## 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_{ k}z^{k} on the unit disk D with absolutely convergent Taylor series, and let {norm of matrix}f{norm of matrix}A_{ +}=σ_{ k≥0} |ακ|; define φ{symbol}_{ n} on {Mathematical expression} by {Mathematical expression}. We show that k_{ n}=sup {φ{symbol}_{ n}(λ_{1},..., λ_{ n} ); (λ_{1},..., 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 J_{n}(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 language | English |
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Pages (from-to) | 225-242 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 87 |

Issue number | 1-3 |

DOIs | |

State | Published - Feb 1994 |