Integral representations of continuous linear operators on \(p\)-adic function spaces (Q2751731)

omt \(E\) and \(F\) be non-Archimedean locally convex spaces over a complete valued field \(K\), \({\mathcal L}(E, F)\) be the space of continuous linear maps of \(E\) into \(F\), \(X\) be a topological space, \({\mathcal K}(X)\) be the Boolean algebra of the closed and open (clopen) subsets of \(X\) and \(M(X,{\mathcal L}(E, F))\) be the vector space of finitely-additive measures \(m\) of \({\mathcal K}(X)\) into \({\mathcal L}(E,F)\), such that \(m({\mathcal K}(X))\) is an equicontinuous subset of \({\mathcal L}(E,F)\).NEWLINENEWLINENEWLINEBy defining the support of such measures, the authors study three subspaces of \(M(X,{\mathcal L}(E,F))\). The first subspace \(M_t(X,{\mathcal L}(E, F))\) is that of tight measures \(m\), i.e., measures which satisfyNEWLINENEWLINENEWLINE(i) for any continuous semi-norm (c.s.n.) \(q\) of \(F\), there exists a c.s.n. \(p\) of \(E\) such that \((m)_{p,q}(X)< \infty\), andNEWLINENEWLINENEWLINE(ii) for any \(\varepsilon> 0\), there exists \(Y\subset X\), \(Y\) compact such that \((m)_{p,q}(Y^c)<\varepsilon\), where NEWLINE\[NEWLINE(m)_{p,q}(Y)= \sup\Biggl\{{q(m(A)x)\over p(x)}: A\in{\mathcal K}(x), A\subset Y, p(x)\neq 0\Biggr\}.NEWLINE\]NEWLINE The other subspaces \(M_k(X,{\mathcal L}(E, F))\supset M_s(X,{\mathcal L}\subset E, F))\) are defined by replacing in (ii) the \(\varepsilon\)-condition by \((m)_{p,q}(Y^c)= 0\), with \(Y\) compact for \(M_k\) and \(Y\) finite for \(M_s\).NEWLINENEWLINENEWLINEThe authors establish algebraic isomorphism of the above measure spaces onto specific linear operator spaces assuming the completeness of \(F\). They give necessary conditions on the representing measure so that the linear operators are compact and nuclear.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].