  We present a generic operator $J$ simply defined as a linear map not
increasing the degree from the vectorial space of polynomial functions into
itself and we address the problem of finding the polynomial sequences that
coincide with the (normalized) $J$-image of themselves. The technique developed
assembles different types of operators and initiates with a transposition of
the problem to the dual space. It is also provided examples where the results
are applied to the case where $J$'s expansion is limited to three terms.
