Abstract
We prove that all compact operators acting on \( {L^p}(\mathbb{R}) \) belong to the algebra generated by the operator of multiplication by the characteristic function of the positive half-axis and by the convolution operators with continuous generating function. This result, together with the similar classical result on the algebra generated by the operators of multiplication and the singular integral operator, is then used to prove that certain ideals of compact-like operator sequences in infinite products of Banach algebras are included in the algebra generated by convolution and multiplication operators and the finite section projection sequence.
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