Derivations and Spectral Triples on Quantum Domains I: Quantum Disk

We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.


Introduction
Derivations in Banach algebras have been intensively studied, originally inspired by applications in quantum statistical mechanics. Problems such as classification, generator properties, closedness of domains were the focus of the attention. Good overviews are [4] and [19]. More recently derivations were studied in connection with the concept of noncommutative vector fields, partially inspired by Connes work [9].
An abstract definition of a first-order elliptic operator is given by the concept of a spectral triple. A spectral triple is a triple (A, H, D) where H is a Hilbert space on which a C * -algebra A acts by bounded operators, A is a dense * -subalgebra of A, and D is an unbounded self-adjoint operator in H such that [D, a] is bounded for a ∈ A, and (I + D 2 ) −1/2 is a compact operator.
It is therefore natural to look at a situation where the commutator [D, a] is not just bounded but belongs to the algebra A in B(H), i.e., when it is an unbounded derivation of A with domain A. The question is then about the compactness of the resolvent. A good example is the irrational rotation algebra, i.e., the noncommutative two-torus, defined as the universal C * -algebra A φ with two unitary generators u and v such that vu = e 2πiφ uv. It has two natural derivations d 1 , d 2 , defined on the subalgebra A φ of polynomials in u, v and its adjoints, by the following formulas on generators of A φ d 1 (u) = u, d 1 (v) = 0, Those derivations are generators of the torus action on A φ . In fact, according to [5], any derivation d : A φ → A φ can be uniquely decomposed into a linear combination of d 1 , d 2 (invariant part) and an approximately inner derivation. The algebra A φ has a natural representation in the GNS Hilbert space L 2 (A φ ) with respect to the unique tracial state on A φ . Then, as described for example in [9], the combination D = d 1 + id 2 is implemented in the Hilbert space L 2 (A φ ) by an operator with compact parametrices and thus leads to the canonical even spectral triple for the noncommutative torus.
In this paper we look at unbounded invariant and covariant derivations on the quantum disk, the Toeplitz C * -algebra of the unilateral shift U , which has a natural S 1 action given by the multiplication of the generator U → e iθ U . We first classify such derivations and then look at their implementations in various Hilbert spaces obtained from the GNS construction with respect to an invariant state. We answer the question when such implementations are operators with compact parametrices and thus can be used to define spectral triples. Surprisingly, no implementation of a covariant derivation in any GNS Hilbert space for a faithful normal invariant state has compact parametrices for a large class of reasonable boundary conditions. This is in contrast with classical analysis, described in the following section, where for a d-bar operator, which is a covariant derivation on the unit disk, subject to APS-like boundary conditions, the parametrices are compact. Similar analysis for the quantum annulus is contained in the follow-up paper [18].
The paper is organized as follows. In Section 2 we describe two commutative examples of the circle and the unit disk which provide motivation for the remainder of the paper. In Section 3 we review the quantum unit disk. Section 4 contains a classification of invariant and covariant derivations in the quantum disk. In Section 5 we classify invariant states on the quantum disk and describe the corresponding GNS Hilbert spaces and representations, while in Section 6 we compute the implementations of those derivations in the GNS Hilbert spaces of Section 5. In Section 7 we analyze when those implementations have compact parametrices. Finally, in Appendix A, we review some general results about operators with compact parametrices needed for the analysis in Section 7.

Commutative examples
The subject of this paper is derivations in operator algebras. If A is a * -algebra, A is a dense * -subalgebra of A and if d(a * ) = (d(a)) * , then d is called a * -derivation. Definition 2.2. Let A be a Banach algebra and A be a dense subalgebra of A such that A A and d is a derivation with domain A. The derivation d is called closed if whenever a n , a ∈ A, a n → a and d(a n ) → b, then we have d(a) = b. Moreover, d is called closable if a n → 0 and d(a n ) → b implies b = 0.
Closable derivation d can be extended (non-uniquely) to a closed derivation, the smallest of which is called the closure of d and denoted by d. In the following we will describe in some detail two commutative examples that have some features of, and provide a motivation for, our main object of study, the noncommutative disk.
is an example of an unbounded * -derivation that is closable.
Let ρ θ : A → A be the one-parameter family of automorphisms of A obtained from rotation x → x + θ on the circle. The map τ : A → C given by is the unique ρ θ -invariant state on A and, up to a constant, d is the unique ρ θ -invariant derivation on A. The Hilbert space H τ , obtained by the Gelfand-Naimark-Segal (GNS) construction on A using the state τ , is naturally identified with L 2 (S 1 ), the completion of A with respect to the usual inner product The representation π τ : A → B(H τ ) is given by multiplication: π τ (a)f (x) = a(x)f (x). Then the operator for a ∈ A. The operator D τ is rotationally invariant and has compact parametrices because its spectrum is Z and thus (A, H τ , D τ ) is a spectral triple.
Example 2.4. The second example is the d-bar operator on the unit disk, and it is the motivating example for the rest of the paper. Let A = C(D) be the C * -algebra of continuous functions on the disk D = {z ∈ C : |z| ≤ 1}. If A is the algebra of polynomials in z andz then is an unbounded, closable derivation in A.
Let ρ θ : A → A be the one-parameter family of automorphisms of A given by the rotation z → e iθ z on the disk. Notice that ρ θ : A → A. Moreover, d is a covariant derivation in A in the sense that it satisfies d(ρ θ (a)) = e −iθ ρ θ (d(a)), a ∈ A.
The map τ : A → C given by is a ρ θ -invariant, faithful state on A. The GNS Hilbert space H τ , obtained using the state τ , is naturally identified with L 2 (D, d 2 z), the completion of A with respect to the usual inner product The representation π τ : A → B(H τ ) is given by multiplication: Then the covariant operator a)), for all a ∈ A. The operator D τ however has an infinite-dimensional kernel, so (I +D * τ D τ ) −1/2 is not compact. This is not a surprise; the disk is a manifold with boundary and we need to impose elliptic type boundary conditions to make D τ elliptic, so that it has compact parametrices.
Denote by D max τ the closure of D τ , since there are no boundary conditions on its domain. On the other hand, let D min τ be the closure of D τ defined on C ∞ 0 (D). While D min τ has no kernel, its cokernel now has infinite dimension. The question then becomes of the existence of a closed operator D τ with compact parametrices, such that D min τ ⊂ D τ ⊂ D max τ ; this is answered in positive by Atiyah-Patodi-Singer (APS) type boundary conditions, see [3]. Spectral triples for manifolds with boundary using operators with APS boundary conditions were constructed in [2]. References [10] and [11] contain constructions of spectral triples for the quantum disk. Recent general framework for studying spectral triples on noncommutative manifolds with boundary is discussed in [12].

Quantum disk
Let {E k } be the canonical basis for 2 (N), with N being the set of nonnegative integers, and let U be the unilateral shift, i.e., U E k = E k+1 . Let A be the C * -algebra generated by U . The algebra A is called the Toeplitz algebra and by Coburn's theorem [8] it is the universal C *algebra with generator U satisfying the relation U * U = I, i.e., U is an isometry. Reference [16] argues that this algebra can be thought of as a quantum unit disk. Its structure is described by the following short exact sequence where K is the ideal of compact operators in 2 (N). In fact K is the commutator ideal of the algebra A.
We will use the diagonal label operator KE k = kE k , so that, for a bounded function a : N → C, we can write a(K)E k = a(k)E k . We have the following useful commutation relation for a diagonal operator a(K) a(K)U = U a(K + 1). (3.1) We call a function a : N → C eventually constant, if there exists a natural number k 0 , called the domain constant, such that a(k) is constant for k ≥ k 0 . The set of all such functions will be denoted by c + 00 . Let Pol(U, U * ) be the set of all polynomials in U and U * and define We have the following observation. Proof . Using the commutation relations (3.1) it is easy to see that a product of two elements of A and the adjoint of an element of A are still in A. It follows that A is a * -subalgebra of A.
Since U and U * are in A, it follows that Pol(U, V ) ⊂ A. To prove the reverse inclusion it suffices to show that for any a ∈ c + 00 the operator a(K) is in Pol(U, V ), as the remaining parts of the sum are already polynomials in U and U * . To show that a(K) ∈ Pol(U, V ), we decompose any a(k) ∈ c + 00 in the following way where a ∞ = lim k→∞ a(k), P k is the orthogonal projection onto the one-dimensional subspace generated by E k and P ≥k 0 is the orthogonal projection onto span{E k } k≥k 0 . A straightforward calculation shows that U k (U * ) k = P ≥k and P k = P ≥k − P ≥k+1 . This completes the proof.
Let c be the space of convergent sequences, and consider the algebra This is precisely the subalgebra of all diagonal operators in A and we can view the quantum disk as the semigroup crossed product of A diag with N acting on A diag via shifts (translation by n ∈ N), that is Several versions of the theory of semigroup crossed products exist, see for example [21].

Derivations on quantum disk
For each θ ∈ [0, 2π), let ρ θ : A → A be an automorphism defined by ρ θ (U ) = e iθ U and ρ θ (U * ) = e −iθ U * . It is well defined on all of A because it preserves the relation U * U = I. Alternatively, the action of ρ θ can be written using the label operator K as ρ θ (a) = e iθK ae −iθK .
It follows that ρ θ (a(K)) = a(K) for a diagonal operator a(K) and ρ θ : A → A.
Any derivation d : A → A that satisfies the relation ρ θ (d(a)) = d(ρ θ (a)) will be referred to as a ρ θ -invariant derivation. Similarly, any derivation d : A → A that satisfies the relation d(ρ θ (a)) = e −iθ ρ θ (d(a)) for all a ∈ A will be referred to as a ρ θ -covariant derivation.
Notice that, as a consequence of Proposition 3.1, we have the identifications {a ∈ A : ρ θ (a) = a} = a(K) : {a(k)} ∈ c + 00 =: A diag , and similarly We will also use the following terminology: we say that a function β :  Proof . Let d(U * ) = f ∈ A and since U * U = I we get which implies that U * d(U ) = −f U . This in turn implies that d(U ) = −U f U + g for some g ∈ A such that U * g = 0. Notice that 0 = U U * g = (1 − P 0 )g.
Applying ρ θ to f we get the following A similar calculation shows that ρ θ (g) = e iθ g. Those covariance properties imply that f = −α(K)U * for some α(K) ∈ A diag and similarly g = U γ(K). However, since g = P 0 g, and P 0 U = 0, we must have g = 0.
The following description of covariant derivations is proved exactly the same as the proposition above. for all a ∈ A.
Reference [5] brought up the question of decomposing derivations into approximately inner and invariant, not approximately inner parts, see also [13,14]. Below we study when invariant derivations in the quantum disk are approximately bounded/approximately inner. Recall that d is called approximately inner if there are a n ∈ A such that d(a) = lim n→∞ [a n , a] for a ∈ A. If d(a) = lim n→∞ d n (a) for bounded derivations d n on A then d is called approximately bounded. Note also that any bounded derivation d on A can be written as a commutator d(a) = [a, x] with x in a weak closure of A; see [15,19]. In fact x must belong to the essential commutant of the unilateral shift, which is not well understood [1].
for all a ∈ A.
Proof . Given an element a ∈ A we define its ρ θ average a av ∈ A by Since by assumption d is approximately bounded, there exists a sequence of bounded opera- since (b n ) av is ρ θ -invariant for every θ and hence by Proposition 4.1 it is given by the commutator with a diagonal operator µ n (K − 1) with the property {µ(k)} ∈ ∞ because of the assumption of boundedness. It is enough to verify (4.1) on the generators of the algebra A; we show the calculation for a = U . We have, equivalently as n → ∞, and this means that for every ε > 0 there exists N such that for all n > N we have and thus we get the estimate This completes the proof.
, the space of sequences converging to zero.
Proof . By the previous lemma there exists {µ n (k)} ∈ ∞ such that for all a ∈ A. Without loss of generality assume β(k) and µ n (k) are real, or else consider the real and imaginary parts separately.
We can assume L > 0; an identical argument works for L < 0. The above equation implies that Therefore for k and n large enough we have and, by telescoping µ n (k), we get for some fixed k 0 . Together this implies that µ n (k) ≥ (L−ε)k +µ n (k 0 −1) which goes to infinity as k goes to infinity. This contradicts the fact that {µ n (k)} ∈ ∞ which ends the proof.
We also have the following converse result.
Proof . We show that there exists a sequence {µ n (k)} ∈ c such that [a, µ n (K − 1)] converges to [a, β(K − 1)] for all a ∈ A. As before, it is enough to verify this on the generators; we show the calculation for a = U . Thus we want to construct µ n such that The above equation is true if and only if the following is true The above in turn is true if and only if is true. Define the sequence {µ n } ∈ c by the following formulas Thus the proof is complete.
Notice that in the above theorem the derivation d need not be bounded. For example, if β(k) = √ k + 1 then β(k)−β(k −1) → 0 as k → ∞, so, by the above theorem, d is approximately inner. However, d is unbounded.

Invariant states
Next we describe all the invariant states on A. If τ : A → C is a state, then τ is called a ρ θinvariant state on A if it satisfies τ (ρ θ (a)) = τ (a). Since and a 0 (K) ∈ A diag . Since A diag is the fixed point algebra for ρ θ , we immediately obtain the following lemma: where E is the natural expectation. Conversely given the natural expectation E and a state t : To parametrize all invariant states we need to first identify the pure states.
Lemma 5.2. The pure states on A diag denoted by t k for k ∈ N and t ∞ are given by Proof . A diag is a commutative C * -algebra that is isomorphic to the algebra of continuous functions on the one-point compactification of N, that is So by general theory, see [15] for details, the pure states are the Dirac measures (or point mass measures).
As a consequence, we have the following classification theorem of the ρ θ -invariant states on A.
Proof . By continuity it is enough to compute τ (a) on the dense set A. Then, by ρ θ -invariance and equation (5.1), we have Set τ (P k ) = ω(k) and notice that ω(k) ≥ 0 since τ (P k ) = τ (P 2 k ) = τ (P * k P k ) ≥ 0. It is clear that ω(k) ≤ 1 since P k are projections. Next decompose any a(K) ∈ A as in the proof of Proposition 3.1 where L is the domain constant and a ∞ is the value of a(k) for k ≥ L . Applying τ to this decomposition we get On the other hand we have Plugging this equation into the previous one we obtain The last equation provides a convex combination of two states τ ∞ (a) = a ∞ and τ w (a) = tr(w(K)a) with w(k) = ω(k) j ω(j) as λ 0 + λ ∞ = 1. This completes the proof. Given a state τ on A let H τ be the GNS Hilbert space and let π τ : A → B(H τ ) be the corresponding representation. We describe the three Hilbert spaces and the representations coming from the following three ρ θ -invariant states: τ w with all w(k) = 0, τ 0 , and τ ∞ . The states τ w with all w(k) = 0 are general ρ θ -invariant faithful normal states on A.
Proposition 5.4. The three GNS Hilbert spaces with respect to the ρ θ -invariant states τ w with all w(k) = 0, τ 0 , and τ ∞ can be naturally identified with the following Hilbert spaces, respectively: 1. H τw is the Hilbert space whose elements are power series Proof . The first Hilbert space is just the completion of A with respect to the inner product given by (5.2). It was discussed in [7], and also [17]. It is the natural analog of the classical space of square-integrable functions L 2 (D) for the quantum disk. The Hilbert space H τ 0 comes from the state τ 0 (a) = E 0 , aE 0 . To describe it we first need to find all a ∈ A such that τ 0 (a * a) = 0. A simple calculation yields Thus if τ 0 (a * a) = 0 we get that a + n (0) = 0 for all n ∈ N.
and a 2 τ 0 = τ 0 (a * a). So, using the canonical basis {E n := U n P 0 } for n ≥ 0, we can naturally identify A/A τ 0 with a dense subspace of 2 (N).
It is easy to describe the representation π τ 0 : and π τ 0 (a(K))E n = a(K)U n P 0 = U n a(K + n)P 0 = U n a(n)P 0 = a(n)E n .
Notice also that A/A τ 0 [I] → P 0 := E 0 . In other words, π τ 0 is the defining representation of the Toeplitz algebra A.
Next we look at the GNS space associated with τ ∞ (a) = lim Again we want to find the subalgebra A τ∞ of a ∈ A such that τ ∞ (a * a) = 0. A direct computation shows that so τ ∞ (a * a) = 0 if and only if a ± n,∞ = 0 for all n. Now A/A τ∞ can be identified with a dense subspace of L 2 (S 1 ) by Moreover we have The representation π τ∞ : A → B(H τ∞ ) is easily seen to be given by This completes the proof.

Implementations of derivations in quantum disk
Let H τ be the Hilbert space formed from the GNS construction on A using a ρ θ -invariant state τ and let π τ : A → B(H τ ) be the representation of A in the bounded operators on H τ via left multiplication, that is π τ (a)f = [af ]. We have that A ⊂ H τ is dense in H τ and [1] ∈ H τ is cyclic.
Proof . Again we need to find U τ 0 ,θ . Since ρ θ (U n P 0 ) = e inθ U n P 0 , we have By using Proposition 4.1 in the above equation we get A short calculation verifies that D β,τ 0 is indeed an implementation of d β . This completes the proof.
Proposition 6.4. There exists a number c such that the implementations D β,τ∞ : D τ∞ → L 2 (S 1 ) of d β are of the form Proof . Like in the other proofs we need to understand what the value of D β,τ∞ on 1 is. A simple calculation shows that It is clear by the invariance properties that there exists a constant c such that D β,τ∞ (1) = c · 1.

Compactness of parametrices 7.1 Spectral triples
We say that a closed operator D has compact parametrices if the operators (I + D * D) −1/2 and (I + DD * ) −1/2 are compact. Other equivalent formulations are summarized in the appendix. Below we will reuse the same notation for the closure of the operators constructed in the previous section. In most cases it is very straightforward to establish when those operators have compact parametrices. Proof . The operators D β,τ 0 are diagonal with eigenvalues β(k − 1) + c, which must go to infinity for the operators to have compact parametrices. The operatorsD β,τ 0 differ from the operators D β,τ 0 by a shift, so they behave in the same way. Proof . Similar to the proof of the proposition above, the operators D β,τ∞ are diagonal with eigenvalues β ∞ n + c, which go to infinity if and only if β ∞ = 0. Proof . The operators D β,τw can be diagonalized using the Fourier series It follows that the numbers β(k + n − 1) − α(k) and β(k − 1) − α(k + n) are the eigenvalues of the diagonal operator, and must diverge for the operator to have compact parametrices.

Covariant derivations and normal states
Here we study the parametrices of the ρ θ -covariant operators which implement derivations in GNS Hilbert spaces H τw corresponding to faithful normal states. In this section we enhance the notation forD β,τw ; we will use instead D β,α,w f = U β(K)f − f U α(K), a notation that clearly specifies the coefficients of the operator. Denote by D max β,α,w the closure of D β,α,w defined on D max τ = π τ (A) · [1]. Define the * -algebra where c 00 are the sequences with compact support, i.e., eventually zero, and let D min β,α,w be the closure of D β,α,w defined on D min τ = π τ (A 0 ) · [1]. Finally, will use the symbol D β,α,w for any closed operator in H τw such that D min β,α,w ⊂ D β,α,w ⊂ D max β,α,w . The main objective of this section is to prove the following no-go result.
Proof . It is assumed below that β ∞ = 0. The outline of the proof is as follows. First, by a sequence of equivalences, we show that the operator D β,α,w has compact parametrices if and only if a simplified version of it has compact parametrices. Since in particular an operator with compact parametrices has to be Fredholm, the finiteness of the kernel and cokernel implies certain growth estimates on the parameters. Those estimates in turn let us compute parts of the spectrum of the Fourier coefficients of D β,α,w and that turns out to be not compatible with compactness of the parametrices.
First we show that β(k) can be replaced by its absolute values. We will need the following information.
Proof . Define the unitary operator V (K) by and consider the following map f → V (K)f for f ∈ H τw . This map preserves the domains D min τ and D max τ . A direct computation gives that This shows that D β,α,w and D |β|,α,w are unitarily equivalent, thus completing the proof.
Proof . Notice that the difference D β+γ 1 ,α+γ 2 ,w − D β,α,w is bounded, hence the two operators both either have or do not have compact parametrices simultaneously, see Appendix A.
It follows from those lemmas that, without loss of generality, we may assume that β(k) > 0, where β(k) satisfies inequalities c 1 and c 2 positive. Next we look at properties of α. For a finite sum Lemma 7.8. If the operator D β,α,w such that D min β,α,w ⊂ D β,α,w ⊂ D max β,α,w has compact parametrices, then dim coker(D max β,α,w ) < ∞ and α(k) has at most f initely many zeros.
As a consequence of the above lemma and also Lemma 7.7 we will assume from now on that α(k) = 0 for every k.
We find it convenient to work with unweighted Hilbert spaces. This is achieved by means of the following lemma. Lemma 7.9. Let H τw be the weighted Hilbert space of Proposition 5.4 (1), and let H be that Hilbert space for which the weight w(k) = 1. The operator D β,α,w such that D min β,α,w ⊂ D β,α,w ⊂ D max β,α,w has compact parametrices if and only if the operator D β,α,1 such that D min β,α,1 ⊂ D β,α,1 ⊂ D max β,α,1 has compact parametrices, wherẽ Proof . In H τw write the norm as and set ϕ(f ) = f w(K) 1/2 : H τw → H. Then ϕ is a bounded operator with bounded inverse, and is in fact an isomorphism of Hilbert spaces. Moreover, we have and ϕD β,α,w ϕ −1 : H → H. So D β,α,w and D β,α,1 are unitarily equivalent, thus completing the proof. Notice also thatα(k) = 0, because α(k) = 0.
From now on we will work with operators D min β,α,1 ⊂ D β,α,1 ⊂ D max β,α,1 in the unweighted Hilbert space H. For convenience we define a sequence {µ(k)} such that µ(0) = 1 and Such µ(k) is completely determined by the above equation in terms of α and β and will be used as a coefficient instead of α. We rewrite the four main operators as follows Next, using Fourier components above, we study the kernel and the cokernel of D β,α,1 .
Lemma 7.10. The formal kernels of D + n and (D − n ) * are one-dimensional and are spanned by, correspondingly The operators D − n and (D + n ) * have no algebraic kernel; consequently they have no kernel at all.
Other calculations are similar. This completes the proof.
The computations above were formal; to actually compute the kernel and the cokernel of D β,α,1 we need to look at only those solutions which are in the domain/codomain of D β,α,1 . It is important to keep in mind the following inclusions ker D min β,α,1 ⊂ ker D β,α,1 ⊂ ker D max β,α,1 , and coker D max β,α,1 ⊂ coker D β,α,1 ⊂ coker D min β,α,1 . The following lemma exhibits the first key departure from the analogous classical analysis of the d-bar operator.
Lemma 7.11. If the operator D β,α,1 such that D min β,α,1 ⊂ D β,α,1 ⊂ D max β,α,1 has compact parametrices, then both ker D max β,α,1 and coker D min β,α,1 are finite-dimensional. Moreover, the sums Proof . Let f + n and f − n be solutions to the equations D + n f = 0 and (D − n ) * f = 0 respectively, as described in Lemma 7.10. First we study D + n f = 0. There are two options (1) f + n < ∞ for all n, or (2) there exists n 0 ≥ 0 such that f + n 0 = ∞. Consider the first option first, i.e., for every n, which implies that D max β,α,1 has an infinite-dimensional kernel. We argue below that in this case the kernel of D min β,α,1 is also infinite-dimensional, which is not true in classical theory. Consider the sequence Notice that, because it is eventually zero, the sequence f N (k) is in the domain of (D + n ) min and Moreover, a direct calculation shows that From this we see that D + n f N → 0 as N → ∞ since f + n < ∞ for all n. This shows that the formal kernel of (D + n ) is contained in the domain of (D + n ) min . This implies that D β,α,1 has an infinite-dimensional kernel contradicting the fact that D β,α,1 is Fredholm. A similar argument produces an infinite-dimensional cokernel for D β,α,1 by studying option (1) for (D − n ) * f = 0. Consequently, option (1) does not happen in our case, and option (2) must be true. It is clear from the growth conditions (7.1) that if there exists n 0 such that f ± n 0 = ∞ then f ± n = ∞ for all n ≥ n 0 . But that means that the 2 (N) kernels of (D ± n ) and (D ± n ) * are all zero for n large enough. This implies that both ker D max β,α,1 and coker D min β,α,1 are finite-dimensional. Moreover, f ± n = ∞ for all n ≥ n 0 gives the divergence of the sums in the statement of the lemma. Thus the proof is complete.
It follows from the above lemma, and from the remarks right before it, that all three operators D min β,α,1 ⊂ D β,α,1 ⊂ D max β,α,1 have compact parametrices. Next we discuss the inverses of D ± n and their formal adjoints. Operators D − n and (D + n ) * have no formal kernels and can be inverted on any domain of sequences. The other operators preserve c 0 ⊂ 2 (N) and can be inverted on c 0 . The corresponding formulas are if n = 1, Using those formulas we obtain key growth estimates on coefficients µ(k) in the following lemma.
Using the growth conditions (7.1) the inequality above yields ∞ k=0 1 (k + 1) 2 · · · (k + n) 2 |µ(k + n)| 2 < ∞ for n ≥ n 0 , which gives the left hand side of the inequality (7.3). To the get the right-hand side, we use ( Now that we have control over the coefficients of D β,α,1 we can compute the spectrum of its Fourier coefficients. Notice that, using Proposition A.6 in the appendix, we have that since D min β,α,1 and D max β,α,1 are Fredholm, and D β,α,1 has compact parametrices, then both D min β,α,1 and D max β,α,1 also have compact parametrices. The following calculations are similar to the calculations in [6] for the Cesaro operator. Lemma 7.13. The continuous spectrum σ c , the point spectrum σ p , and the residual spectrum σ r , of the operator (D + n ) max have the following properties: n ) max ) = ∅ or has at most finitely many spectral values. Proof . The Fredholm property of D max β,α,1 implies that Ran((D + n ) max − λI) is closed, meaning that σ c ((D + n ) max ) = ∅. Next we study the eigenvalue equation (D + n f )(k) = λf (k), that is This equation can be easily solved, yielding a one-parameter solution generated by The question then is when does f λ ∈ 2 (N)? To study estimates on f λ (k) we use the following three simple inequalities 3) there exists a constant C ε such that C ε e (1−ε)x ≤ 1 + x, for 0 < ε < 1 and small |x|.
First we estimate from above each factor in the formula for f λ as follows where we used inequality 1) of equation (7.4). This implies that Notice that for fixed λ we have |β(j + n) − β(j) − λ| ≤ const by (7.1) and, because c 2 (j + 1) ≤ β(j) ≤ c 1 (j + 1), we obtain const (j + 1) 2 < const, which accounts for the second term in the exponent. To estimate the first term we also use 2) of equation (7.4) to get where we applied (7.3) to estimate µ(k). This last inequality implies that f λ (k) ∈ 2 (N) if Re λ 0. This shows that Next we estimate f λ from below by using part 3) of equation (7.4) with which, by previous discussion, is small for j large enough. We get the following estimate valid for large j. By using the conditions on β(j) and µ(k), and also 2) of equation (7.4), we get This inequality shows that if Re λ 0 then f (k) / ∈ 2 (N). This in turn implies that Finally, to determine the residual spectrum of (D + n ) max , we consider the eigenvalue equation (D + n ) * f (k) = λf (k), which is the same as Rearranging the terms in the above equation yields This equation has non-trivial solutions if and only if β(k + n) − λ = 0 for some k, which can only happen for specific values of λ. Namely, if λ l = β(l + n), then the above equation recursively gives f (0) = f (1) = · · · = f (l − 1) = 0 and f (k + l) = const β(k) · · · β(k + l − 1)µ(l + k).
If l is large enough then f (k) / ∈ 2 (N). This means that the residual spectrum of D + n has at most finitely many values or is empty, proving the remaining part of the lemma, thus completing the proof of the lemma.
We can now easily finish the proof of the theorem. As explained in appendix, if D max β,α,1 has compact parametrices then its spectrum is either empty, the whole plane C, or consists of eigenvalues going to infinity. Clearly this is not consistent with Lemma 7.13, and hence D β,α,1 does not have compact parametrices.

A Appendix
The main objective of this appendix is to review some generalities about unbounded operators with compact parametrices. Presumably all of the statements below are known, however they don't seem to appear together in any one reference.
Throughout this appendix D is a closed unbounded operator in a separable Hilbert space. Recall that D is called a Fredholm operator if there are bounded operators Q 1 and Q 2 such that Q 1 D−I and DQ 2 −I are compact. The operators Q 1 and Q 2 are called left and right parametrices respectively. Equivalently, D is a Fredholm operator if the kernel and the cokernel of D are finitedimensional. A Fredholm operator always has a single parametrix, i.e., a bounded operator Q such that QD − I and DQ − I are compact. In the literature the case of unbounded Fredholm operators is usually not discussed directly, however a closed operator can be considered as a bounded operator on its domain equipped with the graph inner product ||x|| 2 D = ||x|| 2 +||Dx|| 2 . A good reference on Fredholm operators is [20].
We say that a closed, Fredholm operator D has compact parametrices if at least one of the parametrices Q 1 and Q 2 is compact. By applying Q 1 to DQ 2 − I on the left and Q 2 to Q 1 D − I on the right, we see that if one of the parametrices Q 1 and Q 2 is compact so is the other. Similarly, if Q 1 and Q 2 is another set of parametrices of an operator with compact parametrices, then both Q 1 and Q 2 must be comopact. It is sometimes easier to construct separate left and right parametrices rather then a two-sided parametrix.
Our first task is to work out several equivalent definitions of an operator with compact parametrices.
Proof . This immediately follows from the resolvent identity.
First we rephrase the concept of an operator with compact parametrices in terms of resolvents.
Proposition A.2. Suppose ρ(D) = ∅ and R D (λ) is compact for some λ ∈ ρ(D), then D is a Fredholm operator with compact parametrices. Conversly, if D is a Fredholm operator with compact parametrices and ρ(D) = ∅ then R D (λ) is compact.
Proof . Consider the following calculations and So, if R D (λ) is compact for λ ∈ ρ(D) then D is Fredholm with parametrix λR D (λ) which is compact. Conversely, if D is a Fredholm operator with compact parametrices and ρ(D) = ∅ then R D (λ) is compact as a parametrix of D. This completes the proof.
Next we give a characterization of operators with compact parametrices in terms of selfadjoint operators D * D and DD * . Notice that, since (I + DD * ) −1/2 is compact, (I + DD * ) −1 is compact. Moreover, we have by the functional calculus that operator D * (I + DD * ) −1/2 is bounded. Consequently we have Writing Q as Q = D * (I + DD * ) −1/2 (I + DD * ) −1/2 , we see that Q is compact and so D has compact right parametrix. Similar argument shows that Q is also a left parametrix.
Conversely, let Q be a compact parametrix of D, i.e., DQ = I + K 1 and QD = I + K 2 , where K 1 and K 2 are compact. Then consider Since D(I + D * D) −1/2 and (I + D * D) −1/2 are bounded and Q and K 2 are compact, it follows that the right hand side of the above equation is compact. A similar decomposition works for showing the compactness of (I + DD * ) −1/2 , thus completing the proof. Operators with compact parametrices have the following simple stability property.
Proposition A.5. Suppose D is a Fredholm operator with compact parametrices. If a is a bounded operator, then D + a is Fredholm with compact parametrices.
Proof . If Q is a compact parametrix of D then it is also a parametrix of D + a.
We have the following "sandwich property" for operators with compact parametrices.
Proposition A.6. Let D i be closed operators for i = 1, 2, 3, such that D 1 ⊂ D 2 ⊂ D 3 . If D 1 and D 3 are Fredholm operators and D 2 has compact parametrices, then both D 1 and D 3 have compact parametrices.
Proof . Since D 2 has compact parametrices, there exists a compact Q such that QD 2 = I + K for some compact operator K. Since D 1 ⊂ D 2 we have dom(D 1 ) ⊆ dom(D 2 ) and therefore QD 1 = I + K. Since D 1 is Fredholm it has both left and right parametrices. The above shows that D 1 has a compact left parametrix. Consequently the right parametrix of D 1 must also be compact. A similar argument works for D 3 . This completes the proof.
Next we turn our attention to spectral properties of operators with compact parametrices. As an example consider operators D 1 , D 2 , and D 3 all equal to 1 Proposition A.7. Let D be a closed operator with compact parametrices. There are exactly three possibilities for the spectrum of D: 1) σ(D) = C, 2) σ(D) = ∅, 3) σ(D) = σ p (D), the point spectrum of D. In the last case, either σ(D) is finite or countably infinite with eigenvalues going to infinity.

Proof .
The examples above demonstrate all three possibilities. Suppose σ(D) = C, then there exists a λ 0 such that R D (λ 0 ) exists. Since D has compact parametrices, R D (λ 0 ) is compact. By spectral theory of compact operators we have σ(R D (λ 0 )) = {0} ∪ σ p (R D (λ 0 )) with three possibilities for the point spectrum: it's empty, finite or countably infinite tending to zero.
Using the bijection we can get all the information about the spectrum of D, since we have σ(D − λ 0 I) = σ(D) − λ 0 . If σ p (R D (λ 0 )) = ∅ then we get that σ(D) = ∅, if σ p (R D (λ 0 )) is finite then σ(D) = σ p (D) and is finite, and finally if σ p (R D (λ 0 )) is countably infinite with eigenvalues tending to zero, then σ(D) = σ p (D) is countably infinite with eigenvalues going to infinity. This completes the proof.
The last topic covered in this appendix is an analysis of operators of the form which appear in the definition of an even spectral triple.
Proposition A.8. The operator D has compact parametrices if and only if the operator D has compact parametrices.
Proof . If Q is a compact parametrix of D, let K 1 and K 2 be the compact operators such that DQ = I + K 1 and QD = I + K 2 . Using Q we can construct a parametrix of D by and similarly for the multiplication in the reverse order. These imply that D has compact parametrices. Conversely, if D has compact parametrices, its resolvent is compact. We can write down the resolvent for imaginary −iλ as follows Inspecting the diagonal elements of the above matrix we see that D has compact parametrices by Proposition A.3.