In this work, we give a description of Taylor spectrum of commuting 2-
contractions in terms of characteristic functions of such contractions.The case
of a single obtained by B.Sz.Nagy and C. Foias is generalised in this work.Comment: 10 page

Bendoukha, Berrabah

English

arXiv

2000 Mathematics Subject Classification: 47A10, 47A13.In this paper, we give a description of Taylor spectrum of commuting 2-contractions in terms of characteritic functions of such contractions. The case of a single contraction obtained by B. Sz. Nagy and C. Foias is generalied in this work

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Serdica Math. J. 34 (2008), 543–556TAYLOR SPECTRUM AND CHARACTERISTICFUNCTIONS OF COMMUTING 2-CONTRACTIONSBerrabah BendoukhaCommunicated by S. L. TroyanskiAbstract. In this paper, we give a description of Taylor spectrum ofcommuting 2-contractions in terms of characteritic functions of such con-tractions. The case of a single contraction obtained by B. Sz. Nagy and C.Foias is generalied in this work.1. Introduction. Let H be a Hilbert space. A 2-tuple A = (A1, A2) ofbounded operators onH is called contractive ( or 2-contraction ) if ‖A1h1+A2h2‖2≤ ‖h1‖2 + ‖h2‖2 for all h1, h2 in H. It is equivalent [1] to the condition: A1A?1 +A2A?2 ≤ 1H. If additionaly, operators A1 and A2 commute, then A is calleda commuting 2-contraction. To every commuting 2-contraction A = (A1, A2)corresponds an analytic operator-valued function θA : ID2 → B (DA , DA∗)called characteristic function of A and defined by :(1.1) θA (z1, z2) = −A + DA∗ (1H − z1A∗1 − z2A∗2)−1 (z1.1H, z2.1H)DAwhere2000 Mathematics Subject Classification: 47A10, 47A13.Key words: Characteritic function, 2-contraction, Taylor spectrum.544 Berrabah Bendoukha(a) D2 ={(z1, z2) ∈ C2 : |z1|2 + |z1|2 < 1},(b) B (DA, DA∗) is the set of all bounded operators from DA into DA∗ ,(c) DA∗ = (I −A1A?1 −A2A?2)12 : H → H and DA∗ is the closure of the range ofDA∗ ,(d) DA =[1H −A∗1A1 −A∗1A2−A∗2A1 1H −A∗2A2] 12: H2 → H2 and is the closure of therange of DA,(e) (z11H, z21H) : H2 →H; (z11H, z21H)(h1h2)= z1h1 + z2h2.Characteristic function of commuting n-contraction has been introducedin [3] as a generalization of characteristic function of a single contraction [9].A lot of its remarquable properties have been established in ([2], [4],[9]). Inparticular, it is shown (like in the single case) that the characteristic function is aunitary invariant. It means that characteristic functions of two pure or completelynoncoisometric n-contractions ([2], [4]) A = (A1, . . . , An) and A′ = (A′1, . . . , A′n)coincide if and only if there exists a unitary operator U : H → H such thatA′i = U−1AiU for every i = 1, . . . , n.In the case of a completly nonunitary single contraction, the spectrumcan be described in terms of the characteristic function (see theoerm 4.1 [9]).The aim of this paper is to give a description of Taylor spectrum in the case ofcommuting pure 2-contractions by means of characteristic function (1.1).In section 2 we briefly remind the definition of Taylor spectrum. Section 3contains characterizations of different components of Taylor spectrum. In section4, we investigate the behavior of Taylor spectrum under the action of involutiveautomorphims of unit ball.2. Taylor spectrum. Let A = (A1, A2, . . . , An) be a pure n-contrac-tion. According to ([5], [6], [7], [10]), the Taylor spectrum of A can be defined asfollows. Let Λ (H) be the exterior algebra on n generators e1,. . . , en with identitye0 = 1 and coefficients in H. In other words,Λ (H) ={x⊗ ei1 ∧ · · · ∧ eip : x ∈ H; 1 ≤ i1 · · ·l ip ≤ n ; 1 ≤ p ≤ n}Taylor spectrum 545with the collapsing property : ei ∧ ej + ej ∧ ei = 0. One hasΛ (H) = ⊕nk=1Λk (H) ;Λk (H) = {x⊗ ei1 ∧ · · · ∧ eik : x ∈ H, 1 ≤ i1 · · ·l ik ≤ n} ;Λ0 (H) = H.Consider in Λ (H) operator:BA : Λ (H) → Λ (H) : BA(x⊗ ei1 ∧ · · · ∧ eip)=n∑k=1Ak (x)⊗ek∧ei1∧· · ·∧eip .It is not difficult to see that B2A = 0 and RanBA ⊆ Ker BA . DecompositionΛ (H) = ⊕nk=1Λk (H) gives a rise to a cochain K (A,H), the so-called Koszulcomplex K (A,H) associated to A on H as follows:K (A1, A2,H) : {0} → H = Λ0 (H)B0A−→ · · ·Bn−1A−−−→ Λn (H) → {0}where BkA is the restriction of BA to the subspace Λk (H). Complex K (A,H) issaid to be exact (or regular) if:{0} = ker B0A, RanB0A = ker B1A, . . . ,RanBn−2A = ker Bn−1A , RanBn−1A = Λn (H) .Definition 1. The Taylor spectrum of n-contraction is the set:σT (A) = {z = (z1, z2,..., zn) ∈ Cn : K (A1 − z1, . . . , An − zn;H) is not exact} .Let us now suppose that n = 2. Then,Λ (H) = Λ0 (H)⊕ Λ1 (H)⊕ Λ2 (H)= (H⊗ e0)⊕ ((H⊗ e1)⊕ (H⊗ e2))⊕ (H⊗ e1 ∧ e2) .According to this direct sum, operator BA admits the matrix representa-tionBA =0 0 0 0A1 0 0 0A2 0 0 00 −A2 A1 0546 Berrabah Bendoukhaand then operators B0(A1,A2) and B1(A1,A2)have the forms:(2.1)B0(A1,A2) (x) = A1 (x)⊕A2 (x) , (x, y ∈ H) ,B1(A1,A2) (x⊕ y) = −A2 (x) + A1 (y) , (x, y ∈ H) .According to Definition 1 and formula (2.1), one hasσT (A1, A2) = σ(1)T (A1, A2) ∪ σ(2)T (A1, A2) ∪ σ(3)T (A1, A2) ,where(2.2) (z1, z2) ∈ σ(1)T (A1, A2) ⇔∃ x ∈ H : x 6= 0, (A1 − z1) x = (A2 − z2)x = 0,(2.3) (z1, z2) ∈ σ(2)T (A1, A2) ⇔∃ (x1, x2) ∈ H2 :{(A1 − z1) x1 − (A2 − z2) x2 = 0(x1, x2) 6= ((A1 − z1)h , (A2 − z2) h) , ∀h ∈ H,(2.4) (z1, z2) ∈ σ(3)T (A1, A2) ⇔∃ y ∈ H : y 6= (A1 − z1) x1 − (A2 − z2) x2, ∀ (x1, x2) ∈ H2.Remark 1. σ(1)T (A1, A2) is called the ponctual joint Taylor spectrum.Taylor joint spectrum generalizes the one variable notion of spectrum. It is anonempty compact subset of Cn. The reader can find an excellent account ofthe Taylor spectrum and its relations with other multiparameter spectral theo-ries in [6]. Note also that in [2] a description of Harte spectrum by means ofcharacteristic function is given.Throughout this paper, we will suppose that if A = (A1, A2) is a com-muting 2-contraction, then operator DA∗ = (I −A1A?1 −A2A?2)12 is one to one.Note that pure ( and more generally completely non coisometric) commutig 2-contractions ([2], [3], [4]) satisfy this condition. Indeed, if A = (A1, A2) is apure 2-contraction then, the decreasing sequence of positive bounded operators((A1A?1 + A2A?2)n)n∈IN admits a strong limit A∞ = 0. Because of that,DA∗ (x) = 0 ⇒ D2A∗ (x) = (I −A1A?1 −A2A?2) x = 0⇒ x = (A1A?1 + A2A?2)x ⇒ x = (A1A?1 + A2A?2)n x , ∀n = 0, 1, 2, . . .⇒ x = A∞ (x) = 0.Taylor spectrum 547Using relations ADA = DA∗A and A∗DA∗ = DAA∗ ([3]), it can beproven that:DA∗ = (I −A1A?1 −A2A?2)12 is one to one⇔DA =[1H −A∗1A1 −A∗1A2−A∗2A1 1H −A∗2A2] 12is one to one.3. Characterization of Taylor spectrum.Lemma 1. Let A = (A1, A2) be a commuting 2-contraction such thatDA∗ is one to one. Then,A (x, y) = z1x+z2y,((z1, z2) ∈ C2, (x, y) ∈ H2)⇔ θA (z1, z2)(DA(xy))= 0.P r o o f. It follows directly from relation(3.1) θA (z1, z2)DA(xy)= DA∗ (1H − z1A∗1 − z2A∗2)−1 [(z1x + z2y)− (A1x + A2y)] .Lemma 2. Let A = (A1, A2) be a commuting pure 2-contraction suchthat DA∗ is one to one, (z1, z2) ∈ C2 and x ∈ H. Then,A∗ (x) =(z1.xz2.x)⇔ (θA (z1, z2))∗DA∗ (x) = 0.P r o o f. Since in this case D2A∗ (x) = (1H − z1A1 − z2A2) x, then thenecessary condition is a direct conequence of relation,(3.2) (θA (z1, z2))∗DA∗ (x)= DA(−(A∗1xA∗2x)+(z1z2)((1H − z1A1 − z2A2)−1)D2A∗ (x)).P r o o f o f t h e s u f f i s a n t c o n d i t i o n. Hence (θA (0, 0))∗DA∗ (x) =−(A∗1xA∗2x), one can whithout loosing the generality suppose that (z1, z2) 6=(0, 0).548 Berrabah Bendoukha(θA (z1, z2))∗DA∗ (x) = 0⇒ DA(−(A∗1xA∗2x)+(z1z2)((1H − z1A1 − z2A2)−1)D2A∗ (x))= 0⇒−A∗1x + z1((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0−A∗2x + z2((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0⇒−A∗1x + z1((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0−A∗2x + z2((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0⇒−z2A∗1x + z2z1((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0−z1A∗2x + z1z2((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2) x = 0⇒{z1A∗2x = z2A∗1x−z2A∗1x + z2z1((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2)x = 0⇒{z1A∗2x = z2A∗1x−z2A∗1x + z2((1H − z1A1 − z2A2)−1)(z1I − z1A1A∗1 − z2A2A∗1) x = 0⇒{z1A∗2x = z2A∗1xz2((1H − z1A1 − z2A2)−1)((z1I −A∗1))x = 0⇒ ((z1I −A∗1))x = 0 ⇒ A∗1x = z1x.On the other hand,1. z1A∗2x = z2A∗1x, A∗1x = z1x and z1 6= 0 ⇒ A∗2x = z2x.2. Putting A∗1x = z1x and z1 = 0 in the relation−A∗2x + z2((1H − z1A1 − z2A2)−1)(I −A1A∗1 −A2A∗2)x = 0,and multiplying by (1H − z2A2)−1, one obtains(1H − z2A2) A∗2x = z2 (I −A2A∗2)x.This last relation is equivalent to A∗2x = z2x .Taylor spectrum 549Proposition 1. Let A = (A1, A2) be a commuting pure 2-contractionsuch that DA∗ is one to one. Then, (z1, z2) ∈ σ(1)T (A1, A2) if and only if equationθA (z1, z2) X = 0 admits at least two nontrivial solutions DA (X1) and DA (X2)such that, X1 =(x0), X2 =(0x), x ∈ H.P r o o f. One has(z1, z2) ∈ σ(1)T (A1, A2)⇔ ∃x ∈ H : x 6= 0, (A1 − z1)x = (A2 − z2) x = 0⇔ ∃x ∈ H : x 6= 0, A1 (x) = z1.x and A2 (x) = z2.x⇔ ∃x ∈ H : x 6= 0, A(x0)= z1.x + z2.0 and A(0x)= z1.0 + z2.x⇔ θA (z1, z2)(DA(x0))= 0 and θA (z1, z2)(DA(0x))= 0.To end the proof, it is sufficient to remark thatx 6= 0 ⇔ DA(x0)6= 0 ⇔ DA(0x)6= 0.Proposition 2. Let A = (A1, A2) be a commuting pure 2-contractionsuch that DA∗ is one to one. Then, (z1, z2) ∈ σ(2)T (A1, A2) if and only if theequation θA (z1, z2) Y = 0 admits at least one non trivial solution Y = DA (X)such that X 6=((A1 − z1)h(z2 −A2)h), ∀h ∈ H.P r o o f. One has,(z1, z2) ∈ σ(2)T (A1, A2) ⇔∃ (x1, x2) ∈ H2 : (A1 − z1) x1 − (A2 − z2) x2 = 0,(x1, x2) 6= ((A1 − z1)h , (A2 − z2)h) ∀h ∈ H,(x1, x2) 6= (0, 0) .It means that(z1, z2) ∈ σ(2)T (A1, A2) ⇔∃ (x1, x2) ∈ H2 : A1 (x1) + A2 (−x2) = z1.x1 + z2. (−x2) ,(x1, x2) 6= ((A1 − z1)h , (A2 − z2)h) ; ∀h ∈ H,(x1, x2) 6= (0, 0) .550 Berrabah BendoukhaAccording to Lemma 1, one has finally(z1, z2) ∈ σ(2)T (A1, A2) ⇔∃ (x1, x2) ∈ H2 : θA (z1, z2)(DA(x1−x2))= 0,(x1, x2) 6= ((A1 − z1) h , (A2 − z2) h) ; ∀h ∈ H,(x1, x2) 6= (0, 0) .Proposition 3. Let A = (A1, A2) be a commuting 2-contraction such thatDA∗ is one to one and (z1, z2) ∈ C2. Assume that equation (θA (z1, z2))∗DA∗ (y)= 0 admits at least one non trivial solution. Then,(z1, z2) ∈ σ(3)T (A1, A2) .P r o o f. Suppose that (θA (z1, z2))∗DA∗ (y) = 0 admits at least one nontrivial solution y. According to Lemma 2, it means thatA∗ (y) =(z1.yz2.y).Thus, for every (x1, x2) ∈ H2, one has0 =〈A∗ (y)−(z1.yz2.y),(x2−x1)〉H2=〈((A∗1 − z1) .y(A∗2 − z2) .y),(x2−x1)〉H2= 〈(A∗1 − z1) .y, x2〉H + 〈(A∗2 − z2) .y,−x1〉H= 〈y, (A1 − z1) x2〉H − 〈.y, (A2 − z2)x1〉H= 〈y, (A1 − z1) x2 − (A2 − z2) x1〉H .Thus,y /∈ RanD1(A1−z1,A2−z2)and finally(z1, z2) ∈ σ(3)T (A1, A2) .Taylor spectrum 5514. Taylor spectrum and involutive automorphisms of unitball. In this section we invetigate the Taylor spectrum under action of involutiveautomorphisms of unit ball D2. Such automorphisms are defined in [8] byΦλ (z) = λ−√1− ‖λ‖21− 〈z, λ〉(z −(1−√1− ‖λ‖2)〈z, λ〉‖λ‖2.λ),λ ∈ D2, λ = (λ1, λ2) 6= 0.Connection between automorphisms of unit ball and multicontractionshas been made in [2] and [4] where some very interesting properties have beenestablished. In particular, if Φλ is an involutive automorphism of unit ball andA = (A1, A2) is a commutative 2-contraction then, (see [2], sections 4 and 5) onecan define operatorΦλ (A) = Λ−DΛ∗ (1H −AΛ∗)−1 ADΛwhere the operator Λ = (λ1.1H , λ2.1H) is defined from H2 into H by :Λ (x1, x2) = λ1.x1 + λ2.x2.Proposition 4 and Theorem 1 below summarise important for us results obtainedin [2] and [4].Proposition 4. Let Φλ an involutive automorphism of unit ball andA = (A1, A2) a commutative 2-contraction. Then, Φλ (A) is a commutative 2-contraction such that,I − Φλ (A)∗Φλ (A) = DΛ (1H −A∗Λ)−1 (I −A∗A) (1H − Λ∗A)−1 DΛ,(4.1)I − Φλ (A) Φλ (A)∗ = DΛ∗ (1H −AΛ∗)−1 (I −AA∗) (1H − ΛA∗)−1 DΛ∗ .(4.2)Theorem 1. Let Φλ be an involutive automorphism of unit ball andA = (A1, A2) a commutative 2-contraction. Then,1. Operators Ω :–DΦλ(A) →–DA and Ω∗ :–DΦλ(A)∗ →–DA∗ defined byΩ(DΦλ(A) (X))= DA (1H − Λ∗A)−1 DΛ (X) ,Ω∗(DΦλ(A)∗ (X))= DA∗ (1H − ΛA∗)−1 DΛ∗ (X)are unitaries.552 Berrabah Bendoukha2. θΦλ(A) and θA are connected by the relationΩ∗θΦλ(A) (z1, z2) = θA (Φλ (z1, z2)) Ω.It can be shown that:(4.3) (1H −AΛ∗)−1 = (1H − z1A1 − z2A2)−1 ,(4.4) (1H2 − Λ∗A)−1 =[(1H − λ1A1 − λ2A2)−1 (1H − λ2A2)λ2A1.(1H − λ1A1 − λ2A2)−1λ1A2.(1H − λ1A1 − λ2A2)−1 (1H − λ1A1 − λ2A2)−1 (1H − λ1A1)](4.5) DΛ =1|λ1|2 + |λ2|2×|λ2|2 + |λ1|2√1− |λ2|2 − |λ1|2 λ1λ2(√1− |λ2|2 − |λ1|2 − 1)λ1λ2(√1− |λ2|2 − |λ1|2 − 1)|λ1|2 + |λ2|2√1− |λ2|2 − |λ1|2 .Using (4.3), one can show that(4.6) Φλ (A) = (B1 (λ) , B2 (λ))whereB1 (λ) = λ1.1H −√1− |λ2|2 − |λ1|2 (1H − λ1A1 − λ2A2)−1×{A1(|λ2|2 + |λ1|2√1− |λ2|2 − |λ1|2)+ λ1λ2A2(√1− |λ2|2 − |λ1|2 − 1)},B2 (λ) = λ2.1H −√1− |λ2|2 − |λ1|2 (1H − λ1A1 − λ2A2)−1×{λ1λ2A1(√1− |λ2|2 − |λ1|2 − 1)+ A2(|λ1|2 + |λ2|2√1− |λ2|2 − |λ1|2)}.Taylor spectrum 553Formulas (4.3), (4.4), and (4.5) allow us to find the explicit forms ofoperators Φλ (A), Ω and Ω∗. On the other hand, from (4.2) follows that if DA∗is one to one, then DΦλ(A)∗ is also one to one. Using Theorem 1, one can obtainthe following caracterization for Taylor spectrum of Φλ (A) in terms of solutionsof equationsθA (z1, z2) DA (X) = 0 and (θA (z1, z2))∗DA∗ (y) = 0.Proposition 5. Φλ (z1, z2) ∈ σ(1)T (Φλ (A)) if and only if equationθA (z1, z2) DA (X) = 0 admits at least two nontrivial solutions X1 and X2 suchthat,X1 = (1H − Λ∗A)−1 DΛ(y0), X2 = (1H − Λ∗A)−1 DΛ(0y), y ∈ H.P r o o f. According to Proposition 3, Φλ (z1, z2) ∈ σ(1)T (Φλ (A)) if andonly if there exists a nonnul vector y ∈ H such that,Y1 =(y0), Y2 =(0y)are solutions of equationθΦλ(A) (Φλ (z1, z2)) DΦλ(A) (Y ) = 0.Using Theorema 1 and the fact that Φλ is involutive, it is equivalent to theexistence of a nonnul vector y ∈ H such that,Y1 =(y0), Y2 =(0y)are solutions of equationΩ−1∗ θA (z1, z2)ΩDΦλ(A) (Y ) = Ω−1∗ θA (z1, z2)DA((1H − Λ∗A)−1 DΛ (Y ))= 0which is equivalent to the equationθA (z1, z2)DA((1H − Λ∗A)−1 DΛ (Y ))= 0.554 Berrabah BendoukhaCorollary 1. Let (z1, z2) ∈ C2. Then, (z1, z2) ∈ σ(1)T (A) and Φλ (z1, z2)∈ σ(1)T (Φλ (A)) if and only if there exists two nonnul vectors x and y in H suchthat vectors:X1 =(x0), X2 =(0x),Y1 = (1H − Λ∗A)−1 DΛ(y0)and Y2 = (1H − Λ∗A)−1 DΛ(0y)are both solutions of equationθA (z1, z2) DA (X) = 0.Proposition 6. Let (z1, z2)∈C2. Assume that equation (θA (z1, z2))∗DA∗(y)= 0 admits at least one non trivial solution. Then,Φλ (z1, z2) ∈ σ(3)T (Φλ (A)) .P r o o f. Note at first that operator DΛ∗ is invertible. Since Φλ is involu-tive then, according Theorem 1,(θA (z1, z2))∗DA∗ (y) = 0⇔ (θA (Φλ (Φλ (z1, z2))))∗DA∗ (y) = 0⇔(Ω∗θΦλ(A) (Φλ (z1, z2)) Ω−1)∗DA∗ (y) = 0⇔ Ω(θΦλ(A)Φλ (z1, z2))∗Ω−1∗ DA∗ (y) = 0⇔(θΦλ(A)Φλ (z1, z2))∗Ω−1∗ DA∗ (y) = 0⇔(θΦλ(A)Φλ (z1, z2))∗Ω−1∗ DA∗((1H − ΛA∗)−1 DΛ∗D−1Λ∗ (1H − ΛA∗) y)= 0⇔(θΦλ(A)Φλ (z1, z2))∗DΦλ(A)∗(D−1Λ∗ (1H − ΛA∗) y)= 0⇔(θΦλ(A)Φλ (z1, z2))∗DΦλ(A)∗ (X) = 0,whereX = D−1Λ∗ (1H − ΛA∗) y.Since y is nonnul then, X = D−1Λ∗ (1H − ΛA∗) y is also nonnul and accordingProposition 3, it follows thatΦλ (z1, z2) ∈ σ(3)T (Φλ (A1, A2)) .Taylor spectrum 555Proposition 7. Φλ (z1, z2) ∈ σ(2)T (A1, A2) if and only if the equationθA (z1, z2) DA (X) = 0 admits at least one solution X such thatX 6=(B1 (λ)h− w1.hB2 (λ)h− w2.h), ∀h ∈ Hwhere (w1, w2) = Φλ (z1, z2).P r o o f. It follows immeditely from Proposition 2. REFERE NCES[1] W. Arveson. Subalgebras of C∗−algebras. III. Multivariable operatortheory. Acta Math. 181 (1998), 159–228.[2] C. Benhida, D. Timotin. Characteristic functions for multicontractionsand automorphisms of the unit ball, Preprint.[3] T. Bhattacharyya, J. Eschmeier, J. Sarkar. Characteristic functionof a pure contractive tuple. Integral Equations Operator theory 53, 1 (2005),23–32.[4] T. Bhattacharyya, J. Eschmeier, J. Sarkar. On completely non coiso-metric tuples and their characteristic functions, Preprint.[5] R. E. Curto, G. C. Hernandez. Ajoint spectral caracterisation of prime-ness for C∗−algebras. Proc. Amer. Math. Soc. 125, 11 (1997), 3299–3301.[6] R. Curto. Application of several complex variables to multiparameter spec-tral theory (Eds J. B. Conway and B. B.Morell) Surveys of Some recentResults in Operator Theory, Vol. II, Longman Publ. co., London (1988),25–90.[7] J. Gime´nez. Joint spectrum of subnormal n-tuples of composition operators.Proc. Amer. Math. Soc. 130, 7 (1669), 2015–2023.[8] W. Rudin. Function Theory in the unit ball of Cn. Springer-Verlag, NewYork-Berlin,1980.556 Berrabah Bendoukha[9] B. Sz. Nagy, C. Foias. Harmonic analysis of operators on Hilbert sace.North-Hollan, 1970.[10] J. L. Taylor. A joint spectrum for several commuting operators. J. Funct.Anal. 6 (1970) 72–191.University of MostaganemDepartment of MathematicsB.O. 227, Mostaganem (27000)Algeriae-mail: bbendoukha@gmail.com Received December 23, 2006

Taylor spectrum and characteristic functions of commuting 2-contractions

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