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br Using the same sort of
Using the same sort of methods, we can also address the proof of Theorem 1.2. The first part of the proof is made much easier if we use the “sieving” construction. Concretely, given a CMV operator , let denote the CMV operator with This induces a simple change in the spectrum; namely, if denotes the two-fold cover , then In other words, one obtains by taking two scaled copies of and putting them on the top and bottom halves of . To see this, one can verify by hand that . The calculation is known to experts, but may not be obvious to the uninitiated, so we will sketch the outline for the reader\'s convenience. First, let denote the factorization of as in (2.1). Then, straightforward calculations using the definitions yield and Therefore, one can verify that Defining subspaces the calculations in Eqs. (3.6)–(3.9) show that HAMI3379 synthesis and invariant and that and . In particular, the claim about the spectrum holds. Moreover, we see that has purely a.c. spectrum if and only if has purely a.c. spectrum.
Additionally, notice that the Szegő matrices (defined in (1.3)) obey In particular, since the spectrum of is given by the closure of the set of at which the Szegő recursion enjoys a polynomially bounded solution [9], (3.10) also suffices to establish (3.5). Beyond that, (3.10) clearly implies for , where denotes the Lyapunov exponent corresponding to . Consequently, since for every set S, one has where we have used hats to denote the sets associated to the sieved CMV operators. The outcome of this discussion is that it suffices to work with the sieved CMV operators, and hence, one may as well assume that all even Verblunsky coefficients vanish.
As a concluding remark to this section, we note that the natural analogue of Theorem 1.2 holds for Jacobi and Schrödinger operators with precisely the same proof.
Well-approximated limit-periodic CMV matrices are reflectionless
We now apply our Theorem 1.2 to prove pure absolute continuity of the spectrum for Pastur–Tkachenko class CMV matrices.
One common thread relating extended CMV matrices to the apparently quite different class of Schrödinger and Jacobi operators on the whole line is Weyl–Titchmarsh theory, by which one studies the whole-line operator via its cyclic restrictions to the half-lines and [18]. Critical to this study are the Weyl–Titchmarsh coefficients, defined for by Restricted to the unit disk, the coefficients are Caratheodory functions; that is, functions holomorphic from to the right half-plane . Consequently, the functions have well-defined radial limits at for (Lebesgue) almost-every . We denote these limits when they exist.
We say that is reflectionless when By design, the reflectionless condition allows one to construct consistent analytic continuations of the Weyl–Titchmarsh coefficients beyond the unit disk. This in turn implies a strong determinism between half-line restrictions of the CMV matrix. The reflectionless property is intimately related to absolute continuity of the spectrum [4], [19], [27] via the following fundamental results:
We will exploit these results in our proof of Theorem 1.3. We first prove via our previous results that the a.c. spectrum is full; from there, we will use Theorem 4.1 to conclude reflectionlessness. In fact, we have the following broader result, which is itself a straightforward application of Theorem 1.2:
Theorem 4.3 allows us to conclude that Pastur–Tkachenko class CMV matrices are reflectionless on their spectrum. Previous spectral estimates from [15] will provide that the spectrum is homogeneous, and we will finally appeal to Theorem 4.2 to conclude purely a.c. spectrum.
Background
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