On the Approximation Numbers of Large Toeplitz Matrices

The $k$th approximation number $\skp(A_n)$ of a complex $\ntn$ matrix $A_n$ is defined as the distance of $A_n$ to the $\ntn$ matrices of rank at most $n-k$. The distance is measured in the matrix norm associated with the $l^p$ norm $(1<p<\iy)$ on $\bC^n$. In the case $p=2$, the approximation numbers coincide with the singular values.

We establish several properties of $\skp(A_n)$ provided $A_n$ is the $\ntn$ truncation of an infinite Toeplitz matrix $A$ and $n$ is large. As $n\to\iy$, the behavior of $\skp(A_n)$ 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 as well as of the scalar-valued version of some recent results by S. Roch and B. Silbermann concerning block Toeplitz matrices on $l^2$.

1991 Mathematics Subject Classification: Primary 47B35; Secondary 15A09, 15A18, 15A60, 47A75, 47A58, 47N50, 65F35

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