Let \(A({\mathbb{C}}^ n)\) denote the entire functions on \({\mathbb{C}}^ n\) and let \({\mathbb{P}}=(p_ k)_{k\in {\mathbb{N}}}\) be an increasing sequence of plurisubharmonic functions on \({\mathbb{C}}^ n\) which satisfy some mild technical conditions. Then \[ A_{{\mathbb{P}}}({\mathbb{C}}^ n):=\{f\in A({\mathbb{C}}^ n);\quad \exists k\in {\mathbb{N}}:\sup_{z\in {\mathbb{C}}^ n}| f(z)| | \exp (-p_ k(z))<\infty\}, \] endowed with its natural inductive limit topology is a (DFN)-algebra.
Using results of C. A. Berenstein and B. A. Taylor [J. Anal. Math.38, 188-254 (1980; Zbl 0464.42003)] and the nuclearity of \(A_{{\mathbb{P}}}({\mathbb{C}}^ n)\) we show:
Let \((F_ 1,...,F_ N)\in (A_{{\mathbb{P}}}({\mathbb{C}}^ n))^ N\) be slowly decreasing (which implies that the zero-variety of \((F_ 1,...,F_ N)\) is discrete) and let \(I_{loc}(F_ 1,...,F_ N)\) denote the localized ideal generated by \((F_ 1,...,F_ N)\) in \(A_{{\mathbb{P}}}({\mathbb{C}}^ n)\). Then \(A_{{\mathbb{P}}}({\mathbb{C}}^ n)/I_{loc}(F_ 1,...,F_ N)\) is isomorphic to \(\lambda(B)'_ b\), the strong dual of \(\lambda(B)\), where the Köthe matrix B can be described explicitly in terms of \({\mathbb{P}}\) and the zero variety of \((F_ 1,...,F_ N).\)
This sequence spaces representation makes it possible to apply results from the structure theory of nuclear Fréchet spaces in order to decide whether \(I_{loc}(F_ 1,...,F_ N)\) is complemented in \(A_{{\mathbb{P}}}({\mathbb{C}}^ n)\). We mention the following particular examples which extend results of B. A. Taylor [Mich. Math. J. 29, 185-197 (1982; Zbl 0471.30014)]. For \({\mathbb{P}}=(k| z|^{\alpha}(\log (1+| z|^ 2))^{\beta})_{k\in {\mathbb{N}}}\), \(0<\alpha <\infty\), \(0\leq \beta <\infty\), every closed ideal in \(A_{{\mathbb{P}}}({\mathbb{C}})\) is complemented. For \({\mathbb{P}}=(k(\log (1+| z|^ 2))^ s)_{k\in {\mathbb{N}}}\), \(1<s<\infty\) and \({\mathbb{P}}:=(| z|^{s_ k})_{k\in {\mathbb{N}}}\), where \((s_ k)_{k\in {\mathbb{N}}}\) is a strictly increasing sequence in \(]0,\infty [\), every proper infinite codimensional closed ideal in \(A_{{\mathbb{P}}}({\mathbb{C}})\) is not complemented. Moreover, one gets information on the translation invariant closed linear subspaces of certain Fréchet spaces of entire functions on \({\mathbb{C}}\). In particular, the following holds:
Every proper closed linear subspace W of \(A({\mathbb{C}})\) which is translation invariant and infinite dimensional has a Schauder basis of exponential polynomials, is isomorphic to a power series space of infinite type and is complemented in \(A({\mathbb{C}})\).