Contractions with Polynomial characteristic functions I. Geometric approach

In this note we study the completely non unitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form T = S&*&* 0&N&* 0&0&C, where $S$ and C^* are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that dimension of ker S^* and dimension of ker C are unitary invariants of $T$ and that N, up to a quasi-similarity is uniquely determined by T. Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.


Introduction
One of the central problems in operator theory is the classification (up to unitary equivalence) of bounded linear operators on an infinite dimensional separable Hilbert space. In all this generality the problem may not have a satisfactory solution. Therefore, the focus of research on this problem was (and it still is) on classes of operators for which one can identify useful (unitary) invariants (e.g. [2], [4], [7], [8], [9], [11] and [12]). The topic of this article is the class of all c.n.u. contractions for which the characteristic functions are operator-valued polynomials.
To be more specific, we recall the following basic concepts. A contraction T on a Hilbert space H, (i.e. T h ≤ h for all h in H) is completely non unitary (or also, c.n.u.) if there is no non-trivial T -reducing subspace M of H such that T | M is unitary. An operator valued analytic function Θ : D → L(E, E * ) for some Hilbert spaces E and E * is said to be contractive if Θ(z)h ≤ h , h ∈ E, and purely contractive when it satisfies also f ∈ E and f = 0.
It is known (see [9], Chapter VI) that there is a bijective correspondence between the class of purely contractive operator valued analytic functions "modulo coincidence" and the class of c.n.u. contractions "modulo unitary equivalence". More precisely, for a given contraction T , contractions T on H and R on K are unitarily equivalent (that is, there is a unitary operator U from H to K such that T = URU), if and only if the characteristic functions Θ T (z) and Θ R (z) coincide. The converse of this fact is also true (see [9], Theorem 3.4). Moreover, for a given L(E, E * )-valued purely contractive analytic function Θ(z) defined on D, there exists a c.n.u. contraction T on some Hilbert space such that Θ T (z) coincide with Θ(z).
In this paper we prove that a c.n.u. contraction T has a polynomial characteristic function of degree n if and only if T admits a matrix representation where S and C * are unilateral shifts of arbitrary multiplicities and N is a nilpotent operator of order n. We stress that the spaces on which one (or two) of the diagonal entries of (1.2) acts may be {0} and the representation (1.2) becomes one of the following For convenience, we will refer to these representations as degenerate forms of (1.2). One of our aims is to identify some properties of the diagonal entries in (1.2) which are unitary invariant of T .
In the next section, we study some relatively simple examples which inspired our research, namely the c.n.u. contractions for which the characteristic function is a scalar polynomial of degree ≤ 1. We point out that even for this simple case there are natural questions which are not trivial.
In Section 3, we give the results already mentioned before, namely -the c.n.u. contractions with a polynomial characteristic function are precisely those which have a upper triangular representation (1.2). Moreover, we define and study two canonical such representations.
In Section 4 we show that the multiplicities of the shift S and the co-shift C in the triangular representation (1.2) are unitary invariants of T .
In Section 5 we define the "minimal" versions of the canonical upper triangular representations of type (1.2) considered in Section 3 and show that their "central" nilpotent entries are quasi-similar.
In Section 6 we study the degenerate forms of the representation (1.2), and characterize them in term of the characteristic function of T .
Then, in Section 7, we consider the special case of the monomial characteristic functions.
Acknowledgement: We thank the referee for his observations and suggestions. In particular, the general result in Step 2 of Theorem 4.1 is due to the referee. Our original proof was less transparent.

An Illustrative Example
In this section, we give a concrete example for which the resolution of our general problem presented in the introduction can be obtained by elementary computations. In particular, this example serves as a paradigm for our general considerations, since it provides a complete picture of the c.n.u. contractions for which the characteristic functions coincides with a scalar polynomial of degree one.
Let l 2 (Z) be the Hilbert space of all square summable sequences given by with the standard orthonormal basis {e n , e 0 , e −n } n≥1 . First we construct a bounded linear operator on l 2 (Z) as follows: Definition 2.1. Let a, b, c be inD and T a,b,c be the bounded linear operator on l 2 (Z) defined by The adjoint of T a.b.c is given by It is easy to see that with respect to the decomposition of l 2 (Z) as where H 1 = span{e 1 , e 2 , . . .}, H 0 = span{e 0 } and H −1 = span{e −1 , e −2 , . . .}, T a,b,c can be expressed as the upper triangular matrix where S is the shift of multiplicity one on H 1 , N (= 0) is the nilpotent of order one on H 0 and C is the co-shift of multiplicity one on H −1 .
Observe that T a,b,c is a contraction if and only if the matrix where γ is inD. We assume the conditions (2.4) and (2.5) on a, b and c so that the operator T a,b,c is a contraction.
Notice that T a,b,c is the bilateral shift when |a| = |b| = 1 and c = 0. Thus to avoid this special case, in what follows we will assume that {a, b} ∩ D = φ.
The squares of the defect operators of the contraction T a,b,c are given by Lemma 2.2. The defect spaces D T a,b,c and D T * a,b,c of T a,b,c are one dimensional if and only if In particular, if a, b are in D, then (2.8) is equivalent to |γ| = 1 in the representation (2.5) of c.
According to this last result T a,b,c will have a scalar characteristic functions if and only if (2.8) holds. Therefore, throughout this Section 2 we will assume that (2.8) holds.
Now we study the special case when c = 0. In this case, by (2.5) we have either |a| = 1 or |b| = 1. Let |a| < 1 and |b| = 1. By (2.6) and (2.7) it follows that D T a,b,0 = Ce 0 and D T * a,b,0 = Ce 1 . Moreover, Θ T a,b,0 (z) = aI Ce 0 →Ce 1 for all z in D, where I Ce 0 →Ce 1 maps e 0 into e 1 . Thus Θ T a,b,0 (z) coincides with the scalar constant function Θ(z) = a for all z in D. Similarly, if |a| = 1 then Θ T a,b,0 (z) coincides with the scalar constant function Θ(z) = b for all z in D.
We have thus obtained the following proposition.  We shall now consider the case when c = 0.
Proof. We conclude from (2.5), (2.6) and (2.7) that In particular, the defect spaces are given by D T a,b,c = Cu and D T * a,b,c = Cv, where u = a(1 − |b| 2 )e −1 − ce 0 and v = (1 − |b| 2 )e 0 −bce 1 . Moreover, we have Thus D T * a,b,c D T a,b,c u = cv and hence by virtue of (1.1), the characteristic function of T a,b,c is given by It is elementary to check the following characterization of a purely contractive scalar polynomial of degree one.
Remark 2.7. Since each characteristic function is pure, so α and β given by α = −ab and β = c with a, b, c as in Proposition 2.3 or Proposition 2.5 will satisfy the conditions in (2.9). Conversely if two complex numbers α and β satisfy (2.9) then it is easy to check that there exist two real numbers a and b satisfying α = −ab and (2.5) with c = β and |γ| = 1.
Since the unitary part of T a,b,c can not be connected to the characteristic function Θ T a,b,c (z), we need to determine when that unitary part is not present, that is, when T a,b,c is c.n.u. The answer to this question is given by the following.
Thus the contraction T a,b,c is not c.n.u. Conversely, assume |c| < 1. Consider a T a,b,c -reducing subspace M of l 2 (Z) such that T a,b,c | M is unitary operator and let (α k ) be a vector in M.
Case 1 (when a = 0): We know from (2.1) that α 0 = 0. As T * l a,b,c (α k ) is in M for all l ≥ 1, (2.6) says precisely that α l = 0 for all l ≥ 1. Also α l = 0 for all l < 0 follows from the fact that This with Cauchy-Schwarz inequality gives for all N ≥ 2 and hence since (α k ) is in l 2 (Z), we must have α 0 = 0 and so (α k ) = 0. Therefore, M = {0} and this completes the proof. Summarizing the previous results we obtain the following. A natural question related to Theorem 2.10 is whether the diagonal terms in a triangular representation of where S and C * are shifts and N is a nilpotent operator of order one are unitary invariants of T . To have a good indication that the question is not trivial, let us consider the case α = 0, or to be more specific, the case a = 0. Then b = 0 and the operator T a,b,c has the triangular representation (2.3) and as well the following for which the shift and the co-shift are unitarily equivalent to their analogues in (2.3) but the two "central" nilpotent operators are 0 and 0 0 0 0 , which are not even similar.
In what follows, we will study the general variant of the above question for the class of c.n.u. contractions with operator valued polynomial characteristic functions.

Polynomial characteristic functions
To start the study of the c.n.u. contractions with polynomial characteristic functions, we need to recall that the order of a nilpotent operator N is the smallest power p for which N p = 0.
be a nilpotent of order n and C be a co-isometry in L(H −1 ). Then the characteristic function of any contraction of the form Proof. It is easy to see that where A is a self adjoint operator. Then, Thus the sum of positive operators A 2 + BB * + CC * is the zero operator and consequently A = 0, B = 0 and C = 0, and In a similar way we deduce Moreover, which is a polynomial of degree ≤ n.
Remark 3.2. If in Proposition 3.1 we consider the representation of the operator Although the characteristic function is a polynomial of degree one. Thus in Proposition 3.1, the degree of the characteristic function can be less than the order of N.
Next we consider a converse of Proposition 3.1.
Theorem 3.3. Let T be a c.n.u. contraction on H such that the characteristic function Θ T is a polynomial of degree n. Then there exists three subspaces H 1 , H 0 , H −1 of H such that H = H 1 ⊕H 0 ⊕H −1 and with respect to that decomposition, T admits the matrix representation is a pure co-isometry; in particular, S and C * are shifts of some multiplicities on H 1 and H −1 respectively. Furthermore in the class of representations (3.1) with the above specified properties there exist unique representations such that holds for all other representations (3.1) in the class; moreover H Proof. By the given assumption, it follows that D T * T * (n+k) D T = 0, which is equivalent to It is easy to see that the subspace H 0 is a nilpotent of order n. The claim follows once we observe that 1 and then by Theorem 3.1. Therefore, we obtain that is a pure isometry. Now we prove that the operator C (c) on the T co- , clearly the largest co-invariant subspace of H such that the restriction of T * on it is isometric. It remains to prove the minimality property of H (c) 1 . For this let 0 is a nilpotent operator of order n and C ′ on H ′ −1 is a co-isometry. Then the computation in the proof of Proposition 3.1 yields Then we must have T k+n D T * H ⊆ H 1 for all k ≥ 0 and hence H Combining Proposition 3.1 and Theorem 3.3 we can readily obtain the following.
Moreover the degree of Θ T is the smallest order of N possible in the matrix representations (3.4) of T .
Remark 3.5. Due to their uniqueness, it is convenient to call the representations (3.2) and (3.3) the canonical and respectively * -canonical representations of the operator T considered in Theorem 3.3. Note that the latter can be obtained also from the canonical representation of T * by passing to the adjoint and by flipping the extreme spaces. In the sequel we will omit the superscripts (c) (indicating that we are dealing with a canonical or * -canonical matrix representation) whenever no confusion can occur.
Remark 3.6. Let T ∈ L(H), T ′ ∈ L(H ′ ) be c.n.u. contractions and assume that their characteristic functions are polynomials of degree n. Let then from the definitions (3.6), (3.7) and (3.9) it readily follows that and consequently that is, the diagonal entries S and S ′ , N and N ′ , C and C ′ are unitarily equivalent too.
Obviously the similar fact holds true for * -canonical representations.

Proof. Let (α k ) be a vector in H
By repeated application of T * a,b,c on (α k ) and then using D T * a,b,c T * l (α k ) = 0 we obtain that or, equivalently |a| < |b|.
Consequently, H

Uniqueness of the shift and the co-shift
In this section we discuss the uniqueness problem for the diagonal of the upper triangular representations of a c.n.u. contraction T on H with a polynomial characteristic function of degree n. It will be convenient to denote the family of these contractions by P n .
First, we need to recall the following fact observed by Sz.-Nagy and the first author (see Remark 3.3, page 134 in [5]) and Parrott (see [10]) on the isometric commutant lifting theorem.
Let T in L(H, K) and U in L(H ′ , K ′ ) be two operators. Then we say that U is a lifting of T if H ′ ⊇ H, K ′ ⊇ K and P K U = T P H , or equivalently, U has a matrix representation of the form In addition, if U is an isometric operator (and hence T is a contraction), then we say that U is an isometric lifting of T . The minimal isometric dilation of a contraction is also an isometric lifting of the contraction. With this preliminary, the fact alluded above is the following. If U on K is an isometric lifting of a contraction T in H, then U admits a decomposition of the form U = U m ⊕ U r on K = ( ∞ k=0 U k H) ⊕ K r where U m is the minimal isometric dilation of the contraction T .
where S ′ is a pure shift on H ′ 1 , N ′ is a nilpotent operator of order n on H ′ 0 and C ′ is a pure co-shift on H ′ −1 . Then the pure shifts S, S ′ and S * are unitarily equivalent and the pure co-shifts C, C ′ and C * are unitarily equivalent.
Proof. It is enough to prove that S * and S ′ are unitarily equivalent. The proof will be separated in three steps in which the result in Step 2 has an independent interest.
Step 1: First observe that if we view T as on a dense set in H ′ 2 . But B ≤ 1 and thus the claim follows.
Step 2: Let T ∈ L(H 1 ⊕ H 2 ) be a c.n.u. contractions such that on H 1 ⊕ H 2 where S is an isometry and B ∈ C 0· (that is, B k → 0 strongly as k → ∞). Let moreover M be the largest invariant subspace of T on which the restriction of T is isometric. Then To establish this fact let G = M ⊖ H 1 and X = P G T | G . Then T | M is a c.n.u. isometric lifting of X and Thus X * k → 0 strongly as k → ∞, that is, X ∈ C ·0 . Moreover it is easy to see that for all k ≥ 0. Hence X k → 0 strongly as k → 0. Thus X ∈ C 00 and therefore is the minimal isometric dilation space of X and M 0 ⊆ H 1 . By the uniqueness of the minimal isometric dilation (see [5], page 135), M m admits a decomposition of the form where ϕ is a unitary operator from D X to ker((T Mm )| Mm⊖G ) * and M + (ϕD X ) = ϕD X ⊕ T Mm (ϕD X ) ⊕ · · · . Hence M = (G ⊕ M + (ϕD X )) ⊕ M 0 , which implies that On the other hand (see [5], page 136), there exist a unitary operator ϕ * from D X * to Finally, by (4.3) there exists a unitary operator from D X to D X * . Therefore, the relation (4.2) follows from (4.4). This completes the proof of Step 2.
Step 3: Applying Step 2 to the setting in Step 1 we see that dim ker S ′ * = dim ker S * , where S = T M and M is the maximal subspace invariant to T on which the restriction of T is isometric and this completes the proof of the theorem.

On the uniqueness of the nilpotent operators
In this section we will discuss the uniqueness of the nilpotent operators in the representations of c.n.u. contractions with polynomial characteristic functions. We start with the following proposition.
Proposition 5.1. Let T on H be in P n with two different matricial representations In particular,Ỹ is a quasi-affinity.
(iii)ỸT =T ′Ỹ , wherẽ Consequently, h 0 ∈ M 1 ∩ H 0 and from the condition The second equality follows as above with the observation that Y (ii) kerỸ = {0} follows from the definition ofỸ . The second equality follows from the equalities on H = H 10 ⊕H 00 ⊕H −10 and H = H 1 * 0 ⊕H 0 * 0 ⊕H −1 * 0 , respectively satisfy those hypotheses.  (iii) shows that N 0 is a quasi-affine transform of N * 0 , or, in the standard notation, N 0 ≺ N * 0 . Since these operators are of class C 0 , it follows that N 0 and N * 0 are quasi-similar (see Proposition 5.1 in [2]). This concludes the proof.
To state the next result we recall that T 1 ∈ L(H 1 ) is said to be injected in T 2 ∈ L(H 2 ) which is denoted by Then Proof. We will apply Proposition 5.1 with the representation (5.6) in the role of the first representation in (5.1) and with the second representation in (5.1) replaced by the second representation in (5.5). Then we have Consequently, by applying Proposition 5.1, (ii) and (iii) we obtain , and we infer thatT ′ is a nilpotent operator and then, since bothT andT ′ are of class C 0 , thatT ∼T ′ . Observe now that , by the minimality property of H −1 * (see Remark 3.6). Therefore, and thus we see thatT In particular, (5.7) shows that N ≺ i N * . Also since H 0 * 0 = H 0 * ⊖ (H 0 * ∩ M −1 ), we have that H 0 * 0 ⊆ H 0 * ⊖ (H 0 * ∩ H −1 ) and

Consequently,
and the relation (5.7) implies that N * 0 ≺ i N.
The following remark follows by applying Theorem 5.3 to T * .
Remark 5.4. Under the conditions of Theorem 5.3 one obtains where N 1 denotes the canonical nilpotent operator as in Theorem 3.3. Note that since N 0 ∼ N * 0 we have N 0 ≺ i N and N * * 0 ≺ i N * 1 . Let T be a c.n.u. contraction with polynomial characteristic function of degree n and let Then N um is said to be ultra-minimal if N um ≺ i N for any other nilpotent operator N of the upper triangular matrix representations of T . By Theorem 5.3, the canonical N * 0 is ultra-minimal. If N is any other ultra-minimal nilpotent operator in a upper triangular matrix representation of T , then we will have Thus But since N * 0 and N * 0 | (ranX) − are C 0 -operators we have (see [2]) Consequently all ultra-minimal nilpotent operators are quasi-similar and hence we have the following theorem.
then for any h −1 ∈ H −1 , we have that Thus we obtain that H −1 = {0} and T takes the degenerate form T = S * 0 N . Finally, T ∈ C · 0 if and only if Θ T is inner (see [9], Chapter VI).
(ii) follows from (i) applied to T * .

The case of monomial characteristic functions
In this section we study the the c.n.u. contraction T such that Θ T (z) is a non-constant monomial (for the case of constant monomials, see Section 6). This particular case, although quite concrete, allow us to show that the nilpotent central entries in the two canonical representations may not be quasi-similar. For this purpose, we explicitly (in terms of Θ T ) calculate the two canonical triangular representations of T . First, we need the following simple fact, which, for instance, can be obtained by using (1.1).
Lemma 7.1. Let m ≥ 1 and U be an unitary operator in L(E, E * ) and Θ(z) = Uz m . Then the c.n.u contraction J m (E * ) for which Θ Jm(E * ) (z) coincides with Θ(z) is a nilpotent operator of order m. Moreover, Let A in L(M, N ) be a non-zero contraction. Then it is easy to see that A| D A ⊥ : D A ⊥ → D A * ⊥ is an unitary operator and A| D A : D A → D A * is pure contraction, that is, Ah < h for all h = 0 in D A . Now, if Θ(z) = Az m is a given non-constant monomial where A in L(M, N ) is a contraction then Θ(z) = A| D A z m ⊕ A| D A ⊥ z m . In what follows, we will assume that m ≥ 1. In view of Lemma 7.1 we have that, T is unitarily equivalent to T p ⊕J m (E * ) where T p is the c.n.u. contraction whose characteristic function coincides with A| D A z m . Now for a given pure contraction A in L(M, N ), we will construct the two canonical upper triangular representations T A andT A with the characteristic function (coinciding with) Θ(z) = Az m for z in D.
where the central block matrix is of order m.
To compute the defect operators of T A , we first observe that Thus, the defect operators are given by and D 2 T A | Ln = 0 : L n → L n , if n = m − 1, −1. From (7.2) we infer that T A is a partial isometry and therefore, the defect spaces of T A can be expressed as Also since A is a pure contraction, we have the following.
Lemma 7.2. The operator T A constructed as above is c.n.u.
On the other hand, it is easy to see that Since T m A R = R, it follows that for any h in R, Consequently, by the facts that T A | R is unitary and A is a strict contraction, (7.4) yields Therefore, we obtain for all h in R, must be zero. Since ker D A * = {0} then h m must be the zero vector. Moreover, using the fact Finally, repeating the above argument replacing the role of T A by T * A , we obtain h k = 0 for all k > m. Thus h = 0. Theorem 7.3. Let A : M → N be a pure contraction (that is, Ah N < h M for all h ∈ M and h = 0). Then T A is a c.n.u. partial isometry and the characteristic function of T A coincides with Θ(z) = Az m . Moreover, the upper triangular representation of T A in (7.1) coincides with the canonical representation introduced in Theorem 3.3.
First, we consider the case when m > 1. Since T A is a partial isometry, we have Consequently for j = m − 1, so that D T * T * (m−1) D T u = P L 0 T * (m−1) u = Av ∈ L 0 , and for j > m − 1, To finish the proof, we will show that τ * Θ T A (z) = Θ(z)τ * for all z ∈ D for some unitary operators τ : M → D T A and τ * : D T * A → N . For this purpose, we define τ * = · · · ⊕ 0 ⊕ I L 0 ⊕ 0 ⊕ · · · : D T * A → N and In the remaining case m = 1, taking u = Av ⊕ D A v ∈ L 0 ⊕ L −1 we have Moreover, for all k ≥ 1, D T * T * k D T u = 0, and the rest of the proof follows in the similar way as in the last part of the proof of (7.5) for the m > 1 case. Finally, recalling that A is a pure contraction, the last part of the theorem follows by inspection.
The proof of the following theorem can be obtained by passing to the adjoint of the matrix representation ofT A and using Theorem 7.3.
Theorem 7.4. Let A : M → N be a pure contraction. ThenT A is a c.n.u. partial isometry and the characteristic function ofT A coincides with Θ(z) = Az m . Moreover, the upper triangular representation ofT A in (7.6) coincides with the canonical representation introduced in Remark 3.5.
Concerning the operators T 0,b,c , it is easy to see that the representation (2.3) is the canonical representation considered in Remark 3.5 and therefore the quasi-similarity class in Theorem 5.3 and Remark 5.6 can be replaced with the dimension of H 0 ; thus for this very particular case, the answer to the problem in Remark 5.6 is positive. The following theorem extends this observation from the scalar case to the vector valued case.
Theorem 7.5. Let T A be the c.n.u. operator with monomial characteristic function Θ(z) = z m A. Then the nilpotent operators N 0 and N * 0 corresponding to the minimal representation considered in Theorem 5.2 are unitarily equivalent.
Proof. Let h = · · · ⊕ h 2 ⊕ h 1 ⊕ h 0 ⊕ h −1 ⊕ h −2 ⊕ · · · be a vector in the maximal T A -invariant subspace M 1 of H where T A is an isometry. First, observe that for all 1 ≤ k < m Thus D 2 T A T m A h = 0 yields D A h −m−1 = 0 and hence h −m−1 = 0. Similarly, for all k > m we have that D 2 T A T k A h = 0 and hence h −k−1 = 0. Then By virtue of D 2 T A T k A h = 0 for each 0 ≤ k < m we obtain that A * h m−(k+1) + D A h −(k+1) = 0.
We conclude with a simple observation on the spectrum of T A which follows from the relation of the spectrum σ(T ) of a contraction T and the invertibility of the characteristic function Θ T (z) (see [9], page 259).
Remark 7.6. The spectrum of the operator T A with the characteristic function Θ(z) coinciding with Az m is ∂D ∪ {0} if A is invertible or D if A is not invertible.