New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.27(2021) 349–362.

The fundamental operator tuples

associated with the symmetrized polydisc

Bappa Bisai and Sourav Pal

Abstract. A commuting tuple of operators (S1, . . . , Sn−1, P), defined on a Hilbert space H, for which the closed symmetrized polydisc is a spectral set, is called a Γn-contraction. To every Γn-contraction, there is a unique operator tuple (A1, . . . , An−1), defined onRan(I−P^∗P), such that

Si−S^∗n−iP=DPAiDP, DP = (I−P^∗P)¹², i= 1, . . . , n−1.

This is called thefundamental operator tupleorFO-tupleassociated with the Γn-contraction. TheFO-tuple of a Γn-contraction completely deter- mines the structure of a Γn-contraction and provides operator model and complete unitary invariant for them. In this note, we analyze the FO-tuples and find some intrinsic properties of them. Given a Γn-contraction (S1, . . . , Sn−1, P) and n−1 operatorsA1, . . . , An−1 de- fined onRanDP, we provide a necessary and sufficient condition under which (A1, . . . , An−1) becomes theFO-tuple of (S1, . . . , Sn−1, P). Also for given tuples of operators (A1, . . . , An−1) and (B1, . . . , Bn−1), defined on a Hilbert spaceE, we find a necessary condition and a sufficient con- dition under which there exist a Hilbert spaceHand a Γn-contraction (S1, . . . , Sn−1, P) onHsuch that (A1, . . . , An−1) becomes theFO-tuple of (S1, . . . , Sn−1, P) and (B1, . . . , Bn−1) becomes the FO-tuple of the adjoint (S1^∗, . . . , Sn−1^∗ , P^∗).

Contents

1. Introduction and preliminaries 350

2. Properties of the fundamental operator tuples (F_O-tuples) 351

3. Admissible fundamental operator tuples 357

References 361

Received April 11, 2019.

2010Mathematics Subject Classification. 47A13, 47A20, 47A25, 47A45.

Key words and phrases. Symmetrized polydisc, fundamental operator tuple.

The first named author is supported by a Ph.D fellowship of the University Grants Com- missoin (UGC). The second named author is supported by the Seed Grant of IIT Bombay with Grant No. RD/0516-IRCCSH0-003 (15IRCCSG032), the INSPIRE Faculty Award (Award No. DST/INSPIRE/04/2014/001462) of DST, India and the MATRICS Grant (Award No. MTR/2019/001010) of Science and Engineering Research Board (SERB) of DST, India.

ISSN 1076-9803/2021

349

1. Introduction and preliminaries

The symmetrized n-disk or simply the symmetrized polydisc Γ^◦_n=Gn=









 X

1≤i≤n

zi, X

1≤i<j≤n

zizj, . . . ,

n

Y

i=1

zi



: |z_i|<1, i= 1, . . . , n





 is a polynomially convex but non-convex domain which has attracted con- siderable attentions in past two decades because of its rich function theory, complex geometry, operator theory and its connection with the appealing and difficult problem of µ-synthesis. We mention here only a few of the numerous references that are relevant to the operator theory on the sym- metrized polydisc of several dimensions, [1, 2, 8, 5, 6, 7, 9, 10, 11, 12, 14, 15].

An interested reader can also see the references there. A tuple of commut- ing operators (S1, . . . , Sn−1, P) defined on a complex Hilbert spaceHis said to be a Γn-contraction if Γn is a spectral set for (S1, . . . , Sn−1, P), i.e., if the Taylor joint spectrumσ_T(S₁, . . . , Sn−1, P) ⊆Γ_n and the von Neumann inequality

kf(S1, . . . , Sn−1, P)k ≤ kfk_∞,Γn = sup

(z1,...,zn)∈Γn

|f(z1, . . . , zn)|

holds for all rational functions f with singularities off Γ_n. It was shown by the second named author of this article ([12], Theorem 3.3) that to every Γ_n-contraction (S₁, . . . , Sn−1, P), there is a unique operator tuple (A1, . . . , An−1) such that

S_i−S_n−i^∗ P =D_PA_iD_P, for each i= 1, . . . , n−1.

For its pivotal role in deciphering the structure of a Γ_n-contraction [11], pro- ducing an operator model and constituting a complete unitary invariant for Γ_n-contractions [5, 6, 12, 13, 14] (A₁, . . . , An−1) is called the fundamental operator tuple or the F_O-tuple of (S1, . . . , Sn−1, P).

There are three main results in this paper. First, Theorem 2.1 pro- vides a necessary and sufficient condition under which an operator tuple (A₁, . . . , An−1) becomes the F_O-tuple of a given Γ_n-contraction. A natural question arises; given two tuples of operators (A1, . . . , An−1) and (B1, . . . , Bn−1) defined on some certain Hilbert spaces, does there exist a Γ_n-contracti- on (S1, . . . , Sn−1, P) such that (A1, . . . , An−1) becomes theF_O-tuple of (S1, . . . , Sn−1, P) and (B1, . . . , Bn−1) becomes theF_O-tuple of its adjoint (S₁^∗, . . . , S_n−1^∗ , P^∗) ? We answer this question in Lemma 3.1 and Theorem 3.4 by finding separately a necessary condition and a sufficient condition. This is considered to be the second main result of this paper. In [14], it was shown that the commutators [A^∗_i, Aj], where [P, Q] = P Q−QP, are the key ingredients in representing the distinguished varieties in Γ_n. Also in [12] and [13], we have seen that the same commutators determined the conditional dilation on Γn and produced a complete unitary invariant for

C_.0 Γ_n-contractions. In this article, we choose a couple of Γ_n-contractions (S1, . . . , Sn−1, P),(S₁⁰, . . . , S_n−1⁰ , P) instead of taking only one Γn-contraction and study the commutators [A_i, A⁰_j],[A^∗_i, A⁰_j] and also [A_i, A_j],[A^∗_i, A_j], where (A⁰₁, . . . , A⁰_n−1) is the F_O-tuple of the Γ_n-contraction (S₁⁰, . . . , S_n−1⁰ , P). As a consequence, we obtain a few new interrelations between a pair of Γn- contractions and their F_O-tuples which are presented in the third main re- sult Theorem 2.9 and its corollaries. En route we find few more interesting properties of theF_O-tuple of a Γn-contraction. Our results on one hand gen- eralize the existing similar results for Γ₂-contractions [4], and on the other hand reflect new light on the possibility of extending operator theory from Γ₂ to Γ_n forn > 2. Indeed, there are notable major differences in operator theory when we move from 2 to higher dimensions and the main underly- ing reason is that rational dilation succeeds on the symmetrizedn-disk for n= 2, ([5]) but fails for n≥3 ([13]).

Note. The arxiv (https://arxiv.org/archive/math) reference [14] has been split into several parts for publications and the present article is one of them.

2. Properties of the fundamental operator tuples (F_O-tuples) We begin this section with a necessary and sufficient condition under which a tuple of operator becomes the F_O-tuple of a Γn-contraction. This is one of the main results of this paper.

Theorem 2.1. A tuple of operators (A1, . . . , An−1) defined on D_P is the F_O-tuple of aΓ_n-contraction(S₁, . . . , Sn−1, P) if and only if (A₁, . . . , An−1) satisfy the following operator equations in X1, . . . , Xn−1:

D_PS_i =X_iD_P +X_n−i^∗ D_PP , i= 1, . . . , n−1.

Proof. First let (A₁, . . . , An−1) be theF_O-tuple of (S₁, . . . , Sn−1, P). Then Si−S_n−i^∗ P =DPAiDP fori= 1, . . . , n−1.

Now

DP(AiDP +A^∗_n−iDPP) = (Si−S_n−i^∗ P) + (S_n−i^∗ −P^∗Si)P

= (I−P^∗P)S_i

=D_P²S_i.

Therefore, if J = D_PS_i−A_iD_P −A^∗_n−iD_PP then J : H → D_P and also DPJ = 0. Now

hJ h, D_Ph⁰i=hD_PJ h, h⁰i= 0 for all h, h⁰ ∈ H.

This shows that J = 0 and henceAiDP +A^∗_n−iDPP =DPSi.

Conversely, let (X₁, . . . , Xn−1) be a tuple of operators on D_P such that D_PSi=XiD_P +X_n−i^∗ D_PP fori= 1, . . . , n−1.

Also suppose that (F₁, . . . , Fn−1) is the F_O-tuple of (S₁, . . . , Sn−1, P). We need to show that (X1, . . . , Xn−1) = (F1, . . . , Fn−1). Since we just proved that (F₁, . . . , Fn−1) satisfies the above mentioned operator equations, we have

FiDP +F_n−i^∗ DPP =DPSi =XiDP +X_n−i^∗ DPP.

and consequently

(Xi−F_i)DP+(Xn−i−Fn−i)^∗DPP = (Xn−i−Fn−i)DP+(Xi−F_i)^∗DPP = 0.

Let for eachi

Yi=Xi−Fi, Yn−i=Xn−i−Fn−i. Then for each i

YiDP +Y_n−i^∗ DPP =Yn−iDP +Y_i^∗DPP = 0. (2.1) To complete the proof, we need to show thatY1 =· · ·=Yn−1 = 0. We have

YiD_P +Y_n−i^∗ D_PP = 0 or Y_iD_P =−Y_n−i^∗ D_PP or DPYiDP =−D_PY_n−i^∗ DPP

or DPY_i^∗DP =P^∗DPY_i^∗DPP =P^∗2DPY_i^∗DPP² =· · ·

We obtained the equalities in the last line by applying (2.1). Thus we have DPY_i^∗DP =P^∗nDPY_i^∗DPPⁿ (2.2) for all n= 1,2, . . .. Now consider the series

∞

X

n=0

kD_PPⁿhk² =

∞

X

n=0

hD_PPⁿh, D_PPⁿhi

=

∞

X

n=0

hP^∗nD²_PPⁿh, hi

=

∞

X

n=0

hP^∗n(I−P^∗P)Pⁿh, hi

=

∞

X

n=0

h(P^∗nPⁿ−P^∗n+1Pⁿ⁺¹h, hi

=

∞

X

n=0

(kPⁿhk²− kPⁿ⁺¹hk²)

= khk²− lim

n→∞kPⁿhk².

khk ≥ kP hk ≥ kP²hk ≥ · · · ≥ kPⁿhk ≥ · · · ≥0.

So lim

n→∞kPⁿhk² exists. Therefore, the series is convergent and thus we have

n→∞lim kD_PPⁿhk² = 0. So

kD_PY_i^∗D_Phk = kP^∗nD_PY_i^∗D_PPⁿhk by (2.2)

≤ kP^∗nkkD_pY_i^∗kkD_PPⁿhk

≤ kD_pY_i^∗kkD_PPⁿhk →0.

So DPY_i^∗DP = 0 and henceYi = 0 for each i= 1, . . . , n−1.

The next few results will provide some useful algebraic relations between Γ_n-contractions and their F_O-tuples.

Theorem 2.2. Suppose (S1, . . . , Sn−1, P) and (S₁⁰, . . . , S_n−1⁰ , P) are two commuting Γ_n-contractions on a Hilbert space H. Let (A₁, . . . , An−1) and (A⁰₁, . . . , A⁰_n−1)be the commutingF_O-tuples of(S1, . . . , Sn−1, P)and(S₁⁰, . . . , S_n−1⁰ , P), respectively and suppose A_iA⁰_j =A⁰_jA_i for any i, j= 1, . . . , n−1.

Then for each i, j= 1, . . . , n−1 we have

S_i^∗S_j⁰ −S_n−j^0∗ Sn−i =DP(A^∗_iA⁰_j−A^0∗_n−jAn−i)DP.

Proof. We have that (A₁, . . . , An−1) is a commuting tuple satisfying Si−S_n−i^∗ P =DPAiDP, fori= 1, . . . , n−1.

and (A⁰₁, . . . , A⁰_n−1) is a commuting tuple satisfying

S_j⁰ −S_n−j^0∗ P =D_PA⁰_jD_P, forj= 1, . . . , n−1.

Then

S_i^∗S_j⁰ =S_i^∗ S_n−j^0∗ P+DPA⁰_jDP

=S_i^∗S_n−j^0∗ P+S_i^∗D_PA⁰_jD_P

=S_n−j^0∗ S^∗_iP+S_i^∗D_PA⁰_jD_P

=S_n−j^0∗ (Sn−i−DPAn−iDP) +S_i^∗DPA⁰_jDP

=S_n−j^0∗ Sn−i−S_n−j^0∗ D_PAn−iD_P +S_i^∗D_PA⁰_jD_P. Now from Theorem 2.1 we have

S_i^∗D_P =D_PA^∗_i +P^∗D_PAn−i

and

S_n−j^0∗ D_P =D_PA^0∗_n−j+P^∗D_PA⁰_j. Then

S_i^∗D_PA⁰_j−S_n−j^0∗ D_PAn−i

= DPA^∗_iA⁰_j +P^∗DPAn−iA⁰_j

− DPA^0∗_n−jAn−i+P^∗DPA⁰_jAn−i

=D_PA^∗_iA⁰_j−D_PA^0∗_n−jAn−i. Therefore, S_i^∗S_j⁰ −S_n−j^0∗ Sn−i =D_P

A^∗_iA⁰_j −A^0∗_n−jAn−i

D_P.

A direct consequence of the previous theorem is the following.

Corollary 2.3. Let (S₁, . . . , Sn−1, P) be a Γ_n-contraction with commuting F_O-tuple (A1, . . . , An−1). Then for each i , j = 1, . . . , n−1 we have

S_i^∗S_j−S_n−j^∗ Sn−i =D_P(A^∗_iA_j−A^∗_n−jAn−i)D_P.

Lemma 2.4. Let(S1, . . . , Sn−1, P)be aΓn-contraction on a Hilbert spaceH and let(A₁, . . . , An−1)and(B₁, . . . , Bn−1)be theF_O-tuples of(S₁, . . . , Sn−1, P) and (S₁^∗, . . . , S_n−1^∗ , P^∗), respectively. Then

D_PA_i = (S_iD_P −D_P^∗Bn−iP)|_DP for i= 1, . . . , n−1.

Proof. For h∈ H, we have

(SiDP −DP^∗Bn−iP)DPh = Si(I−P^∗P)h−(DP^∗Bn−iDP^∗)P h

= Sih−SiP^∗P h−(S_n−i^∗ −SiP^∗)P h

= S_ih−S_iP^∗P h−S_n−i^∗ P h+S_iP^∗P h

= (S_i−S_n−i^∗ P)h= (D_PA_i)D_Ph.

Hence,

D_PA_i = (S_iD_P −D_P^∗Bn−iP)|_DP.

Lemma 2.5. Let(S₁, . . . , Sn−1, P)be aΓ_n-contraction on a Hilbert spaceH and let(A1, . . . , An−1)and(B1, . . . , Bn−1)be theF_O-tuples of(S1, . . . , Sn−1, P) and (S₁^∗, . . . , S_n−1^∗ , P^∗), respectively. Then

P Ai =B_i^∗P|D_P, for i= 1, . . . , n−1.

Proof. We observe here that the operators on both sides are defined from D_P toD_P^∗. Leth, h⁰∈ H be any two elements. Then

h(P A_i−B_i^∗P)D_Ph, D_P^∗h⁰i

= hD_P^∗P AiDPh, h⁰i − hD_P^∗B_i^∗P DPh, h⁰i

= hP(D_PAiD_P)h, h⁰i − h(D_P^∗B_i^∗D_P^∗)P h, h⁰i

= hP(S_i−S_n−i^∗ P)h, h⁰i − h(S_i−P S_n−i^∗ )P h, h⁰i

= h(P S_i−P S_n−i^∗ P −S_iP+P S_n−i^∗ P)h, h⁰i= 0.

Hence P Ai=B_i^∗P|_DP for i= 1, . . . , n−1 and the proof is complete.

Lemma 2.6. Let(S1, . . . , Sn−1, P)be aΓn-contraction on a Hilbert spaceH and let(A₁, . . . , An−1)and(B₁, . . . , Bn−1)be theF_O-tuples of(S₁, . . . , Sn−1, P) and (S₁^∗, . . . , S_n−1^∗ , P^∗) respectively. Then for i= 1, . . . , n−1,

(A^∗_iD_PD_P^∗−An−iP^∗)|_DP∗ =D_PD_P^∗Bi−P^∗B_n−i^∗ .

Proof. For h∈ H, we have

(A^∗_iD_PD_P^∗−An−iP^∗)D_P^∗h

= A^∗_iDP(I−P P^∗)h−An−iP^∗DP^∗h

= A^∗_iD_Ph−A^∗_iD_PP P^∗h−An−iD_PP^∗h

= A^∗_iD_Ph−(A^∗_iD_PP+An−iD_P)P^∗h

= A^∗_iDPh−DPSn−iP^∗h [ by Lemma (2.1)]

= (SiDP −DP^∗Bn−iP)^∗h−DPSn−iP^∗h [by Lemma 2.4]

= D_PS_i^∗h−P^∗B_n−i^∗ D_P^∗h−D_PSn−iP^∗h

= D_P(S_i^∗−Sn−iP^∗)h−P^∗B_n−i^∗ D_P^∗h

= DPDP^∗BiDP^∗h−P^∗B^∗_n−iDP^∗h

= (D_PD_P^∗Bi−P^∗B_n−i^∗ )D_P^∗h.

The following theorem is another main result of this section.

Theorem 2.7. Suppose (S1, . . . , Sn−1, P) and (S₁⁰, . . . , S_n−1⁰ , P) are two commuting Γ_n-contractions on a Hilbert space H. Let (A₁, . . . , An−1) and (A⁰₁, . . . , A⁰_n−1)be the commutingF_O-tuples of(S1, . . . , Sn−1, P)and(S₁⁰, . . . , S_n−1⁰ , P), respectively and suppose A_iA⁰_j =A⁰_jA_i for any i, j= 1, . . . , n−1.

Suppose (B₁, . . . , Bn−1) and (B₁⁰, . . . , B_n−1⁰ ) are the F_O-tuples of (S₁^∗, . . . , S_n−1^∗ , P^∗) and (S₁^0∗, . . . , S_n−1^0∗ , P^∗) respectively. If P has dense range, then the following identities hold for i, j= 1, . . . , n−1:

(i) [A_i, A^0∗_j] = [A⁰_n−j, A^∗_n−i] (ii) [B_i, Bn−j] = [B_i⁰, B_n−j⁰ ] = 0 (iii) [B_i^∗, B_j⁰] = [B_n−j^0∗ , Bn−i].

Proof. (i) By Theorem 2.1, we have for each i= 1, . . . , n−1 thatD_PS_i= A_iD_P +A^∗_n−iD_PP and D_PS⁰_i = A⁰_iD_P +A^0∗_n−iD_PP. Multiplying D_PS_i = A_iD_P +A^∗_n−iD_PP byD_PA⁰_n−j from the left we get,

D_PA⁰_n−jD_PSi=D_PA⁰_n−jAiD_P +D_PA⁰_n−jA^∗_n−iD_PP

⇒(S_n−j⁰ −S_j^0∗P)S_i =D_PA⁰_n−jA_iD_P +D_PA⁰_n−jA^∗_n−iD_PP

⇒(S_n−j⁰ Si−S_j^0∗SiP) =DPA⁰_n−jAiDP +DPA⁰_n−jA^∗_n−iDPP.

Similarly, multiplyingD_PS_n−j⁰ =A⁰_n−jD_P+A^0∗_jD_PP byD_PA_ifrom the left we get

DPAiDPS_n−j⁰ =DPAiA⁰_n−jDP +DPAiA^0∗_jDPP

⇒(S_i−S_n−i^∗ P)S_n−j⁰ =D_PA⁰_n−jA_iD_P +D_PA_iA^0∗_jD_PP

⇒(S_n−j⁰ S_i−S_n−i^∗ S_n−j⁰ P) =D_PA⁰_n−jA_iD_P +D_PA_iA^0∗_jD_PP.

On subtraction we get

(S_j^0∗S_i−S_n−i^∗ S_n−j⁰ )P =D_P(A_iA⁰_n−j−A⁰_n−jA_i)D_P +D_P(A_iA^0∗_j −A⁰_n−jA^∗_n−i)D_PP

⇒D_P(A^0∗_jAi−A^∗_n−iA⁰_n−j)DPP =DP(AiA^0∗_j −A⁰_n−jA^∗_n−i)DPP

⇒D_P(A_iA^0∗_j −A^0∗_j A_i+A^∗_n−iA⁰_n−j−A⁰_n−jA^∗_n−i)D_PP

⇒D_P [Ai, A^0∗_j] + [A^∗_n−i, A⁰_n−j]

D_PP = 0

⇒D_P [Ai, A^0∗_j] + [A^∗_n−i, A⁰_n−j]

DP = 0 [since RanP is dense inH]

⇒[A_i, A^0∗_j] = [A⁰_n−j, A^∗_n−i].

(ii) From Lemma 2.5, we have thatP Ai =B_i^∗P|_DP, fori, j= 1, . . . , n−1.

Therefore,

P A_iAn−jD_P =B_i^∗P An−jD_P

⇒P An−jAiDP =B_i^∗P An−jDP

⇒B_n−j^∗ B_i^∗P DP =B^∗_iB_n−j^∗ P DP

⇒[B_i^∗, B_n−j^∗ ]D_P^∗P = 0

⇒[B_i^∗, B_n−j^∗ ] = 0

⇒[B_i, Bn−j] = 0.

Similarly, for each i, j= 1, . . . , n−1 we have [B_i⁰, B_n−j⁰ ] = 0.

(iii) Applying Theorem 2.2 for Γ_n-contractions (S₁^∗, . . . , S_n−1^∗ , P^∗) and (S₁^0∗, . . . , S_n−1^0∗ , P^∗) we getSiS_j^0∗−S_n−j⁰ S_n−i^∗ =DP^∗(B_i^∗B_j⁰ −B_n−j^0∗ Bn−i)DP^∗. From Lemma 2.4,DPAi= (SiDP−D_P^∗Bn−iP)|_DP. Multiplying byA⁰_n−jDP

from right we get

DPAiA⁰_n−jDP = (SiDP −DP^∗Bn−iP)A⁰_n−jDP

⇒D_PAiA⁰_n−jDP =SiDPA⁰_n−jDP −DP^∗Bn−iP A⁰_n−jDP

⇒D_PA⁰_n−jA_iD_P =S_i(S_n−j⁰ −S_j^0∗P)−D_P^∗Bn−iP A⁰_n−jD_P

⇒D_PAiA⁰_n−jDP =SiS_n−j⁰ −SiS_j^0∗P−DP^∗Bn−iB_n−j^0∗ P DP.

Similarly, multiplying DPA⁰_n−j = (S_n−j⁰ DP −DP^∗B_j⁰P)|_DP by AiDP from right we get

D_PA⁰_n−jA_iD_P = (S_n−j⁰ D_P −D_P^∗B⁰_jP)A_iD_P

⇒D_PA⁰_n−jAiDP =S_n−j⁰ DPAiDP −DP^∗B_j⁰P AiDP

⇒D_PA⁰_n−jA_iD_P =S_n−j⁰ (S_i−S_n−i^∗ P)−D_P^∗B_j⁰P A_iD_P

⇒D_PA⁰_n−jA_iD_P =S_n−j⁰ S_i−S_n−j⁰ S_n−i^∗ P −D_P^∗B_j⁰B_i^∗P D_P.

Subtracting those two equations we get

D_P(A_iA⁰_n−j −A⁰_n−jA_i)D_P = (S_n−j⁰ S^∗_n−i−S_iS_j^0∗)P

+D_P^∗(B_j⁰B_i^∗−Bn−iB_n−j^0∗ )P D_P

⇒(SiS_j^0∗−S_n−j⁰ S_n−i^∗ )P +DP^∗(Bn−iB^0∗_n−j−B_j⁰B_i^∗)DP^∗P = 0

⇒D_P^∗(B_i^∗B_j⁰ −B_n−j^0∗ Bn−i)D_P^∗P+D_P^∗(Bn−iB_n−j^0∗ −B_j⁰B_i^∗)D_P^∗P = 0

⇒D_P^∗([B_i^∗, B⁰_j] + [Bn−i, B_n−j^0∗ ])D_P^∗P = 0

⇒[B_i^∗, B⁰_j] = [B^0∗_n−j, Bn−i].

A direct consequence of the previous theorem is the following.

Corollary 2.8. Let(S1, . . . , Sn−1, P)be aΓn-contraction acting on a Hilbert space H and let (A₁, . . . , An−1) and (B₁, . . . , Bn−1) be the F_O-tuples of (S1, . . . , Sn−1, P) and (S₁^∗, . . . , S_n−1^∗ , P^∗), respectively. If [Ai, An−j] = 0 for eachi, j= 1, . . . , n−1and ifP has dense range, then the following identities hold for i, j= 1, . . . , n−1:

(i) [A^∗_j, A_i] = [A^∗_n−i, An−j] (ii) [Bi, Bn−j] = 0

(iii) [B_i^∗, B_j] = [B_n−j^∗ , Bn−i].

Lemma 2.9. Let(S₁, . . . , Sn−1, P)and(S₁⁰, . . . , S_n−1⁰ , P)be twoΓ_n-contract- ions on a Hilbert space H such that P is invertible. Let (A₁, . . . , An−1), (A⁰₁, . . . , A⁰_n−1), (B1, . . . , Bn−1) and (B₁⁰, . . . , B_n−1⁰ ) be as in previous theo- rem. Then [Ai, Aj] = 0 = [A⁰_i, A⁰_j], for i, j = 1, . . . , n−1 if and only if [Bi, Bj] = 0 = [B⁰_i, B_j⁰], for i, j= 1, . . . , n−1.

Proof. Suppose that [Ai, Aj] = 0 = [A⁰_i, A⁰_j] fori, j= 1, . . . , n−1. Since P has dense range, by part (ii) of above theorem, we get [Bi, Bj] = 0 = [B_i⁰, B_j⁰] fori, j= 1, . . . , n−1.

Conversely, let [Bi, Bj] = 0 = [B_i⁰, B_j⁰] for i, j = 1, . . . , n−1. Since P is invertible, P^∗ has dense range too. So applying previous theorem for Γn- contractions (S₁^∗, . . . , S_n−1^∗ , P^∗) and (S₁^0∗, . . . , S_n−1^0∗ , P^∗), we get [A_i, A_j] = 0 = [A⁰_i, A⁰_j] fori, j= 1, . . . , n−1.

Corollary 2.10. Let (S₁, . . . , Sn−1, P) be a Γ_n-contraction on a Hilbert spaceHsuch that P is invertible. Let (A1, . . . , An−1) and(B1, . . . , Bn−1) be as in previous theorem. Then [A_i, An−j] = 0, for i, j = 1, . . . , n−1 if and only if[Bi, Bn−j] = 0, for i, j= 1, . . . , n−1.

3. Admissible fundamental operator tuples

We recall from [16], the notion of characteristic function of a contraction introduced by Sz.-Nagy and Foias. For a contractionP defined on a Hilbert space H, let Λ_P be the set of all complex numbers for which the operator

I−zP^∗ is invertible. For z∈Λ_P, the characteristic function ofP is defined as

ΘP(z) = [−P+zDP^∗(I−zP^∗)⁻¹DP]|_DP. (3.1) By virtue of the relation P D_P =D_P^∗P (Section I.3 of [16]), Θ_P(z) maps D_P = RanD_P into D_P^∗ = RanD_P^∗ for every z in Λ_P. Since for each z ∈ D, ΘP(z) maps D_P into D_P^∗, ΘP induces a multiplication operator M_ΘP from H²(D)⊗ D_P intoH²(D)⊗ D_P^∗, defined by

M_ΘPf(z) = Θ_P(z)f(z), for all f ∈H²(D)⊗ D_P and z∈D. Note thatMΘP(Mz⊗IDP) = (Mz⊗ID_P∗)MΘP.

Lemma 3.1. Let (A₁, . . . , An−1) and (B₁, . . . , Bn−1) be the F_O-tuples of a Γn-contraction (S1, . . . , Sn−1, P) and its adjoint (S₁^∗, . . . , S_n−1^∗ , P^∗), respec- tively. Then for each i= 1, . . . , n−1,

(A^∗_i +An−iz)Θ_P^∗(z) = Θ_P^∗(z)(B_i+B_n−i^∗ z) for all z∈D. (3.2) Proof. We have that

(A^∗_i +An−iz)Θ_P^∗(z)

= (A^∗_i +An−iz)(−P^∗+

∞

X

n=0

zⁿ⁺¹D_PPⁿD_P^∗)

= (−A^∗_iP^∗+

∞

X

n=1

zⁿA^∗_iDPPⁿ⁻¹DP^∗) +(−zA_n−iP^∗+

∞

X

n=2

zⁿAn−iD_PPⁿ⁻²D_P^∗)

= −A^∗_iP^∗+z(A^∗_iD_PD_P^∗−An−iP^∗) +

∞

X

n=2

zⁿ(A^∗_iDPPⁿ⁻¹DP^∗+An−iDPPⁿ⁻²DP^∗)

= −A^∗_iP^∗+z(A^∗_iDPDP^∗−An−iP^∗) +

∞

X

n=2

zⁿ(A^∗_iD_PP+An−iD_P)Pⁿ⁻²D_P^∗

= −P^∗Bi+z(DPDP^∗Bi−P^∗B_n−i^∗ ) +

∞

X

n=2

zⁿDPS2Pⁿ⁻²DP^∗.

The last equality follows by using Theorem 2.1, Lemma 2.5 and Lemma 2.6.

Also we have

Θ_P^∗(z)(B_i+B_n−i^∗ z)

= (−P^∗+

∞

X

n=0

zⁿ⁺¹D_PPⁿD_P^∗)(B_i+B_n−i^∗ z)

= (−P^∗Bi+

∞

X

n=1

zⁿDPPⁿ⁻¹DP^∗Bi) +(−zP^∗B_n−i^∗ +

∞

X

n=2

zⁿD_PPⁿ⁻²D_P^∗B^∗_n−i)

= −P^∗B_i+z(D_PD_P^∗B_i−P^∗B_n−i^∗ ) +

∞

X

n=2

zⁿ(D_PPⁿ⁻¹D_P^∗B_i+D_PPⁿ⁻²D_P^∗B_n−i^∗ )

= −P^∗Bi+z(DPDP^∗Bi−P^∗B_n−i^∗ ) +

∞

X

n=2

zⁿDPPⁿ⁻²(P DP^∗Bi+DPB_n−i^∗ )

= −P^∗B_i+z(D_PD_P^∗B_i−P^∗B_n−i^∗ ) +

∞

X

n=2

zⁿD_PPⁿ⁻²Sn−iD_P^∗

= −P^∗Bi+z(D_PD_P^∗Bi−P^∗B_n−i^∗ ) +

∞

X

n=2

zⁿD_PSn−iPⁿ⁻²D_P^∗.

Hence fori= 1, . . . , n−1 we have (A^∗_i+An−iz)Θ_P^∗(z) = Θ_P^∗(z)(Bi+B^∗_n−iz)

for all z∈Dand the proof is complete.

Note 3.2. Under the hypotheses of Theorem 3.1, the following equations hold:

(B_i^∗+Bn−iz)Θ_P(z) = Θ_P(z)(A_i+A^∗_n−iz), for all z∈D. (3.3) We are now in a position to present one of the main results of this paper.

We first state a result from the literature which provides a characterization of Γn-unitaries. We shall use this result in the proof of the main theorem.

Theorem 3.3 ([7], Theorem 4.2). Let (S₁, . . . , Sn−1, P) be a commuting tuple of bounded operators. Then the following are equivalent.

(1) (S1, . . . , Sn−1, P) is a Γn-unitary,

(2) P is a unitary,(ⁿ⁻¹_n S₁,ⁿ⁻²_n S₂, . . . ,_n¹Sn−1)is aΓn−1-contraction and Si=S_n−i^∗ P for i= 1, . . . , n−1.

Theorem 3.4. Let P be a C.0 contraction on a Hilbert space H. Let A1, . . . , An−1∈ B(D_P) and B1, . . . , Bn−1∈ B(D_P^∗) be such that they satisfy

equations (3.3) and n−1

n (B₁^∗+Bn−1z),n−2

n (B₂^∗+Bn−2z), . . . ,1

n(B^∗_n−1+B1z)

are Γn−1-contractions for all z ∈ T. Then there exists a Γ_n-contraction (S1, . . . , Sn−1, P)such that(A1, . . . , An−1)is theF_O-tuple of(S1, . . . , Sn−1, P) and (B1, . . . , Bn−1) is the F_O-tuple of (S₁^∗, . . . , S_n−1^∗ , P^∗).

Proof. Let us define W :H →H²(D)⊗ D_P^∗ by W(h) =

∞

X

n=0

zⁿ⊗D_P^∗P^∗nh for allh∈ H.

It is evident that

||W h||² =

∞

X

n=0

||D_P^∗P^∗nh||² =

∞

X

n=0

||P^∗nh||²− ||P^∗n+1h||²

=||h||²− lim

n→∞||P^∗nh||².

ThereforeW is an isometry if P is a pure contraction. It is obvious that W^∗(zⁿ⊗ξ) =PⁿDP^∗ξ for all ξ∈ D_P^∗ and n≥0.

Also if M_z is the multiplication operator onH²(D) and if M =M_z⊗I on H²(D)⊗ D_P^∗, then we have

M^∗W h=T_z^∗

∞

X

n=0

zⁿD_P^∗P^∗nh

!

=

∞

X

n=0

zⁿD_P^∗P^∗n+1h=W P^∗h.

ThereforeM^∗W =W P^∗. Since n−1

n (B₁^∗+Bn−1z),n−2

n (B₂^∗+Bn−2z), . . . ,1

n(B^∗_n−1+B1z)

is a Γn−1-contraction for all z ∈ T, it follows from Theorem 3.3 that the multiplication operator tuple (MB₁^∗+Bn−1z, . . . , MB^∗_n−1+B1z, Mz) onL²(D_P^∗) is a Γ_n-unitary. Obviously the Toeplitz operator tuple

(T_B1^∗_+Bn−1z, . . . , T_Bn−1^∗ _+B1z, Tz) onH²(D_P^∗), by being the restriction of (M_B^∗

1+Bn−1z, . . . , M_B^∗

n−1+B1z, M_z) to the joint invariant subspace H²(D_P^∗), is a Γ_n-isometry. Again since H²(D_P^∗) and H²(D)⊗ D_P^∗ are isomorphic, the Γn-isometry on the space H²(D)⊗ D_P^∗ that corresponds to (T_B^∗

1+Bn−1z, . . . , T_B^∗_n−1+B1z, T_z) is

(I⊗B^∗₁+Mz⊗Bn−1, . . . , I⊗B_n−1^∗ +Mz⊗B1, Mz⊗I).

Let us define

S_i =W^∗M_iW fori= 1, . . . , n−1, where

Mi=I ⊗B_i^∗+Mz⊗Bn−i fori= 1, . . . , n−1.