INVARIANT SUBSPACES FOR DOUBLY COMMUTING

J. OPERATOR THEORY 40(1998), 373–384

c Copyright by Theta, 1998

INVARIANT SUBSPACES FOR DOUBLY COMMUTING CONTRACTIONS WITH RICH TAYLOR SPECTRUM ERNST ALBRECHT and MAREK PTAK

Communicated by Florian-Horia Vasilescu Abstract. In the present paper we prove the existence of nontrivial common invariant subspaces for N -tuples of doubly commuting contractions with dominating Taylor spectrum for the H ∞ algebra of the polydisc. We also obtain the reflexivity of the WOT-closed subalgebra generated by finite systems of completely non-unitary doubly commuting contractions with dominating essential Taylor spectrum. Keywords: Invariant subspaces, reflexive operators, N -tuple of contractions, Taylor spectrum. MSC (2000): Primary 47A15; Secondary 47D27.

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS

One of the big steps in the invariant subspace problem for a single operator was the paper [4], where the existence of nontrivial invariant subspaces was proved for contractions having the dominating spectrum for the H ∞ algebra. The corresponding reflexivity result was shown in [2] and [14], where this property for completely non-unitary (c.n.u.) contractions with dominancy of left essential and essential spectrum, respectively, were obtained. It is natural to ask if the above results are also valid for pairs or N -tuples of contractions. One of the approaches was made in [12], where reflexivity, thus also existence of nontrivial common invariant subspaces for N -tuples of doubly commuting c.n.u. contractions with dominancy of joint left essential spectrum was proved. In [11] a different result for this spectrum for pairs of contractions was shown. See also [6] and [8] for further results. However, the natural notion of

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Ernst Albrecht and Marek Ptak

a spectrum for N -tuples is the Taylor spectrum ([18], [19]). Some progress in that direction was made in [1]. In this note we assume double commutativity for the contractions and obtain a natural extension of the result from [4]. The definitions of the notation in the following statements will be given below. Theorem 1.1. Let T = (T1 , . . . , TN ) be an N -tuple of doubly commuting contractions. If the intersection of the Taylor spectrum with the open polydisc σ(T ) ∩ DN is dominating for H ∞ (DN ), then T = (T1 , . . . , TN ) has a common nontrivial invariant subspace. As we shall see, because of the double commutativity, it will be sufficient in the proof to consider the case where the Taylor spectrum of T coincides with the essential Taylor spectrum. This allows us to reduce the proof of Theorem 1.1 to the following: Theorem 1.2. Let T = (T1 , . . . , TN ) be an N -tuple of doubly commuting completely non-unitary contractions. If the intersection of the Taylor essential spectrum with the open polydisc σe (T ) ∩ DN is dominating for H ∞ (DN ), then T = (T1 , . . . , TN ) is reflexive. Of course, the reflexivity is a much stronger property than having a nontrivial common invariant subspace. Theorem 1.2 is a generalization of the reflexivity result for a single contraction case from [2] and [14]. It also improves the result of [12], Theorem 5.4, with respect to the type of spectrum considered. Recently, the existence of a nontrivial common invariant subspace for spherical contracting instead of N -tuples of contractions is also considered, see [7]. Throughout this paper let H be a separable infinite-dimensional complex Hilbert space. If R is a subset of the algebra L(H) of all bounded linear operators on H, we denote by W(R), respectively A(R), the WOT (= weak operator topology)-closed, respectively the weak-star closed subalgebras of L(H) generated by R and the identity Id. Lat R will be the lattice of all (closed) invariant subspaces for R, and Alg Lat R is as usual the algebra of all T ∈ L(H) such that T L ⊂ L for all L ∈ Lat R. R is said to be reflexive if W(R) = Alg Lat R. We say that a commutative set R ⊂ L(H) is doubly commuting (respectively almost doubly commuting) if S T ∗ − T ∗ S is a zero (respectively compact) operator for all S, T ∈ R with S 6= T . In particular sets consisting of a single operator are doubly commuting. Recall, that a set E contained in the open polydisc DN is dominating for the algebra H ∞ (DN ) of all bounded analytic functions on DN , if for all h ∈ H ∞ (DN ) we have khk∞ = sup |h(z)|. A(DN ) denotes the closed subalgebra of all those z∈E

h ∈ H ∞ (DN ) having continuous extensions to the closed polydisc.

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Invariant subspaces for contractions 2. PRELIMINARIES FROM SEVERAL VARIABLES SPECTRAL THEORY

Recall from [18], [19] that the Koszul (cochain) complex K(T, H) for a N -tuple T = (T1 , . . . , TN ) of commuting operators in L(H) with respect to H is given by δ 0 (T )

δ 1 (T )

0 −→ Λ0 (H) −→ Λ1 (H) −→ · · ·

δ N −1 (T )

−→

ΛN (H) −→ 0,

where Λp (H) denotes the set of all p-forms with coefficients in H and the cochain mapping δ p (T ) : Λp (H) → Λp+1 (H) is defined by δ p (T )

X0

xI sI :=

N X j=1

|I|=p

X0

Tj xI sj ∧ sI ,

|I|=p

where s1 , . . . , sN is a fixed basis of Λ1 (C) and

P

0

denotes that the sum is taken

|I|=p

over all I = (i1 , . . . , ip ) ∈ Np with 1 6 i1 < · · · < ip 6 N , sI := si1 ∧ · · · ∧ sip . Let us notice that Λp (H) can be endowed with the natural scalar product ! X0 X0 X0 xI sI , yI sI := (xI , yI ), |I|=p

|I|=p

|I|=p

N which gives us a canonical isomorphism with a direct sum of copies of H. p Following [18], [19], λ belongs to the Taylor spectrum σ(T ) ⊂ CN if, by definition, the complex K(λ − T, H) is not exact, and λ belongs to the Taylor essential spectrum σe (T ) ⊂ CN if, by definition, at least one of the cohomology groups H p (λ − T ) := ker δ p (λ − T )/ran δ p−1 (λ − T ) has infinite dimension. Following [1], N S we can decompose σe (T ) = σep (T ), where σep (T ) is the set of all λ ∈ CN such p=0

that the induced mapping δˆp (λ − T ) : Λp (H)/ ran δ p−1 (λ − T ) → Λp+1 (H) has non-closed range or infinite dimensional kernel. The points of σep (T ) have the following property: Lemma 2.1. Let T = (T1 , . . . , TN ) be an N -tuple of commuting operators. p If λ is a point in σep (T ), there exists an orthonormal sequence {ηn }∞ n=1 in Λ (H) such that (2.1)

∗

∗

δ p−1 (λ − T )δ p−1 (λ − T ) ηn + δ p (λ − T ) δ p (λ − T )ηn → 0

for n → ∞. Proof. Indeed, by the definition of σep (T ) and a standard argument we can p p−1 find an orthonormal sequence {ηn }∞ (λ−T ) such that δ p (λ− n=1 in Λ (H) ran δ T )ηn → 0. Obviously, this sequence satisfies (2.1).

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We will need the following fact (see [5], Corollary 3.7). Lemma 2.2. Let T = (T1 , . . . , TN ) be a N -tuple of almost doubly commuting operators. Then there is a compact operator K on Λp (H) such that ∗

∗

δ p−1 (T )δ p−1 (T ) + δ p (T ) δ p (T ) = K +

N MX F

TF (j),

j=1

where the orthogonal sum is taken over all functions F : {1, . . . , N } → {0, 1}, card{j : F (j) = 0} = p and TF (j) = Tj Tj∗ for F (j) = 0 and TF (j) = Tj∗ Tj for F (j) = 1. Moreover, if the N -tuple doubly commutes, then K = 0. We apply this lemma in the following form. Lemma 2.3. Let T = (T1 , . . . , TN ) be a N -tuple of doubly commuting operators in L(H). If, for some λ ∈ CN and some p ∈ {1, . . . , N }, we have ⊥

ker δ p (λ − T ) ∩ ran δ p−1 (λ − T ) 6= {0}, then there is i0 ∈ {1, . . . , N } such that the operator Ti0 has nontrivial hyperinvariant subspaces or the tuple consists of scalar operators. Proof. By our assumptions, there exists some 0 6= ω ∈ Λp (H) with δ p (λ − ∗ T )ω = 0 = δ p−1 (λ − T ) ω. Hence ∗

∗

δ p−1 (λ − T ) δ p−1 (λ − T ) ω + δ p (λ − T ) δ p (λ − T ) ω = 0. By Lemma 2.2 there are disjoint sets S, T with S ∪ T = {1, . . . , N } and a vector 0 6= x ∈ H such that X i∈S

(λi − Ti )∗ (λi − Ti )x +

X

(λk − Tk )(λk − Tk )∗ x = 0.

k∈T

Hence x ∈ ker (λi − Ti ) for i ∈ S and x ∈ ker (λk − Tk )∗ for k ∈ T . From this we obtain a nontrivial hyperinvariant subspace or Tj = λj for all j ∈ {1, . . . , N }. Since the space H is infinite dimensional, we have the following. Corollary 2.4. Let T = (T1 , . . . , TN ) be a N -tuple of doubly commuting operators in L(H). If σ(T ) \ σe (T ) 6= ∅, then Ti0 has nontrivial hyperinvariant subspace for some i0 ∈ {1, . . . , N }. It seems to be a difficult problem to remove the double commutativity assumptions from the preceding statement.

Invariant subspaces for contractions

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3. REDUCTION TO THE APPROXIMATION PROBLEM

We start with Reduction from Theorem 1.1 to Theorem 1.2. If one of the contractions T1 , . . ., TN , say Ti0 , has a nontrivial unitary part, then from the formula for the subspace Hu i0 on which the contraction Ti0 is unitary (cf. [17], Theorem I.3.2), we see that Hu i0 is invariant for all operators S doubly commuting with Ti0 and Lat(T ) will be nontrivial. Hence we may assume that all T1 , . . . , TN are c.n.u.. Also, because of Corollary 2.4, we may assume that σ(T ) = σe (T ) and we are in the situation of Theorem 1.2. For further reductions we need some notation. Recall that a representation (a unital algebra homomorphism) Φ : A(DN ) → L(H) is called contractive if kΦ(f )k 6 kf k∞ for f ∈ A(DN ). Standard techniques show that for every x, y ∈ H N there is a complex, Borel, regular measure µx,y on D , called the elementary R measure, satisfying (Φ(f )x, y) = f dµx,y for all f ∈ A(DN ). The representation Φ is called absolutely continuous (a.c.) if it has a system of elementary measures {µx,y }x,y∈H such that each measure µx,y is absolutely continuous with respect to some positive representing measure νz for some z ∈ DN . ∗ We shall also need the language of dual algebras. Recall that L(H) = C1 (H) , where C1 (H) is the ideal of trace class operators and the duality is given by the form hT, Si := tr(T S) for T ∈ L(H), S ∈ C1 (H). Hence, every ultraweakly closed subalgebra A of L(H) is a dual Banach space with predual space QA ∼ = C1 (H)/⊥ A ⊥ via hT, [S]i := tr(T S) for T ∈ A, [S] ∈ C1 (H)/ A. Thus, for a rank one operator x ⊗ y (z 7→ (z, y)x), we have hT, [x ⊗ y]i = (T x, y). Now we can start the Proof of Theorem 1.2. Under the assumptions of the theorem we can construct in a standard way, using a dilation ([17], Proposition I.9.2), a contractive representation Φ : A(DN ) → L(H) with Φ(zi ) = Ti for i = 1, . . . , N . Since T1 , . . . , TN are c.n.u., this representation is a.c. ( see [10], Theorem 1), in which case we say that the N -tuple T = (T1 , . . . , TN ) is a.c. By [12], Proposition 4.1 the functional calculus can be extended to H ∞ (DN ) as a homeomorphism for the weak-star topologies as Φ : H ∞ (DN ) → L(H). We will write h(T ) instead of Φ(h) for h ∈ H ∞ (DN ). Since the representation is contractive, to have an isometric functional calculus, we need the following known fact. Lemma 3.1. Suppose that the assumptions of Theorem 1.2 are satisfied. If f ∈ H ∞ (DN ) then kf k∞ 6 kf (T )k. For the sake of completeness we include the elementary

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Proof. Let λ ∈ σe (T ) ∩ DN ⊂ σ(T ) ∩ DN . By the Gleason property of N P the polydisc f (T ) − f (λ) = (λi − Ti )ui (T ) for some ui ∈ H ∞ (DN ). Then i=1

the operator

N P

(λi − Ti )ui (T ) can not be invertible. Hence f (λ) ∈ σ(f (T )) and

i=1

kf k∞ 6 r(f (T )) 6 kf (T )k, which completes the proof of the lemma. Because of the isometry and the weak-star continuity of the functional calculus, the predual of H ∞ (DN ) and that of A(T ) (which will be denoted by Q) are isometrically isomorphic. Denote by [Cλ ] the element of Q corresponding to the point evaluation at λ ∈ DN . Since σe (T ) ∩ DN is dominating, standard techniques (see [4]) show that the set aco{[Cλ ] : λ ∈ σe (T ) ∩ DN } contains the closed unit ball in Q. We shall need the well known fact from [3] that every dual algebra with property X0,1 is reflexive. Recall that A has property X0,1 if the unit ball of QA is contained in X0,1 — the set of all those [L] ∈ QA such that there exist sequences ∞ {xn }∞ n=1 , {yn }n=1 , ⊂ H with kxn k 6 1, kyn k 6 1 for all n, such that lim k[xn ⊗ yn ] − [L]kQ = 0,

(3.1)

n→∞

(3.2)

lim k[xn ⊗ w]kQ = 0

for all w ∈ H,

lim k[w ⊗ yn ]kQ = 0

for all w ∈ H.

n→∞

(3.3)

n→∞

Since X0,1 is absolutely convex and closed ([3]), it suffices to show that X0,1 contains all [Cλ ], λ ∈ σe (T ) ∩ DN . Hence, let us take λ ∈ σe (T ) ∩ DN . Then, by Lemma 2.1 there is a number p p ∈ {1, . . . , N } and an orthonormal sequence {ηn }∞ n=1 in Λ (H) such that (2.1) holds. Passing, if necessary, to some subsequence, we may assume that for some I = (i1 , . . . , ip ) ∈ Np the coefficients xn of si1 ∧ · · · ∧ sip in ηn satisfy kxn k > α for all n ∈ N and some α > 0. By Lemma 2.2 there are disjoint sets S, T with S ∪ T = {1, . . . , N } such that X i∈S

(λi − Ti )∗ (λi − Ti )xn +

X

(λk − Tk )(λk − Tk )∗ xn → 0.

k∈T

Taking the scalar product with xn we get k(Ti − λi )xn k → 0 for all i ∈ S and k(Tk∗ − λk )xn k → 0 for all k ∈ T . The sequence {ηn }∞ n=1 being orthonormal we have xn → 0 weakly. Moreover, since the numbers kxn k are bounded below, we can assume, without loss of generality, that kxn k = 1. The approximation lemmas in the next section will now prove that [Cλ ] ∈ X0,1 for λ ∈ σe (T ) ∩ DN . This will show the following structure result.

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Proposition 3.2. Let T = (T1 , . . . , TN ) be an N -tuple of doubly commuting completely non-unitary contractions. If the intersection of the Taylor essential spectrum with the open polydisc σe (T ) ∩ DN is dominating for H ∞ (DN ), then the algebra A(T1 , . . . , TN ) has property X0,1 . The proposition implies the reflexivity of A(T1 , . . . , TN ), hence also of T = (T1 , . . . , TN ). 4. THE APPROXIMATION LEMMAS

We will start with the approximation of the point evaluation. Lemma 4.1. Let T = (T1 , . . . , TN ) be an a.c. N -tuple of commuting contractions, S, T be disjoint sets such that S ∪ T = {1, . . . , N }, and λ = (λ1 , . . . , λN ) ∈ DN . Assume that xn → 0 weakly, kxn k = 1 for all n. If k(Ti − λi )xn k → 0 for all i ∈ S and k(Ti∗ − λi )xn k → 0 for all i ∈ T then lim k[xn ⊗ xn ] − [Cλ ]kQ = 0. n→∞

Proof. By the Hahn-Banach theorem, for each n, there exist some fn ∈ H (DN ) such that kfn (T )k = kfn k = 1 and k[xn ⊗ xn ] − [Cλ ]kQ = |hfn (T ), [xn ⊗ xn ] − [Cλ ]i|. The polydisc has the Gleason property, thus there are gin ∈ H ∞ (DN ) N P satisfying kgin k 6 Mλ for i = K + 1, . . . , N and fn (z) = fn (λ) − (zi − λi )gin (z), ∞

i=1

where z = (z1 , . . . , zN ) ∈ DN . Hence k[xn ⊗ xn ] − [Cλ ]kQ N D E X = fn (λ) + (Ti − λi )gin (T ), [xn ⊗ xn ] − [Cλ ] i=1 N X (Ti − λi )gin (T )xn , xn = i=1

6

X

6

X

|(gin (T )(Ti − λi )xn , xn )| +

i∈S

|(gin (T )xn , (Ti∗ − λi )xn )|

i∈T

kgin (T )k k(Ti

− λi )xn k +

i∈S

6 Mλ

X

X

kgin (T )k k(Ti∗ − λi )xn k

i∈T

X i∈S

k(Ti − λi )xn k +

X

k(Ti∗ − λi )xn k .

i∈T

Thus, the proof of the lemma is finished since k(Ti − λi )xn k → 0 for all i ∈ S and k(Ti∗ − λi )xn k → 0 for all i ∈ T . Next, the approximate orthogonality (3.2) will be shown.

380


Lemma 4.2. Let T = (T1 , . . . , TN ) be an a.c. N -tuple of commuting contractions. Let S, T be disjoint sets such that S ∪ T = {1, . . . , N }. Assume that {Ti : i ∈ S} is doubly commuting. Let λ = (λ1 , . . . , λN ) ∈ DN and xn → 0 weakly, kxn k = 1 for all n. If k(Ti − λi )xn k → 0 for all i ∈ S and k(Ti∗ − λi )xn k → 0 for all i ∈ T then lim k[y ⊗ xn ]kQ = 0 for all y ∈ H. n→∞

Proof. Without loss of generality we can assume that S = {1, . . . , K}. There can be found some functions fn ∈ H ∞ (DN ) such that kfn (T )k = kfn k = 1 and k[y ⊗ xn ]kQ = |(fn (T )y, xn )|. Notice that k[y ⊗ xn ]kQ 6 |(fn (T1 , . . . , TK , λK+1 , . . . , λN )y, xn )| + |((fn (T1 , . . . , TN ) − fn (T1 , . . . , TK , λK+1 , . . . , λN ))y, xn )|. By an obvious modification of the Gleason property for polydomains, there are n n functions gK+1 , . . . , gN ∈ H ∞ (DN ) such that kgin k 6 Mλ for i = K + 1, . . . , N and fn (z) − fn (z1 , . . . , zK , λK+1 , . . . , λN ) =

N X

(λi − zi )gin (z),

i=K+1

where z = (z1 , . . . , zN ). The point λ being fixed, let us denote by hn ∈ H ∞ (DK ) the function hn (z1 , . . . , zK ) = fn (z1 , . . . , zK , λK+1 , . . . , λN ) and write Te = (T1 , . . . , TK ). Obviously, we have a natural functional calculus for Te = (T1 , . . . , TK ). Hence, for any ε > 0, we have k[y ⊗ xn ]kQ 6 |(fn (T1 , . . . , TK , λK+1 , . . . , λN )y, xn )| N X + (λi − Ti )gin (T )y, xn i=K+1

6 |(hn (Te)y, xn )| +

N X

kgin (T )k kyk k(λi − Ti∗ )xn k

i=K+1

6 |(hn (Te)y, xn )| + ε, for n sufficiently large, since k(Ti∗ − λi )xn k → 0 for i = K + 1, . . . , N . By the same construction as in [15], p. 1234 (see also [13], Lemma 1), we can construct a doubly commuting K-tuple of isometries V = (V1 , . . . , VK ) ⊂ L(K), ∗ which is a minimal isometric dilation of the K-tuple Te∗ = (T1∗ , . . . , TK ). Then ∗ ∗ ∗ e V = (V1 , . . . , VK ) is an extension of T = (T1 , . . . , TK ). By the minimality, as in [11] using [10], Section 3 (see also [9], [12]) the K-tuple V = (V1 , . . . , VK ) is a.c. and so is V ∗ , see [11], Lemma 2.1. Moreover, by [12], Proposition 4.1 we can construct a functional calculus for each of them.

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For any contraction A and any µ ∈ D, we will denote by Aµ the operator (A− µ)(Id−µA)−1 . Let K = Ksi ⊕Kui be the decomposition of Vi in a shift and a unitary part, respectively. The decomposition coincides with the decomposition of Viλi (see [17], Proposition I.4.3 and its proof). Moreover (V λ )∗ = ((V1 λ1 )∗ , . . . , (VK λK )∗ ) is an extension of Teλ = (T1 λ1 , . . . , TK λK ). Since k(Ti − λi )xn k → 0 for i = 1, . . . , K, we have kTiλi xn k → 0 for i = 1, . . . , K. By the double commutativity of V = (V1 , . . . , VK ), using [16], Theorem 3, we obtain an orthogonal decomposition K = Ks ⊕ Kr = Ks ⊕ Kr1 ⊕ · · · ⊕ KrK , into reducing invariant subspaces for V , such that Vi s = Vi |Ks is a shift operator and Vi |Kri is a unitary operator for all i = 1, . . . , K. We will write Vi r = Vi |Kr . Form xn = xsn ⊕ xrn , y = y s ⊕ y r with respect to this orthogonal decomposition. Let Pi denote the projection onto ker(Viλi )∗ . By the double commutativity one can see that Pi , Pj commute for i, j = 1, . . . , K. We have also P1 · · · PK H ⊂ Ks . Thus kxrn k 6 kxn − P1 · · · PK xn k 6

K X

kP1 · · · Pi−1 (xn − Pi xn )k

i=1

6

K X

kxn − Pi xn k =

i=1

=

K X

K X

kViλi (Viλi )∗ xn k

i=1

k(Viλi )∗ xn k =

i=1

K X

kTiλi xn k → 0.

i=1

There is also a natural functional calculus for Vs∗ = (V1s ∗ , . . . , VKs ∗ ) and for Vr∗ = (V1r ∗ , . . . , VKr ∗ ), since Vs∗ and Vr∗ are the restrictions of V ∗ . Hence, because of kxrn k → 0, we have |(hn (Te)y, xn )| = |(hn (V ∗ )y, xn )| 6 |(hn (Vs∗ )y s , xsn )| + |(hn (Vr∗ )y r , xrn )| 6 |(hn (Vs∗ )y s , xsn )| + khn k ky r k kxrn k 6 |(hn (Vs∗ )y s , xsn )| + ε, for n sufficiently large. Since kxrn k → 0, we have kxsn k → 1, hence we may assume that kxsn k = 1. Now we know that Vis ∗ is a shift, for i = 1, . . . , K. For 2 < M ∈ N let K W RM be the orthogonal projection onto the space ran Vis M . Using the obvious i=1

extension of [16], Theorem 1, from pairs to K-tuples of doubly commuting shifts, there is M (sufficiently large) such that kRM y s k 6 ε/2. Let y1 = (Id − RM )y s and y2 = RM y s . Then Vis ∗ M y1 = 0 for i = 1, . . . , K. We can write hn (z) =

M −1 X |I|=0

anI z I +

K X i=1

ziM qin (z),

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where z = (z1 , . . . , zK ), anI ∈ C, qin ∈ H ∞ (DK ). Moreover, since anI is a Fourier coefficient of hn we can estimate that |anI | 6 1. Since xsn → 0 weakly, we have for n sufficiently large |(VI∗ I y1 , xsn )| 6 ε/2M K , for all I such that |I| 6 M − 1. Hence |(hn (Vs∗ )y s , xsn )| 6 |(hn (Vs∗ )y2 , xsn )| +

M −1 X

|anI | |(Vs∗ I y1 , xsn )|

|I|=0

+

K X

|(qin (Vs∗ )Vis ∗ M y1 , xsn )|

i=1

6 khn k ky2 k kxsn k +

K X i=1

ε + 0 6 2ε. 2M K

Thus, for n sufficiently large k[y ⊗ xn ]kQ 6 4ε. The second orthogonality condition (3.3) turns out to be symmetric to the previous one. Lemma 4.3. Let T = (T1 , . . . , TN ) be an a.c. N -tuple of commuting contractions. Let S 0 , T 0 be disjoint sets such that S 0 ∪ T 0 = {1, . . . , N }. Assume that {Ti : i ∈ T 0 } is doubly commuting. Let λ = (λ1 , . . . , λN ) ∈ DN and xn → 0 weakly, kxn k = 1 for all n. If k(Ti − λi )xn k → 0 for all i ∈ S 0 and k(Ti∗ − λi )xn k → 0 for all i ∈ T 0 then lim k[xn ⊗ y]kQ = 0 for all y ∈ H. n→∞

Proof. The set {Ti∗ : i ∈ T 0 } is also doubly commuting and we can apply Lemma 4.2 to S = T 0 and T = S 0 . Hence we get k[xn ⊗ y]kQ = |[y ⊗ xn ]kQ(A(T ∗ )) → 0.

5. FINAL REMARKS

Remark 5.1. Notice that (with the same proof) the c.n.u. assumptions for contractions in Theorem 1.2 can be replaced by the requirement that the N -tuple is a.c.. Remark 5.2. Let T = (T1 , . . . , TN ) be a N -tuple fulfilling the assumptions of Theorem 1.1 or Theorem 1.2. Let T a = (T1a , . . . , TNa ) be an N -tuple such that Tia = Ti or Tia = Ti∗ for i = 1, . . . , N . Then Lemma 2.2 and easy calculations show that the dominating property still holds for T a . Thus, Theorem 1.1 or Theorem 1.2, respectively, holds for T a . Some of our methods work even under the weaker conditions that the contractions are only almost doubly commuting.


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Proposition 5.3. Let T = (T1 , T2 ) be an a.c. pair of almost doubly commuting contractions. If the middle part σe1 (T ) of the essential Taylor spectrum is dominating for H ∞ (D2 ), then T = (T1 , T2 ) is reflexive. Proof. We only need to make some remarks, which cover the difference to the proof of Theorem 1.2. First, for any pair of contractions there is a unitary dilation ([17], Theorem I.6.4), so we can construct the functional calculus and it will have the same property as before. Next, since compact operators convert weak convergence of sequences of vectors to strong convergence, using Lemma 2.2, one can see that a compact perturbation does not disturb us in choosing the sequence xn in Section 3. Finally, notice that we consider the spectrum σe1 (T1 , T2 ) and that S and T have exactly one element and all the approximation Lemmas 4.1, 4.2, 4.3 remain true. Remark 5.4. Let us observe that the statement of the preceding proposition is still valid if the dominancy assumption is replaced by one of the following conditions: (i) T1 is of class C0· , T2 is of class C·0 and σe (T ) is dominating for H ∞ (D2 ); (ii) one of the operators is of class C0· and σe0 (T ) ∪ σe1 (T ) is dominating for H ∞ (D2 ); (iii) one of the operators is of class C·0 and σe1 (T ) ∪ σe2 (T ) is dominating for H ∞ (D2 ); (iv) T is diagonally extendable (for definition see [11]) and σe0 (T ) ∪ σe1 (T ) is dominating for H ∞ (D2 ). For the proof of the approximation lemmas for λ ∈ σe1 (T ) we proceed as in the proof of Proposition 5.3. For the left case (i.e. λ ∈ σe0 (T )) one uses the methods of [11]. Finally, the right exterior case (i.e. λ ∈ σe2 (T )) follows by duality. Note added in proof. B. Chevreau announced during the conference “Deuxiemes ` Journées Lilloises de Théorie des Opérateurs”, March 1997, that he removed from Theorem 1.2 the doubly commutativity assumption keeping the existence of a unitary dilation. The research was supported in part by KBN and DAAD.

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ERNST ALBRECHT Fachbereich Mathematik Universit¨ at des Saarlandes Postfach 1150 D–66041 Saarbr¨ ucken GERMANY

MAREK PTAK Institute of Mathematics University of Agriculture Kr´ olewska 6 30–045 Krak´ ow POLAND

E-mail: [email protected]

E-mail: [email protected]

Received April 2, 1997; revised February 2, 1998.