Ideal properties and integral extension of convolution operators on L^\infty (G)


Abstract


We investigate operator ideal properties of convolution operators C_\lambda (via measures \lambda) acting in {L^\infty (G)}, with G a compact abelian group. Of interest is when C_\lambda is compact, as this corresponds to \lambda having an integrable density relative to Haar measure \mu, i.e., \lambda \ll \mu . Precisely then is there an \textit{optimal} Banach function space L^1 (m_\lambda) available which contains {L^\infty (G)} properly, densely and continuously and such that C_\lambda has a continuous, {L^\infty (G)}-valued, linear extension I_{m_\lambda} to L^1 (m_\lambda). A detailed study is made of L^1 (m_\lambda) and I_{m_\lambda}. Amongst other things, it is shown that C_\lambda is compact iff the finitely additive, {L^\infty (G)}-valued set function m_\lambda (A) := C_\lambda ({\chi_{_{_{\scriptstyle{A}}}}}) is norm \sigma-additive iff \lambda \in L^1 (G), whereas the corresponding optimal extension I_{m_\lambda} is compact iff \lambda \in C (G) iff m_\lambda has finite variation. We also characterize when m_\lambda admits a Bochner (resp.\ Pettis) \mu-integrable, L^{\infty} (G)-valued density.

DOI Code: 10.1285/i15900932v31n1p149

Keywords: Convolution operator ; vector measure ; optimal domain ; Bochner-Pettis density

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