Remarks on j−$j-$eigenfunctions of operators

The paper is largely concerned with the possibility of obtaining a series representation for a compact linear map T acting between Banach spaces. It is known that, using the notions of $j-$eigenfunctions and $j-$ eigenvalues, such a representation is possible under certain conditions on T. Particular cases discussed include those in which T can be factorized through a Hilbert space, or has certain s-numbers that are fast-decaying. The notion of p-compactness proves to be useful in this context; we give examples of maps that possess this property.