On the approximation numbers of large Toeplitz matrices.(English)Zbl 0873.47014

Summary: The $$k$$th approximation number $$s_k^{(p)}(A_n)$$ of a complex $$n\times n$$ matrix $$A_n$$ is defined as the distance of $$A_n$$ to the $$n\times n$$ matrices of rank at most $$n-k$$. The distance is measured in the matrix norm associated with the $$l^p$$ norm $$(1<p<\infty)$$ on $$\mathbb{C}^n$$. In the case $$p=2$$, the approximation numbers coincide with the singular values.
We establish several properties of $$s_k^{(p)}(A_n)$$ provided $$A_n$$ is the $$n\times n$$ truncation of an infinite Toeplitz matrix $$A$$ and $$n$$ is large. As $$n\to\infty$$, the behavior of $$s_k^{(p)}(A_n)$$ depends heavily on the Fredholm properties (and, in particular, on the index) of $$A$$ on $$l^p$$.
This paper is also an introduction to the topic. It contains a concise history of the problem and alternative proofs of the theorem by G. Heinig and F. Hellinger [Integral Equations Oper. Theory 19, No. 4, 419-446 (1994; Zbl 0817.47036)] as well as of the scalar-valued version of some recent results by S. Roch and B. Silbermann concerning block Toeplitz matrices on $$l^2$$.

MSC:

 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A75 Eigenvalue problems for linear operators 47A58 Linear operator approximation theory 47N50 Applications of operator theory in the physical sciences 65F35 Numerical computation of matrix norms, conditioning, scaling 15A09 Theory of matrix inversion and generalized inverses 15A18 Eigenvalues, singular values, and eigenvectors 47A53 (Semi-) Fredholm operators; index theories

Zbl 0817.47036
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