, and its proof is completed by induction. Therefore lim

k→∞

−1 |gk ◦ L−1 θi , gk ◦ Lθj |

−1 ||gk ◦ L−1 θi || · ||gk ◦ Lθj ||

|gk (L−1 θi θj )| = 0. k→∞ ||gk ||2

= lim

This shows that [Ii ]⊥ and [Ij ]⊥ are asymptotic orthogonal if i = j.



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Remark 3.4. In the proof of Proposition 3.3, it can be seen that if Vθ1 is orthogonal to Vθ2 , then [Jθ1 ]⊥  C is orthogonal to [Jθ2 ]⊥  C, where C denotes the 1-dimensional subspace of constant functions.

4. Boundary Representation of Quotient Modules In this section, we consider the boundary representation of quotient modules. Let us begin with recalling Arveson’s boundary representation theorem given in [3]. Theorem 4.1. Let A be an irreducible set of operators on a Hilbert space H, such that A contains the identity, and C ∗ (A) the C ∗ -algebra generated by A contains the algebra K(H) of compact operators on H. Then the identity representation of C ∗ (A) is a boundary representation for A if and only if the quotient map Q : B(H) → B(H)/K(H) is not completely isometric on the linear span of A ∪ A∗ . To continue, we need to make some understanding on the essential joint spectrum of (S1 , . . . , Sd ). Lemma 4.2. Let V be a distinguished homogenous variety and I is an ideal such that Z(I) = V , then for the quotient module [I]⊥ , V ∩ Td ⊆ d σe (S1 , . . . , Sd ) and σe (S1 , . . . , Sd ) ⊆ V ∩ D . Proof. By Theorem 3.2, λi − Szi (i = 1, . . . , d) are essentially normal. By [10, Corollary 3.9], the tuple (λ1 − Sz1 , . . . , λd − Szd ) is Fredholm if and only if d ∗ i=1 (λi − Szi )(λi − Szi ) is Fredholm. We claim that ∂V ⊆ σe (S1 , . . . , Sd ). Otherwise, there exists λ = (λ1 , . . . , λd ) ∈ ∂V such that T =

d 

(λi − Szi )(λi − Szi )∗

i=1

is Fredholm. Since T is positive, there is an invertible positive operator B and a compact operator K such that T = B + K. Now, take a sequence {μn } ⊂ V ∩ Dd such that μn → λ as n → ∞. Notice that {kμn } converges to 0 weakly, there is a positive number c such that       lim T kμn , kμn = lim (B + K)kμn , kμn = lim Bkμn , kμn ≥ c. n→∞

n→∞

n→∞

⊥

However, since μn ∈ V , kμn ∈ [I] it holds that   lim T kμn , kμn = lim |λ − μn |2 = 0, n→∞

n→∞

(4.1)

contradicting to the former inequality. Hence the claim is proved. Moreover, for f ∈ I, if f (S1 , . . . , Sd ) = 0 then the Spectral Mapping Theorem ensures that σe (S1 , . . . , Sd ) ⊆ Z(f ). Hence σe (S1 , . . . , Sd ) ⊆ V . On d the other hand, since Si  ≤ 1 for each i, we have σe (S1 , . . . , Sd ) ⊆ V ∩D .  The following lemma is easy to verify, and we omit its proof.

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Lemma 4.3. Let X be a Banach space, and T is a bounded operator on X ˙ ˙ 2 , where + with two invariant subspaces M1 and M2 satisfying X = M1 +M denotes topological direct sum. Then σe (T ) = σe (T1 ) ∪ σe (T2 ). Let Vθi (i = 1, 2) be the variety defined by (2.1), and Ii (i = 1, 2) be a homogenous ideal with varitey Vθi . Consider the quotient module N = [I1 I2 ]⊥ . Notice that [I1 ]⊥ ∩ [I2 ]⊥ is of finite dimension, and [I1 ]⊥ + [I2 ]⊥ is closed. Now, let Ni = [Ii ]⊥  ([I1 ]⊥ ∩ [I2 ]⊥ ). Then ˙ 2 ) ⊕ ([I1 ]⊥ ∩ [I2 ]⊥ ) N = (N1 +N By Lemma 4.3 we have σe (Mz∗i |N ) = σe (Mz∗i |[I1 ]⊥ ) ∪ σe (Mzi |[I2 ]⊥ ). Therefore, to study σe (Mz∗i |N ) it suffices to study σe (Mz∗i |[Ii ]⊥ ). Lemma 4.4. Let Vθ be defined by (2.1), and I is a homogenous ideal such that Z(I) = Vθ . Then σe (Mzi |∗[I]⊥ ) ⊂ T. √ Proof. Let Jθ = I and N is a positive integer such that JθN ⊂ I ⊂ Jθ . By Remark 2.8, there is an isometry S and a compact operator K such that Mz∗i |[JθN ]⊥ = S ∗ + K, and hence σe (Mz∗i |[JθN ]⊥ ) = σe (S ∗ ) = T. Since [I]⊥ is an invariant subspace of Mz∗i |[JθN ]⊥ , we get σe (Mzi |∗[I]⊥ ) ⊂ T.  By Lemma 4.3, for distinguished variety V and homogenous ideal I with Z(I) = V , we have σe (Mz∗i |[I]⊥ ) ⊂ T.

(4.2)

Proposition 4.5. Let I be a homogenous ideal with distinguished variety, then σe (Sz1 , . . . , Szd ) = V ∩ Td . Proof. By Lemma 4.2, it suffices to show σe (Sz1 , . . . , Szd ) ⊂ ∂Dd . By (4.2), λ − Szi is Fredholm if |λ| < 1. It follows that for λ = (λ1 , . . . , λd ) ∈ Dd ,  λi − Szi are Fredholm. Therefore σe (Sz1 , . . . , Szd ) ⊂ Td by [10]. Suppose that V is a distinguished homogenous variety, with the decomposition V = V1 ∪ . . . ∪ Vn , such that each Vi is irreducible with dimC Vi = 1. Let I be an ideal such that Z(I) = V , with the associated primary decomposi.∩In , such tion I √ = I0 ∩I1 ∩. .√ √ that √ Z(I0 ) = {0} and Z(Ii ) = Vi for i = 1, . . . , n. Since I0 ∩ I1 = I0 ∩ I1 = I1 , we can assume I = I1 ∩ . . . ∩ In without loss of generality. Theorem 4.6. With the above notations, if some Ii is not prime, then the identity representation of C ∗ ([I]⊥ ) is a boundary representation for B(S1 , . . . , Sd ). Proof. It is easy to see that {S1 , . . . , Sd } is a irreducible set and C ∗ ([I]⊥ ) contains all the compact operators on [I]⊥ . Without loss of generality, suppose I1 is not prime. Then by the proof √ of Theorem 3.2,√we can find a polynomial that ϕ ∈ / J1 . Choose g ∈ I1 − I1 and let f = ϕg, ϕ ∈ I2 ∩ . . . ∩ In such √ then we have f ∈ I1 ∩ I2 ∩ . . . ∩ In and f ∈ I1 ∩ I2 ∩ . . . ∩ In . Then f |V = 0 and Sf = P[I]⊥ Mf |[I]⊥ = 0. By the Spectral Mapping Theorem, σe (Sf ) = f (σe (S1 , . . . , Sd )) = {0}.

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Since Sf is essentially normal, it must be compact. On the other hand, since f ∈ I1 ∩ I2 ∩ . . . ∩ In , we have Sf = 0 and then Sf  > Sf e = 0. By Arveson’s boundary representation theorem, the lemma is proved.  About the inverse proposition of Theorem 4.6, we have the following result. Its proof follows from [3, page 292, Corollary 2] and the proof of Proposition 2.6. Proposition 4.7. Suppose that I is a prime homogenous ideal such that Z(I) is a distinguished variety. Then the identity representation of C ∗ ([I]⊥ ) is not a boundary representation for B(S1 , . . . , Sd ). Example. With the above notations, suppose that all the Ii ’s are prime. If the Z(Ii )’s are orthogonal to each other, then the [Ii ]⊥  C’s are orthogonal to each other by Remark 3.4. It follows that in this case, the identity representation of C ∗ ([I]⊥ ) is not a boundary representation for B(S1 , . . . , Sd ). Acknowledgements Both the authors would like to thank Professor Kunyu Guo for his encouragements and interests. The authors thank the referee for helpful suggestions which make the paper more readable. The authors also thank Li Chen for his comments on the earlier version of this manuscript.

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Penghui Wang School of Mathematics Shandong University Jinan 250100 Shandong The People’s Republic of China e-mail: [email protected]

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Chong Zhao (B) School of Mathematics Shandong University Jinan 250100 Shandong The People’s Republic of China e-mail: [email protected] Received: August 29, 2017. Revised: October 31, 2017.

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