Equivalence classes of block Jacobi matrices

The paper contains two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has A_n coefficients converging to 1, and second, that under an L1-type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2 and 3 are pairwise asymptotic.


Introduction and results
A block Jacobi matrix is an infinite matrix of the form where A n , B n are l × l matrices with A n invertible. The sequences A n and B n are called Jacobi parameters of J. Two block Jacobi matrices J and J are called equivalent if their Jacobi parameters satisfy (1.1) A n = σ * n A n σ n+1 , B n = σ * n B n σ n for unitary σ n 's with σ 1 = 1. The definition comes from the fact that (1.1) holds if and only if the (matrix-valued) spectral measures of J and J coincide (see [3] for the details).
Inductively it is easy to see that where p n are the orthonormal polynomials for J. We say that a block Jacobi matrix is of type 1 if A n > 0 for all n, of type 2 if A 1 A 2 . . . A n > 0 for all n, and of type 3 if every A n is lower triangular with strictly positive elements on the diagonal. Each equivalence class of block Jacobi matrices contains exactly one matrix of type 1, 2 and 3 (follows from the uniqueness of the polar and QR decompositions, see [3] for the proof). We say that J is in the Nevai class if B n → 0, A n A * n → 1. Note that this definition is invariant within the equivalence class of Jacobi matrices. Theorem 1. Assume J belongs to the Nevai class. If J is of type 1, 2 or 3, then A n → 1 as n → ∞.
This result was proven in [3] for the type 1 and 3 cases, and was left open for type 2. It is proven here in Section 2.
Note that the essence of Theorem 1 is to show that σ * n σ n+1 → 1, where σ n 's are the unitary coefficients from (1.1) for J, J of type 1, 2 or 3. Looking at (1.3), it is clear that any result on the asymptotics of p n (see e.g. [1], [4], [5]) would involve the limit lim n→∞ σ n . This explains the need for the following definition.
Definition. Two equivalent matrices J and J with (1.1) are called asymptotic to each other if the limit lim n→∞ σ n exists.
Clearly this is an equivalence relation on the set of equivalent block Jacobi matrices. Thus, establishing Szegő asymptotics (which simply means lim n→∞ z n p n (z + z −1 ) exists) for any block Jacobi matrix immediately implies the corresponding asymptotics for any of the Jacobi matrices asymptotic to the original one.
Then the corresponding Jacobi matrices of type 1, 2 and 3 are pairwise asymptotic.
Remarks. 1. The condition (1.4) doesn't depend on the choice of the representative of the equivalence class of equivalent matrices.
2. The proof also shows that any Jacobi matrix, for which eventually each A n has real eigenvalues, is also asymptotic to type 1, 2, 3.
3. An example of an equivalence class of block Jacobi matrices that fails (1.4) and that has type 1 and type 2 nonasymptotic to each other can be found at the end of Section 2.
Acknowledgements. The author would like to thank Prof Barry Simon for helpful comments.

Proofs of the results
We will be using the following lemma from [6]. For self-containment purposes we give a proof of it in the Appendix.
Lemma 1 (Li [6]). Let φ be the map that takes any invertible matrix T to the unitary factor U in its polar decomposition T = |T |U , where |T | = √ T T * . Then for any invertible l × l matrices B, D the following holds where || · || HS is the Hilbert-Schmidt norm.
Proof of Theorem 1. For type 1 and 3, the statement is proven in [3].
Assume J is of type 2. Denote by J the type 1 Jacobi matrix equivalent to J. Denote its Jacobi parameters by A n , B n , and let for some unitaries σ n . Since A n → 1, we get A n = σ * n A n σ n+1 = σ * n A n σ n σ * n σ n+1 converges to 1 if and only if lim n→∞ σ * n σ n+1 = 1.
Here φ is the same as in Lemma 1.
For the type 3 case of Theorem 2, we will need the following lemma. Recall that the singular values of a matrix A are defined to be the eigenvalues of |A|.
Proof. For sufficiently large matrices A the inequality is clear. It also holds for any compact set on which the right-hand side of (2.2) does not vanish. Therefore, we only need to worry about neighborhoods of matrices with l j=1 (1 − σ j ) 2 = 0, that is, unitary matrices.
Consider any matrix A within distance 1/2 from the unitary group. Let U = φ(A) be the unitary factor in the polar decomposition of A. Since φ(A) is always the closest unitary to A (see e.g. [2]), we get A − U ≤ 1/2 and |A| − 1 ≤ 1/2.
The first-order terms (i.e. those involving only one of ε's or δ's) of the numerator cancel out: Now note that by (2.3), |δ j | ≤ |ε 1 | + |ε l |. Using this and |ε j ε k | ≤ (ε 2 j + ε 2 k )/2 we can bound all of the second-order terms (i.e. those with ε j ε k , ε j δ k and δ j δ k ) by c l j=1 ε 2 j , where c will depend on l only. All of the higher-order terms can be taken care of by using |ε j | < 1, |δ j | < 1 to reduce it to the second-order. Finally, the denominator of the right-hand side of (2.5) is bounded below by 1/2 l . Therefore, we obtain which proves our lemma. (1 − σ j ) 2 .
Since λ j = |λ j |, the previous lemma proves the result.
Proof of Theorem 2. As in Theorem 1, let A n be of type 1, and A n of type 2 with the equivalence (2.1). Then keeping the notation of Theorem 1 and using Lemma 1, we have This implies that σ n is Cauchy, and so converges. An alternative indirect way of proving that type 1 and type 2 are asymptotic to each other is as follows: it is proven in [4] that under condition (1.4) Szegő asymptotics for the type 2 block Jacobi matrix holds. In [5] the same fact is obtained for the type 1 Jacobi matrix. Therefore (1.3) implies that the limit lim n→∞ σ n exists. Now assume that A n is of type 1, and A n of type 3 with the equivalence (2.1). Since all eigenvalues of A n are real and positive, Lemma 3 applies, and we get This shows that σ n is Cauchy, and so converges.