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Spectral properties of operators that characterize \(\ell_\infty^{(n)}\). (English) Zbl 0983.46010

Let \(T\) be an operator on a fixed \(n\)-dimensional Banach space \(V\) (over \(\mathbb C\)). For a variable Banach space \(X\) containing \(V\) let \(e(T, X) := \inf\{\|\widetilde{T}\|: \widetilde{T}:X \mapsto V, \widetilde{T}_{|V} = T \}\) and \(e(T) := \sup\{e(T, X): X\supseteq V\}.\) When \(T = I_{V}\), these numbers are the relative and absolute projection constants for \(V\). A result of L. Nachbin [Trans. Am. Math. Soc. 68, 28-46 (1950; Zbl 0035.35402)] states that \(e(I_V) = 1\) if and only if \(V \cong \ell^{(n)}_{\infty}\) (\(\cong\) means “is isometric to”). In order to consider other operators on \(V\) (and allow for isometric images of \(V\)) a certain subtlety is necessary.
For a fixed matrix \(A\) let \(A(V)\) denote the set of all operators \(T\) on \(V\) for which there is a basis relative to which the matrix for \(T\) is \(A\). Then \(\lambda_A(V):= \inf\{e(T)/\|T\|: T \in A(V)\}\) and \(\lambda_A^u(V):= \inf\{e(T) : T \in A(V)\}\). In an earlier paper [Linear Algebra Appl. 270, 155-169 (1988; Zbl 0898.46011)] the authors showed that \((\lambda_A(V) = 1) \Rightarrow (V \cong \ell^{(n)}_{\infty})\) holds if and only if \(A = I\) and conjectured that the implication \((\lambda_A^u(V) = 1) \Rightarrow (V \cong \ell^{(n)}_{\infty})\) holds if and only if the spectrum of \(A\) is contained in the unit circle. The main purpose of the present paper is to prove the truth of the conjecture.

MSC:

46B07 Local theory of Banach spaces
46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
51M20 Polyhedra and polytopes; regular figures, division of spaces
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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