×

Trace formulas for a class of vector-valued Wiener-Hopf like operators. I. (English) Zbl 1262.47043

Let \(h\) be a summable \(d\times d\) matrix-valued function on \({\mathbb R}\) and \[ \Delta(\mu)=I_d+\int_{\mathbb R} e^{i\mu x}h(x)\,dx. \] By \(L_2^d({\mathbb R},\Delta)\) we denote the space of \(d\times 1\) vector-valued functions \(f\) such that \[ \int_{\mathbb R}f(\mu)^*\Delta(\mu)f(\mu)\,d\mu<\infty, \] and let \({\mathcal E}_T^d(\Delta)\) be its subspace consisting of entire functions of exponential type less than or equal to \(T\). The orthogonal projection from \(L_2^d({\mathbb R},\Delta)\) to \({\mathcal E}_T^d(\Delta)\) is denoted by \(P_T\), and the operator of multiplication by a suitably restricted \(d\times d\) matrix-valued function is denoted by \(G\). It is shown that, under certain assumptions, the operators \[ (P_T GP_T)^n-P_TG^nP_T \] are of trace class for every \(n\in{\mathbb N}\) whenever \(T\) is sufficiently large and the limits \[ k_n(G)=\lim_{T\to\infty}\text{trace}\{(P_TGP_T)^n-P_TG^nP_T\} \] exist. Moreover, these limits are independent of \(h\) when \(G\) commutes with certain factors of \(\Delta\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhiezer, N. I., A functional analogue of some theorems on Toeplitz matrices, Ukrain. Math. Žh., 16, 445-462 (1964)
[2] Arov, D. Z.; Dym, H., (J-Contractive Matrix Valued Functions and Related Topics. J-Contractive Matrix Valued Functions and Related Topics, Encyclopedia of Mathematics and its Applications, vol. 116 (2008), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1159.30001
[3] Arov, D. Z.; Dym, H., (Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations. Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations, Encyclopedia of Mathematics and its Applications, vol. 145 (2012), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1264.34004
[4] Basor, E., A brief history of the strong Szegő limit theorem, (Mathematical Methods in Systems, Optimization, and Control. Mathematical Methods in Systems, Optimization, and Control, Oper. Theory Adv. Appl., vol. 222 (2012), Birkhäuser, Springer: Birkhäuser, Springer Basel, AG), 73-83 · Zbl 1300.47037
[5] Böttcher, A.; Silbermann, B., (Analysis of Toeplitz Operators. Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2006), Springer-Verlag: Springer-Verlag Berlin), Prepared jointly with Alexei Karlovich · Zbl 1098.47002
[6] Böttcher, A.; Widom, H., Szegő via Jacobi, Linear Algebra Appl., 419, 2-3, 656-667 (2006) · Zbl 1116.47024
[7] Devinatz, A., On Wiener-Hopf operators, (Functional Analysis (Proc. Conf., Irvine, Calif., 1996) (1967), Academic Press: Academic Press London), 81-118
[8] Devinatz, A., The strong Szegő limit theorem, Illinois J. Math., 11, 160-175 (1967) · Zbl 0166.40301
[9] Dym, H., Trace formulas for a class of Toeplitz-like operators, Israel J. Math., 27, 1, 21-48 (1977) · Zbl 0379.47019
[10] Dym, H., Trace formulas for a class of Toeplitz-like operators. II, J. Funct. Anal., 28, 1, 33-57 (1978) · Zbl 0449.47023
[11] Dym, H., Trace formulas for blocks of Toeplitz-like operators, J. Funct. Anal., 31, 1, 69-100 (1979) · Zbl 0418.47014
[12] Dym, H., Hermitian block Toeplitz matricees, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, (Orthogonal Matrix-Valued Polynomials and Applications (Tel Aviv, 1987-1988). Orthogonal Matrix-Valued Polynomials and Applications (Tel Aviv, 1987-1988), Oper. Theory Adv. Appl., vol. 34 (1988), Birkhäuser: Birkhäuser Basel), 79-135
[13] Dym, H., On reproducing kernels and the continuous covariance extension problem, (Analysis and Partial Differential Equations. Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., vol. 122 (1990), Dekker: Dekker New York), 427-482
[14] Dym, H., On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares, Comm. Pure Appl. Math., 47, 2, 207-256 (1994) · Zbl 0819.47013
[15] Dym, H.; Ta’asan, S., An abstract version of a limit theorem of Szegő, J. Funct. Anal., 43, 3, 294-312 (1981) · Zbl 0499.47017
[16] Ellis, R. L.; Gohberg, I.; Lay, D. C., Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials, (Continuous and Discrete Fourier Transforms, Extension Problems, and Wiener-Hopf Equations. Continuous and Discrete Fourier Transforms, Extension Problems, and Wiener-Hopf Equations, Oper. Theory Adv. Appl., vol. 58 (1992), Birkhäuser: Birkhäuser Basel), 26-70 · Zbl 0786.47015
[17] Gohberg, I.; Goldberg, S.; Kaashoek, M. A., (Classes of Linear Operators. Vol. II. Classes of Linear Operators. Vol. II, Operator Theory: Advances and Applications, vol. 63 (1993), Birkhäuser Verlag: Birkhäuser Verlag Basel)
[18] Gohberg, I.; Koltracht, I., Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation, Numer. Math., 47, 2, 237-288 (1985) · Zbl 0589.65087
[19] Gohberg, I.; Krein, M. G., (Introduction to the Theory of Linear Nonselfadjoint Operators. Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, vol. 18 (1969), American Mathematical Society: American Mathematical Society Providence, RI), Translated from the Russian by A. Feinstein. · Zbl 0181.13504
[20] Hirschman, I. I., On a theorem of Szegő, Kac, and Baxter, J. Anal. Math., 14, 225-234 (1965) · Zbl 0141.07001
[21] Hirschman, I. I., On a formula of Kac and Achiezer, J. Math. Mech., 16, 167-196 (1966) · Zbl 0154.37203
[22] Hirschman, I. I., On a formula of Kac and Achiezer. II, Arch. Ration. Mech. Anal., 38, 189-223 (1970) · Zbl 0211.41804
[23] Kac, M., Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J., 21, 501-509 (1954) · Zbl 0056.10201
[24] Simon, B., (Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory, American Mathematical Society Colloquium Publications, vol. 54 (2005), American Mathematical Society: American Mathematical Society Providence RI)
[25] Simon, B., (Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory, American Mathematical Society Colloquium Publications, vol. 54 (2005), American Mathematical Society: American Mathematical Society Providence, RI)
[26] Sz-Nagy, B.; Foiaş, C., Harmonic Analysis of Operators on Hilbert Space (1970), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam, Translated from the French and revised · Zbl 0201.45003
[27] Widom, H., Asymptotic behavior of block Toeplitz matrices and determinants, Adv. Math., 13, 284-322 (1974) · Zbl 0281.47018
[28] Widom, H., On the limit of block Toeplitz determinants, Proc. Amer. Math. Soc., 50, 167-173 (1975) · Zbl 0312.47027
[29] Widom, H., Perturbing Fredholm operators to obtain invertible operators, J. Funct. Anal., 20, 1, 26-31 (1975) · Zbl 0307.47015
[30] Widom, H., Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math., 21, 1, 1-29 (1976) · Zbl 0344.47016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.