Vol. 24, No. 2, 1968

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Convolution operators on Lp(G) and properties of locally compact groups

John Eric Gilbert

Vol. 24 (1968), No. 2, 257–268

A locally compact group G is said to have property (R) if every continuous positive-definite function on G can be approximated uniformly on compact sets by functions of the form s s,s ∈𝒦(G). When μ is a bounded, regular, Borel measure on G, the convolution operator Tμ defined by

(T  )(s) = (μ∗ s)(x) =  s(y−1x)dI(y),s ∈ 𝒦(G),
μ                  G        l

can be extended to a bounded operator on Lp(G) whose norm satisfies Tμp μ. In this paper three characterizations of property (R) are given in terms of the norm Tμp, 1 < p < , for specific operators Tμ. From these characterizations some closely-related, but seemingly weaker properties than (R), are shown to be equivalent to (R). Examples illustrating the results are given also.

Mathematical Subject Classification
Primary: 22.65
Received: 31 January 1967
Published: 1 February 1968
John Eric Gilbert