Tameness in Fréchet spaces of analytic functions (Q2809349)

For a continuous, linear operator \(T\) between graded Fréchet spaces, the characteristic of continuity is defined as NEWLINE\[NEWLINE \sigma_T(n) = \inf \left\{ s \mid \exists C\, \forall x: \| T(x) \|_n \leq C \| x \|_s \right\}. NEWLINE\]NEWLINE 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 [\textit{E. Dubinsky} and \textit{D. Vogt}, Stud. Math. 93, No. 1, 71--85 (1989; Zbl 0694.46003)], while a result of \textit{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.NEWLINENEWLINEThe 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 \( \Lambda_1((\mathcal E_n)_n) \).NEWLINENEWLINEThe framework is also used to construct pluricomplex Green's functions for some Stein manifolds.