The PDE-preserving operators on nuclearly entire functions of bounded type (Q1878606)

Let \(E\), \(F\) be complex Banach spaces. The pair \((E,F)\) is called a Banach pairing if \(F=E'\) or \(E=F'\). The Banach space of nuclear \(n\)-homogeneous polynomials equipped with the nuclear norm \(\| \,\cdot\,\| _n\) is denoted by \(P_N({}^n E)\), \(\mathcal H_{Nb}(E)\) stands for the space of nuclear entire functions of bounded type, and \(\text{Exp}\, (F)\) for the space of exponential type functions. The set of all convolution operators on \(\mathcal H_{Nb}(E)\), i.e., of all \(T\in \mathcal L(\mathcal H_{Nb}(E))\) which commute with all translations, is denoted by \(\mathcal C(E)\). Let \((E,F)\) be a Banach pairing. An operator \(T\in\mathcal L(\mathcal H_{Nb}(E))\) is PDE-preserving if \(\ker P(D)\subseteq \ker P(D)T \) for every polynomial \(P\in\mathcal P({}^n F)\). The set of all PDE-preserving operators is denoted by \(\mathcal O=\mathcal O(E)\). Denote by \(\Sigma(F)\) the set of all sequences \((\varphi_i)\subseteq \text{Exp}\,(F)\), \(\varphi_i=\sum^\infty_{j=0}\varphi_{ij}\), \(\varphi_{ij}\in \mathcal P({}^jF)\), such that for some \(N,M,r>0\), \(\| \varphi_{ij}\| \leq NM^i r^j/{j!}\). The author shows the following Theorem: Let \((E,F)\) be a Banach pairing and let \(\phi=(\varphi_n)\in\Sigma\). Then \(\phi\) defines a PDE-preserving operator \(\phi(D)\) on \(\mathcal H_{Nb}(E)\) by \[ \phi(D)f:=\sum^\infty_{n=0}\varphi_{nn}(d)f, \quad f\in \mathcal H_{Nb}(E). \] Conversely, if \(T\) is a PDE-preserving operator, then there is a unique \(\phi\in \mathcal L\) such that \(T=\phi(D)\). Therefore the ring structures on \(\Sigma\) can be carried over to \(\mathcal O\) and vice versa. In this way it follows that \(\mathcal O\) is a non-commutative ring (algebra) with unity with respect to composition and \(\mathcal C\) forms a commutative subring (algebra). Moreover, the author investigates range and kernel properties for the operators in \(\mathcal O \) and characterizes the projectors (onto polynomial spaces) in \(\mathcal O\) by determining the corresponding elements in \(\Sigma\).