Finite sections of truncated Toeplitz operators

We describe the $C^*$-algebra associated with the finite sections discretization of truncated Toeplitz operators on the model space $K^2_u$ where $u$ is an infinite Blaschke product. As consequences, we get a stability criterion for the finite sections discretization and results on spectral and pseudospectral approximation.


Truncated Toeplitz operators
Let H 2 denote the standard Hardy space on the unit disk D, i.e. the Hilbert space of all holomorphic functions on D which have square-summable Taylor coefficients. As usual, we identify H 2 with its space of non-tangential boundary functions, which is a closed subspace of the Lebesgue space L 2 (T) with normalized Lebesgue measure m on the unit circle T. The orthogonal projection from L 2 (T) onto H 2 is denoted by P .
Every function a ∈ L ∞ (T) defines an operator of multiplication on L 2 (T), which we denote by aI. The Toeplitz operator induced by a is the operator T (a) := P aI| H 2 , acting from H 2 to H 2 . The Toeplitz operator with generating function a(z) = z is the operator S of forward shift, (Sf )(z) = zf (z). Its adjoint S * , the backward shift operator, is given by (S * f )(z) = z −1 (f (z) − f (0)).
Let u be a non-constant inner function, i.e., u is holomorphic on D and |u(t)| = 1 for t ∈ T (the following becomes trivial when u is constant). The subspace K 2 u := H 2 ⊖ uH 2 is a proper nontrivial invariant subspace of S * . Conversely, every proper nontrivial invariant subspace of S * is of this form by a celebrated theorem of Beurling. The spaces K 2 u are also known as model spaces. We denote the orthogonal projection from L 2 (T) onto K 2 u by P u . If M u and Mū denote the operators of multiplication by u andū on L 2 (T), then P u = P − M u P Mū.
For a ∈ L ∞ (T), the truncated Toeplitz operator (TTO for short) generated by a is the operator T u (a) := P u aI| K 2 u acting from K 2 u to K 2 u . Truncated Toeplitz operators share many of their properties with their relatives, the Toeplitz operators on H 2 , to which we here sometimes refer as classical Toeplitz operators, but there are also some striking differences. For example, the function a is in general not uniquely determined by the operator T u (a) it generates, and the truncated shift S u , i.e., the TTO with generating function a(z) = z, is the sum of a unitary and a compact operator (hence a Fredholm operator with index 0), whereas its classical counterpart S is a proper isometry (and a Fredholm operator of index −1). Moreover, whereas the spectrum of the classical shift S is the closed unit disk D, the spectrum of the truncated shift S u coincides with the so-called spectrum The following collection of positive results is taken from and proved in [2]. We denote by T u (C) the smallest closed C * -subalgebra of L(K 2 u ) which contains the truncated shift S u and the identity operator (the notation will be justified by assertion (d) in the theorem below). Further, we write Comm A for the commutator ideal of a C * -algebra A, i.e., for the smallest closed ideal of A which contains all commutators ab − ba with a, b ∈ A. The essential spectrum of an operator A is denoted by σ ess (A), and its essential norm by A ess .
(h) Every operator in T u (C) is the sum of a normal and a compact operator. Moreover, is a short exact sequence, with the mapping π given by T u (a) + K → a| σ(u)∩T .
Here is an outline of the contents of the paper. In Section 2 we will single out a sequence (P un ) of finite rank projections which converge strongly to the identity operator on K 2 u . The operator P un T u (a)P un is considered as a finite section of the truncated Toeplitz operator T u (a). In Section 4, Theorem 10, we describe the C * -algebra S(T u (C)) generated by all sequences of the form (P un T u (a)P un ) n≥1 with a a continuous function. This description is based of a formula of Widomtype that we will derive in Section 3. As consequences of Theorem 10, we get a stability criterion and results on spectral approximation. The stability criterion (Theorem 11) says that a sequence (A n ) in S(T u (C)) is stable if and only if its strong limit A is invertible. In the case when the A n are the finite sections of a truncated Toeplitz operator, this result is due to Treil [8]. One advantage of Theorem 11 is that it implies (without any additional effort) results on spectral and pseudospectral approximation as well as on the asymptotic behavior of the small singular values of A n ; see the end of Section 4.

A filtration and Widom's identity
Recall that a filtration on a Hilbert space H is a sequence P = (P n ) of orthogonal projections of finite rank on H which converges strongly to the identity operator on H. To define a filtration on the model space K 2 u we specify u to be a Blaschke product, as follows. A single Blaschke factor is a function on the unit disk of the form A Blaschke product is then a function which satisfies the Blaschke condition If u is a finite Blaschke product, i.e., if u is of the form (2) with k(λ) = 0 for all but finitely many λ ∈ D, then (3) is satisfied. Conversely, if (3) holds, then every disk {z ∈ D : |z| ≤ r} with 0 < r < 1 contains only finitely many λ with k(λ) = 0. Thus, if u in (2) is an (infinite) Blaschke product, the number of its non-one factors is countable. We order the λ with k(λ) = 0 in a sequence (λ k ) k≥1 in such a way that |λ k | ≤ |λ k+1 | for all k. Then (2) and (3) can be written as Blaschke products are inner functions. If u is an infinite Blaschke product, we use its product representation (4) to define a filtration on the associated model space K 2 u . For n ≥ 1, set u n := n k=1 b λ k and let P un denote the orthogonal projection from L 2 (T) onto K 2 un . The projections P un own the following properties.
Proposition 2 (a) The projections P un have a finite spatial rank.
(b) P un → P u on L 2 (T) strongly as n → ∞.
(c) P um P un = P un P um = P u min{m, n} for m, n ≥ 1.
(d) P un P u = P u P un = P un and P un P = P P un = P un for n ≥ 1.
Hints to the proof. Assertion (a) is the "Lemma on Finite Dimensional Subspaces" in [4, p. 33]. For assertion (b) observe that the u n converge to u uniformly on compact subsets of D by the "Lemma on Blaschke Products" in [4, p. 280]. Thus, the model spaces K 2 un converge to K 2 u by the "Theorem on Lower Limits" in [4, p. 34] (note that the limit of the u n exists; so the set of its limit points contains only one element, which clearly coincides with the greatest common divisor in the formulation of that theorem). Thus, P un → P u on L 2 (T) strongly by the definition in [4, p. 34].
Finally, u n is a divisor of u and of u m for m ≥ n. Corollary 8 in [4, p. 19] then implies that K 2 un ⊆ K 2 um ⊆ K 2 u ⊆ H 2 , whence assertions (c) and (d). Thus, the restrictions of the projections P un to the model space K 2 u form a filtration on K 2 u by the preceding proposition. We denote this filtration by P u and write P u = (P un ) n≥1 , not distinguishing between a projection P un on L 2 (T) and its restriction to K 2 u . The study of the finite sections discretization (FSD for short) for (classical) Toeplitz operators is dominated by Widom's identity P n T (ab)P n = P n T (a)P n T (b)P n + P n H(a)H(b)P n + R n H(ã)H(b)R n . (5) To explain this identity, we need some notation. Let J : L 2 (T) → L 2 (T) denote the operator (Jf )(t) = t −1 f (t −1 ). One easily checks that, for every function a ∈ L ∞ (T), JaJ is the operator of multiplication by the functionã(t) := a(t −1 ). Then H(a) := P aIJ| H 2 is the (classical) Hankel operator, R n := H(t n ), and P n := R 2 n . Note that P n is just the orthogonal projection onto the linear span of the functions t n with n ∈ {0, 1, . . . , n − 1} and that R n = R * n . Our goal is to achieve a comparable identity for the finite sections of TTO, where the role of the R n in Widom's identity is played by the operators Theorem 3 (Widom's identity for TTO) Let a, b ∈ L ∞ (T). Then Proof. By Proposition 2 (d) and since P un = P − M un P M un , we find P un T u (a)P un T u (b)P un = P un P aP un bP P un = P un P aP bP P un − P un aM un P M un bP un = P un P abP P un − P un P aQbP P un − P un M un aP bM un P un with Q := I − P = P un abP un − P un P aQJ 2 QbP P un − P un M un P aP bM un P un − P un M un QaP bM un P un = P un T u (ab)P un − P un H(a)H(b)P un − P un M un QaP bM un P un .
In the last line we used that P un M un P f ∈ P un (u n H 2 ) = 0 for f ∈ L 2 (T), whence P un M un P = 0. Then P M un P un = (P un M un P ) * = 0, too, and we can proceed with Since JQaP bQJ = H(ã)H(b), this is the assertion

Hankel operators by Blaschke products
The operators R n in Widom's identity (5)  Analogously, R * un can be identified with H(M un ) * = H( M un ). We will see that the operators R un in Widom's identity for TTO play a quite different role compered with the R n in (5). We start with some general properties of Hankel operators generated by inner functions.
Proposition 4 Let u ∈ H ∞ and |u(t)| = 1 for t ∈ T. Then the Hankel operator H(u) = P uQJ is a partial isometry, the range and initial projection of which are given by H(u)H(u) * = P − uPūI and H(u) * H(u) = P −ūPũI.
Using this identity and PūQ = 0 we obtain i.e., H(u) is an isometry. Replacing u in (7) byū (which is also in H ∞ ) and taking into account that H(ū) = H(u) * , the identity for H(u) * H(u) follows.
Corollary 5 R un R * un = P un , R * un R un = P un , P un R un = R un , R * un P un = R * un . The following convergence result for the R un is in sharp contrast with the R n in Widom's identity (5), which converge weakly to zero.
In the proof of this result, we will make use of the following well known assertion.
Lemma 7 Let A n , A be bounded linear operators on a Hilbert space H. If A n → A weakly and A n x → Ax for all x ∈ H, then A n → A strongly.
Proof. It is clearly sufficient to prove the following fact for elements x n , x of H: if x n → x weakly and x n → x , then x n − x → 0. This follows from We start with showing that P un → Pū strongly as n → ∞.
Indeed, let b λ be a single Blaschke factor as in (1). For t ∈ T and λ = 0 we then have For λ = 0, the equality b λ = bλ on T is evident. Moreover, if (λ n ) is a sequence in D satisfying the Blaschke condition, then the sequence (λ n ) also satisfies this condition. So we can apply the assertion of Proposition 2 (b) to the functions

The algebra of the FSD for TTO
With every filtration P = (P n ) on a Hilbert space H, there are naturally associated some algebraic objects. By F P we denote the set of all sequences A = (A n ) of operators A n : im P n → im P n for which the sequence (A n P n ) converges * -strongly to some operator W (A) on H. Provided with element-wise defined operations and the supremum norm, F P becomes a C * -algebra, the set G P of all sequences in F P which converge to 0 in the norm is a closed ideal of F P , and the mapping W , also called the consistency map of the filtration P, is a * -homomorphism from F P to the algebra L(H) of the bounded linear operators on H.
We prepare the proof of Theorem 10 below by an assertion of independent interest.
Proposition 9 Let P = (P n ) be a filtration on a Hilbert space H. Then the ideal G P is contained in the smallest closed subalgebra J of F P which contains all sequences (P n KP n ) with K compact if and only if P is injective.
Proof. The "only if"-part of the assertion is evident. For the "if"-part we are going to show that, for each n 0 ∈ N, there is a sequence (G n ) in J with G n 0 = 0 and G n = 0 for all n = n 0 . Since the matrix algebras C k×k are simple, this fact already implies that each sequence (G n ) with arbitrarily prescribed G n 0 ∈ L(im P n 0 ) and G n = 0 for n = n 0 belongs to J . Since G P is generated by sequences of this special form, the assertion follows.
For n 0 ∈ N, put N < := {n ∈ N : im P n ∩ im P n 0 is a proper subspace of im P n 0 }, and set N > := N \ ({n 0 } ∪ N < ). The set N < is at most countable, and none of the closed linear spaces im P n ∩ im P n 0 has interior points relative to im P n 0 . By the Baire category theorem, ∪ n∈N< (im P n ∩ im P n 0 ) is a proper subset of im P n 0 .
Choose a unit vector f ∈ im P n 0 \ ∪ n∈N< (im P n ∩ im P n 0 ).
Then P n f < 1 for all n ∈ N < by the Pythagoras theorem. (Indeed, otherwise P n f = 1, and the equality 1 = f 2 = P n f 2 + f − P n f 2 implies f = P n f , whence f ∈ im P n .) Let Q n := I − P n . If n ∈ N > , then im P n ∩ im P n 0 = im P n 0 by the definition of N > . Thus, im P n 0 ⊆ im P n , and since no two of the projections P n coincide, this implies that im P n 0 is a proper subspace of im P n and im Q n is a proper subspace of im Q n 0 for n ∈ N > . Again by the Baire category theorem, ∪ n∈N> im Q n is a proper subset of im Q n 0 . Choose a unit vector g ∈ im Q n 0 \ ∪ n∈N> im Q n .
As above, Q n g < 1 for all n ∈ N > . Consider the operator K : x → x, g f on H. Its adjoint is the operator x → x, f g, and P n KQ n K * P n x = P n x, f Q n g, g P n f = x, P n f Q n g 2 P n f.
If n ∈ N < then P n f < 1, and if n ∈ N > then Q n g < 1 by construction. In both cases, P n KQ n K * P n < 1. In case n = n 0 , the operator P n KQ n K * P n x = x, f f is an orthogonal projection of norm 1, which we call P . The sequence K := (P n KQ n K * P n ) belongs to the algebra J since (P n KQ n K * P n ) = (P n KK * P n ) − (P n KP n ) (P n K * P n ).
As r → ∞, the powers K r converge in the norm of F P to the sequence (G n ) with G n 0 = P = 0 and G n = 0 if n = n 0 . Indeed, since P n → I strongly, one has Q n g < 1/2 for n large enough, whence P n KQ n K * P n < 1/2 for these n, whereas P n KQ n K * P n < 1 for the remaining (finitely many) n as seen above. Since K r ∈ J and J is closed, the sequence (G n ) has the claimed properties.
The goal in this section is to study the FSD of TTO with respect to the filtration P u . In accordance with Theorem 1 (e), we define the corresponding (full) algebra of the FSD as the smallest closed subalgebra S(T u (C)) of F Pu which contains all sequences (P un (T u (a) + K)P un ) n≥1 with a ∈ C(T) and K ∈ K(K 2 u ).
Theorem 10 S(T u (C)) consists of all sequences (P un (T u (a) + K)P un + G n ) with a ∈ C(T), K ∈ K(K 2 u ) and (G n ) ∈ G Pu . Proof. The proof runs parallel to that of Theorem [3, 1.53]; so we address to some main steps only.
For a moment, let S 1 denote the set of all sequences of the mentioned form. The sequences (P un (T u (a) + K)P un ) are contained in S(T u (C)) by definition, and since the filtration P u is injective, we conclude from Proposition 9 that the ideal G Pu of the zero sequences is also contained in S(T u (C)). Thus, S 1 ⊆ S(T u (C)).
For the reverse inclusion, we prove that S 1 is a closed subalgebra of S(T u (C)). If a, b ∈ C(T) then, by Theorem 3 (Widom's identity) and Corollary 8, with a compact operator K and a sequence (G n ) ∈ G Pu . Thus, (P un T u (a)P un ) (P un T u (b)P un ) ∈ S 1 .
It follows now in a standard way that S 1 is an algebra. To prove that S 1 is closed, let ((P un (T u (a m ) + K m )P un + G m n ) n≥1 ) m≥1 be a sequence in S 1 which converges in the norm of F Pu . Let W denote the consistency map of the filtration P u , i.e., W ((A n )) = s-lim A n P un . Then is a Cauchy sequence in T u (C). Since T u (C) is a closed algebra, this sequence converges in T u (C). The limit of this sequence is of the form T u (a) + K with a ∈ C(T) and K compact by Theorem 1 (e). (But note that the representation of the limit in that form is not unique by Theorem 1 (d).) It is now easy to see that (P un (T u (a m ) + K m )P un ) n≥1 → (P un (T u (a) + K)P un ) n≥1 in the norm of F Pu as m → ∞. Then, finally, the sequence ((G m n ) n≥1 ) m≥1 converges; its limit is in G Pu .
Theorem 11 A sequence A = (A n ) ∈ S(T u (C)) is stable if and only if the operator W (A) = s-lim A n P un is invertible.
Proof. By Theorem 10, we have to show that the sequence A := (P un (T u (a) + K)P un ) (with a ∈ C(T) and K ∈ K(K 2 u )) is stable if and only if the operator A := T u (a) + K is invertible. Since the stability of A implies the invertibility of A by Polski's theorem, we are left with the reverse implication.
So let A be invertible. By inverse closedness of C * -algebras, A −1 ∈ T u (C); hence, A −1 = T u (b) + L with a certain function b ∈ C(T) and a compact operator L by Theorem 1 (e). Using Widom's identity as in the proof of the previous theorem and employing assertion (a) of Theorem 1, we conclude that there are compact operators R 1 , R 2 and a sequence (G n ) ∈ G Pu such that P un T u (a)P un · P un T u (b)P un − P un = P un (T u (ab) − I)P un − P un R 1 P un + G n = P un R 2 P un − P un R 1 P un + G n .
Thus, the sequence (P un T u (b)P un ) is a right inverse of the sequence (P un T u (a)P un ) modulo the ideal ). A similar computation shows that it is a left inverse modulo J , too. Then (P un T u (b)P un ) is also an inverse of (P un AP un ) modulo J . Now the assertion follows from the Lifting Theorem [3, 5.37] in its simplest form, i.e., with J /G Pu consisting of one elementary ideal only.
The following is certainly the most important consequence of Theorem 11. The definition of a fractal algebra is in [3].
Corollary 12 The algebra S(T u (C)) is fractal.
Note that the consistency map of P u is fractal; so the assertion of the corollary follows from Theorems [3, 1.69] and 11.
Sequences in fractal algebras are distinguished by their excellent convergence properties. To mention only a few of them, let σ(a) denote the spectrum of an element a of a C * -algebra with identity element e, write σ 2 (a) for the set of the singular values of a, i.e., σ 2 (a) is the set of all non-negative square roots of elements in the spectrum of a * a and finally, for ε > 0, let σ (ε) (a) refer to the εpseudospectrum of a, i.e. to the set of all λ ∈ C for which a − λe is not invertible or (a − λe) −1 ≥ 1/ε. Theorem 13 Let (A n ) be a sequence in S(T u (C)) with strong limit A. Then the following set-sequences converge with respect to the Hausdorff distance as n → ∞ : The proof follows immediately from the stability criterion in Theorem 11 and from Theorems 3.20, 3.23 and 3.33 in [3]. Note that in general one cannot remove the assumption (A n ) = (A n ) * in assertion (a), whereas (c) holds without this assumption.
The notion of a Fredholm sequence was introduced in [6]; see also [3,Chapter 6]. In the present setting, the Fredholm property of a sequence (A n ) ∈ S(T u (C)) means nothing but the invertibility of the coset (A n ) + J in the quotient algebra S(T u (C))/J , and the results of [6] specify as follows. Let σ 1 (a) ≤ . . . ≤ σ n (A) = A denote the singular values of the n × n-matrix A.
Theorem 14 Let (A n ) be a sequence in S(T u (C)) with strong limit A. Then