Zeros of analytic functions and norms of inverse matrices

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 _kz^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(λ₁,..., 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 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→+∞.