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Tameness in Fréchet spaces of analytic functions. (English) Zbl 1382.46007

For a continuous, linear operator \(T\) between graded Fréchet spaces, the characteristic of continuity is defined as \[ \sigma_T(n) = \inf \left\{ s \mid \exists C\, \forall x: \| T(x) \|_n \leq C \| x \|_s \right\}. \] A graded Fréchet space \(X\) is tame if there is a common upper bound for the characteristics of continuity of all continuous linear endomorphisms of \(X\). The space of entire functions in any dimension is not tame [E. Dubinsky and D. Vogt, Stud. Math. 93, No. 1, 71–85 (1989; Zbl 0694.46003)], while a result of V. P. Zakharyuta [Sov. Math., Dokl. 22, 631–634 (1980; Zbl 0467.32009); translation from Dokl. Akad. Nauk SSSR 255, 11–14 (1980)] can be used to see that the space of all holomorphic functions on a hyperconvex Stein manifold is tame. In the present paper, it is proved that hyperconvexity is a necessary condition for tameness.
The proof relies on the investigation of the ways a power series space of finite type can be embedded into a nuclear Fréchet space. To state the result, some notation is needed: Let \( V \subset U \) be two suitably chosen neighborhoods of zero, let \( d_n(V,U) \) denote their \(n\)-th Kolmogorov diameter and set \( \mathcal E_n = -\ln d_n(V, U) \). Then \( (\mathcal E_n)_n \) is the exponent sequence associated to \(X\). The hypothesis of the main theorem is that \(X\) is nuclear, has properties (\(\underline{\text{DN}}\)) and (\(\Omega\)) and that the power series space of finite type \( \Lambda_1((\mathcal E_n)_n) \) is nuclear and stable. Then \(X\) is a power series space of finite type if and only if \( X \) is tame and its approximate diametral dimension coincides with the approximate diametral dimension of \( \Lambda_1((\mathcal E_n)_n) \).
The framework is also used to construct pluricomplex Green’s functions for some Stein manifolds.

MSC:

46A61 Graded Fréchet spaces and tame operators
46E10 Topological linear spaces of continuous, differentiable or analytic functions
32A70 Functional analysis techniques applied to functions of several complex variables
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
32U15 General pluripotential theory
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References:

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