Commentary on "Expansions for eigenfunctions and eigenvalues of largen Toeplitz matrices"

 

Torsten Ehrhardt¹*

*Email: tehrhard@ucsc.edu

Department of Mathematics, University of California, Santa Cruz, CA95064, USA.

 

 

Theorem and its generalizations. The reader is advised to consult, e.g. [3] for many more details and references.

For various classes of symbols </>, descriptions have been given for the limiting set (in the Hausdorff metric) of the spectrum of T_n((p) as n goes to infinity. For instance, a result of Widom [5] states that for certain symbols, the eigenvalues of T_n((p) accumulate asymptotically along the curve described by <f>(z), \z\ = 1. The result applies to the symbols considered here, where it is of importance that a(z) is nonsmooth at precisely one point.

Moreover, under certain conditions on </>, one variant of the Szegó Limit Theorem states that

with O < a < |/31 < 1 being real parameters. The function a(z) is smooth (in fact, analytic) except at the point z = 1, where it has a "mild" zero. Its image describes a simple closed curve in the complex plañe as z passes along the unit circle.

The problem of the asymptotics of the eigenvalues of Toeplitz matrices has a long history and is a multifaceted and difiicult topic. It is closedly connected with the asymptotics of the determinants of Toeplitz matrices and thus with the Szegó Limit

where / is a smooth test function and X_k are the eigenvalues of T_n (</>).

One should point out that there are other classes of symbols which show a completely different asymptotics of the Toeplitz eigenvalues. For instance, if <f> is a rational function, then it is proved that the eigenvalues do (in general) accumulate along ares which lie inside the curve described by </>. This case is best understood because there is an explicit formula for the characteristic polynomial of

Furthermore, if </> is a piecewise continuous function with at least two jump discontinuities, then it is conjectured and numerically substantiated that "most", but not all eigenvalues, accumulate along the image. If there is precisely one jump discontinuity, then one expects that all eigenvalues accumulate along the image.

For continuous realvalued symbols, i.e. for Hermitian Toeplitz matrices, the asymptotics of the eigenvalues is again "canonical", i.e. the eigenvalues accumulate along the image and the above formula holds for continuous test functions.

The two aforementioned results give some, but limited information about the eigenvalues of the Toeplitz matrices. The paper under consideration (together with a preceding paper [4]) makes a significant first attempt to determine the asymptotics of the individual eigenvalues of T_n(a). The asymptotics are obtained up to third order and can be described by

In view of the argumentation, it seems plausible that the results can be generalized without too much effort to slightly more general symbols,

where b(z) is a smooth (or analytic) function for \z\ = 1 such that a(z) describes a simple closed curve in the complex plañe. On the other hand, notice that symbols with two or more singularities could produce a more complicated eigenvalue behavior [5].

After a preprint versión of the paper appeared, Bogoya, Bóttcher and Grudsky [2] gave a rigorous proof of the eigenvalue asymptotics in the special case of symbols a(z) with ¡3 = a 1. The general case is (as of now) still open.

asn> oo, with an explicit expression for d = d{x), which is continuous in O < x < 1. (We made some slight changes regarding notation and formulation in comparison with the main formula (25) in [1].) The asymptotics holds uniformly in k under the assumption O < e < k/n < 1 e. The latter means that the description does not catch the eigenvalues accumulating near the point O = a(l) where the curve a(z) is not smooth. At this point, perhaps a different, more complicated asymptotics holds.

The derivation of the results in the paper is not completely rigorous, despite the arguments being quite convincing. The methods are appropriate for dealing with Toeplitz systems. For instance, it is made use of the fact that the finite matrices T_n((p) are naturally related to two semiinfinite Toeplitz systems T(</>) = ((pjk) and T(</>) = (<f>kj), j,k > 0. In the paper, this is reflected by the use of the auxiliary functions (p~ and </>+. The symbol </> is equal to K(z) = a(z) A, where A is an eigenvalue (which is to be determined). The two semiinfinite systems are analyzed by WienerHopf factorization, the factors of which serve as approximations for the auxiliary functions (p~ and </>+. Both auxiliary functions allow to reconstruct the eigenfunction for

Acknowledgements Supported in part by NSF grant DMS0901434.

[1] L P Kadanoff, Expansions for eigenfunctions and eigenvalues of largen Toeplitz matrices, Pap. Phys. 2, 020003 (2010).

[2] J M Bogoya, A Bóttcher, S M Grudsky, Asymptotics of individual eigenvalues of large Hessenberg Toeplitz matrices, Preprint 20108, Fakultát für Mathematik, Technische Universitát Chemnitz, ISSN 16148835.

[3] A Bóttcher, B Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer, New York (1999).

[4] H Dai, Z Geary, L P Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices, J. Stat. Mech. P05012 (2009).

[5] H Widom, Eigenvalue distrihution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, Oper. Theory: Adv. Appl. 48, 387 (1990).