Zeros of analytic functions and norms of inverse matrices

E. Gluskin, M. Meyer, A. Pajor

Research output: Contribution to journalArticlepeer-review

Abstract

Let k n be the smallest constant such that for any n-dimensional normed space X and any invertible linear operator T ∈L(X) we have {Mathematical expression}. Let A + be the Banach space of all analytic functions f(z)=Σ k≥0 a kzk on the unit disk D with absolutely convergent Taylor series, and let {norm of matrix}f{norm of matrix}A + k≥0 |ακ|; define φ{symbol} n on {Mathematical expression} by {Mathematical expression}. We show that k n=sup {φ{symbol} n1,..., λ n ); (λ1,..., λ n )∈ {Mathematical expression}}. Moreover, if S is the left shift operator on the space ℓ∞:S(x 0, x 1, ..., x p, ...)=(x 1,..., x p,...) and if Jn(S) denotes the set of all S-invariant n-dimensional subspaces of ℓ∞ on which S is invertible, we have {Mathematical expression}. J. J. Schäffer (1970) proved that k n≤√en and conjectured that k n=2, for n≥2. In fact k 3>2 and using the preceding results, we show that, up to a logarithmic factor, k n is of the order of √n when n→+∞.

Original languageEnglish
Pages (from-to)225-242
Number of pages18
JournalIsrael Journal of Mathematics
Volume87
Issue number1-3
DOIs
StatePublished - Feb 1994

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