Infinite extensions of Toeplitz matrices. (English. Russian original) Zbl 1091.47023

Al’pin, Yu. A.; Il’in, S. N.

J. Math. Sci., New York 127, No. 3, 1957-1961 (2005); translation from Zap. Nauchn. Semin. POMI 296, 5-14 (2003).

In the paper under review, the authors derive a formula for the smallest of the ranks of infinite Toeplitz extensions of a finite rectangular Toeplitz matrix. More concretely, they show that if \(T_k\) is a \(k\times (n+1-k)\) Toeplitz matrix and \(\text{rt}\,(T_k)\) denotes the smallest of the ranks of infinite Toeplitz extensions of \(T_k\), then \(\text{rt}\,(T_k)\) is the greatest value of \(k\leq n\) such that either the first row of \(T_k\) cannot be expressed as a linear combination of the subsequent rows, or the last row of \(T_k\) cannot be expressed as a linear combination of the preceding rows.

Reviewer: Woo Young Lee (Seoul)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
15B57	Hermitian, skew-Hermitian, and related matrices
47A20	Dilations, extensions, compressions of linear operators
47A57	Linear operator methods in interpolation, moment and extension problems

Keywords:

Toeplitz matrix; Toeplitz extension

References:

[1]	I. S. Iokhvidov, Hankel and Toeplitz Matrices and Forms [in Russian], Nauka, Moscow (1974).
[2]	I. S. Iokhvidov and O. D. Tolstykh, ”On the (r, k, l) characteristics of rectangular Toeplitz matrices,” Ukr. Mat. Zh., 32, 477–482 (1980).
[3]	Yu. A. Al’pin and N. Z. Gabbasov, ”Extension of generalized Hankel matrices,” Izv. Vyssh. Uchebn. Zaved., Matematika, No. 5, 35–39 (1981). · Zbl 0499.15007