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

Citations:

Zbl 0035.35402; Zbl 0898.46011
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