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On tame pairs of Fréchet spaces. (English) Zbl 1162.46010

A pair \((X,Y)\) of Fréchet spaces is called tame if there exists an increasing function \(\psi :\mathbb N\to\mathbb N\) such that for each continuous linear operator \(T : X \to Y\) there exists \(k_0\in\mathbb N\) such that for each \(k\geq k_0\) there exists \(C_k > 0\) such that
\[ \|Tx\|_k\leq C_k\|x\|_{\psi(k)}\text{ for each }x\in X. \]
The author characterizes tame pairs \((X,Y)\) when either \(X\) is a Köthe sequence space of order 1 or \(Y\) is a Köthe sequence space of order \(\infty\). From this, he derives that if \(Y\) (resp., \(X\)) is a stable power series space of finite type then \((X,Y)\) is tame iff \(X\) satisfies the linear topological invariant \((\overline\Omega)\) (resp., \(Y\) satisfies (\(\underline{\text{DN}}\))). A pair \((X,Y)\) is obviously tame if each continuous linear map \(T :X\to Y\) is bounded. The converse is shown to hold whenever \(Y\) or \(X\) is a stable power series space of inifinte type. This result complements an earlier one of K.Nyberg [Trans.Am.Math.Soc.283, 645–660 (1984; Zbl 0521.46007)]. The author also shows that an arbitrary pair \((X,Y)\) of Fréchet spaces can be tame only if \(Y\) is a Banach space or \(X\) is quasinormable. Hence, the pair \((X,X)\) is tame only if \(X\) is quasinormable.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces

Citations:

Zbl 0521.46007
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References:

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