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Transformation semigroups of the space of functions that are square integrable with respect to a translation-invariant measure on a Banach space. (English. Russian original) Zbl 1460.28010

J. Math. Sci., New York 252, No. 1, 72-89 (2021); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 151, 73-90 (2018).
In the paper under review, translation-invariant finitely additive measures on some Banach spaces are constructed and studied. In particular, for the Banach space \(\ell^p\) of real \(p\)-summable sequences, where \(p\in[1,\infty]\), one constructs a nonnegative, locally finite, complete, translation invariant, finitely additive measure \(\lambda_p\) on a suitable ring, which is not \(\sigma\)-finite and moreover is countably additive if and only if \(p=\infty\). Moreover, one studies the Hilbert spaces of equivalence classes of square-integrable functions with respect to these measures and also the one-parameter unitary groups of translation operators in these Hilbert spaces. Interestingly, it is pointed out that all these one-parameter unitary groups are strongly continuous only in the case \(p=1\). One actually establishes a necessary and sufficient criterion on a vector in order that its corresponding one-parameter translation group be continuous, and one studies their infinitesimal generators. Rotation-invariant measures are then discussed and finally, one studies mathematical expectations of operators defined by translation along random vectors by a one-parameter convolution semigroup of Gaussian measures.

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
47D08 Schrödinger and Feynman-Kac semigroups
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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