Solid hulls and cores of weighted \(H^\infty \)-spaces (Q1790943)

Let \(E\) be a vector subspace of \(s:=\mathbb C^\mathbb N\), the space of all complex sequences. Assume that \(E\) contains \(c_{00}\), the set of all sequences with finite support. Then \(E\) is called solid if \((a_n)\in E\) and \(|b_n|\leq |a_n|\) for all \(n\) implies that \((b_n)\in E\). The solid hull \(S(E)\) of \(E\) is the set of all \((c_n)\in s\) such that there exists \((a_n)\in E\) such that \(|c_n|\leq |a_n|\) for all \(n\). Finally, the solid core of \(E\) is the set of all \((c_n)\in s\) such that, for all bounded sequences \((b_n)\), one has \((c_nb_n)\in E\). The authors determine for various spaces \(\mathcal H\) of holomorphic functions in the disk and the plane their solid hull and core. The elements in these spaces are, of course, identified with their associated sequence of Taylor coefficients. These spaces \(\mathcal H\) are of the type \[ H^\infty_\nu=\{f\in H(U): \|f\|_\nu=\sup_{z\in U}|f(z)|\nu(z)<\infty\}, \] where \(U\) is \(\mathbb D\) or \(\mathbb C\) and where \(\nu\) is a certain weight function, as, for example, \(\nu(z)=\nu(r)=w(r)\exp(-a/((1-r^e)^b)\) for \(r=|z|\), \(a,b>0\), \(e\in\{1,2\}\). It is also shown that, if \(\mathcal H:=H^\infty_\nu(\mathbb C)\) coincides with its solid hull \(S\), then the monomials \(\{z^k: k=0,1,\dots\}\) form a Schauder basis for \(H^0_\nu\), the closure of the polynomials in \(H^\infty_\nu\). Note that, for all weights, \(\mathcal H\not=S\) in the disk case. The paper is very technical with a myriad of formulas. It continues research done by \textit{J. Bonet} and \textit{J. Taskinen} in [Rev. Mat. Iberoam. 34, No. 2, 593--608 (2018; Zbl 1403.46023)] and [Ann. Acad. Sci. Fenn., Math. 43, No. 1, 521--530 (2018; Zbl 1398.46021)].