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On an integral transform by R. S. Phillips. (English) Zbl 1202.44002

Author’s abstract: The properties of a transformation \(f \mapsto \tilde f_h\) by R. S. Phillips [Ann. Math. (2) 59, 325–356 (1954; Zbl 0059.10704)], which transforms an exponentially bounded \(C_0\)-semigroup of operators \(T(t)\) to a Yosida approximation depending on \(h\), are studied. The set of exponentially bounded, continuous functions \(f:[0, \infty [\to E\) with values in a sequentially complete \(L_c\)-embedded space \(E\) is closed under the transformation. It is shown that \((\tilde f_h )^{\tilde{}}_k = \tilde f_{h + k}\) for certain complex \(h\) and \(k\), and that \(f(t) = \lim _{h \to 0^+} \tilde f_h (t)\), where the limit is uniform in \(t\) on compact subsets of the positive real line. If \(f\) is Hölder-continuous at 0, then the limit is uniform on compact subsets of the non-negative real line. Inversion formulas for this transformation as well as for the Laplace transformation are derived. Transforms of certain semigroups of non-linear operators on a subset \(X\) of an \(L_c\)-embedded space are studied through the \(C_0\)-semigroups, which they define by duality on a space of functions on \(X\).

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47D06 One-parameter semigroups and linear evolution equations
44A10 Laplace transform

Citations:

Zbl 0059.10704
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Full Text: DOI

References:

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