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-$j-$eigenfunctions and j-$j-$ 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.

Automated summary

The paper investigates the possibility of obtaining a series representation for a compact linear map T acting between Banach spaces. It is shown that this representation is possible under certain conditions on T, using the concepts of j-j-eigenfunctions and j-j-eigenvalues. Various specific cases are discussed, including T being factorized through a Hilbert space or having rapidly decaying s-numbers. The concept of p-compactness is found to be useful in this context, and examples of maps possessing this property are provided.