Spectra of weighted (LB)-algebras of entire functions on Banach spaces (Q655462)

Given a complex Banach space \(X,\) a \textit{weight} on \(X\) is a \(]0,1]\)-valued function \(v\) defined on \(X\) by \(v(x)=\eta(\|x\|),\) where \(\eta\) is a rapidly decreasing continuous function on \([0,+\infty[\). The paper under review deals with the space \(VH(X)\) of analytic functions on \(X\) that become bounded when multiplied by some weight in the decreasing sequence \(V=(v_n)\) and endowed with the inductive limit topology. Along the lines of [``Weighted spaces of holomorphic functions in Banach spaces'', Stud. Math. 138, No. 1, 1--24 (2000; Zbl 0960.46025)] by \textit{D. García, M. Maestre} and \textit{P. Rueda}, conditions are provided for \(VH(X)\) to be an algebra. Attention is paid to generalized Hörmander algebras \(A_p(X)\), a particular case of \(VH(X)\) whose weights are defined through a so-called \textit{growth condition} \(p:[0,+\infty[\to [0,+\infty[\) (see the paper) by means of \(v_n(x)=\exp (-np(\|x\|)).\) When such \(p\) is the identity mapping, it turns out that \(A_p(X)\) is the space of analytic functions of exponential type \(\mathrm{Exp}(X).\) Bearing in mind that the spaces of homogeneous polynomials \(P(^nX)\) suitably decompose \(A_p(X),\) a quite natural condition on sequences of isomorphisms between \(P(^nX)\) and \(P(^nY)\) (\(Y\) another complex Banach space) is shown so that the two Hörmander algebras \(A_p(X)\) and \(A_p(Y)\) are isomorphic. That condition is used to reprove Theorem 2 by \textit{D. Carando} and \textit{P. Sevilla-Peris} in [``Spectra of weighted algebras of holomorphic functions'', Math. Z. 263, No. 4, 887--902 (2009; Zbl 1191.46023)] concerning whether \(\mathrm{Exp}(X)\) is isomorphic to \(\mathrm{Exp}(Y).\) The final section is devoted to the spectrum of \(VH(X),\) \(X\) a symmetrically regular space. For some weights it is shown to be a Riemann domain to which every function in \(VH(X)\) can be extended in an analytic way. Much of the section's development relies on work by \textit{D. Carando} et al. [Topology 48, No. 2--4, 54--65 (2009; Zbl 1198.46026)] where the same kind of questions was treated.