Continuous linear operators commuting with convolution of measures in infinite-dimensional space (Q580657)

The author generalizes a result of [\textit{R. Edwards}, Functional Analysis, (1965; Zbl 0182.161), Point 5.11.3]. Let H be the real infinite dimensional Hilbert space and B be the \(\sigma\)-algebra of Borel subsets of H. Also let (\({\mathcal S},\tau)\) be the space of basic measures and (\(\tilde {\mathcal S}',\sigma)\) be the space of generalized measures (distributions), where \(\tau\) and \(\sigma\) are suitably defined topologies. The convolution of two basic measures \(m_ 1,m_ 2\in {\mathcal S}\) is \(m_ 1*m_ 2\) defined by \[ m_ 1*m_ 2(A)=\int_{H}m_ 1(A- x)m_ 2(dx),\quad A\in B. \] The convolution of \(m\in {\mathcal S}\) with generalized measure L is the generalized measure L*m as defined in [\textit{O. G. Smolyanov}, Dokl. Akad. Nauk SSSR 263, 558-562 (1982; Zbl 0534.35087)]. The author proves the Theorem. If T:(\({\mathcal S},\tau)\to (\tilde {\mathcal S}',\sigma)\) is a continuous linear operator, then the following conditions are equivalent: (i) the operator T commutes with convolutions; (ii) there exists a generalized measure \(L\in \tilde {\mathcal S}'\) such that for all \(\mu\in {\mathcal S}\), \(T\mu =L*\mu\).