This paper is concerned
with the problem of finding all closed invariant subspaces of operators of the form
Tf= Mf−JMfl and the determination of the similarity relationships between such
operators. The operator Tf is defined, for suitable conditions on a complex-valued
function f and its derivative f′, by Tfg(x) = f(x)g(x) −∫0xf′(t)g(t)dt for all g in
Lp. The main result asserts that the closed invariant subspaces of Tf are
precisely those subspaces that are generated by the eigenfunctions of Tf.
Conversely, any operator on Lp whose closed invariant subspaces coincide
with those of M − J (i.e., Tf where f(x) ≡ ixj) must be of the form Tg
for some function g. The closed invariant subspaces of Tf are cyclic and
the generating functions have a rather simple description. The algebra of
operators 𝒞 , generated by M − J, is maximal abelian. A corollary is that 𝒞 is
reflexive. It is shown that Tf and Tg are isometrically equivalent in Lp if
and only if f = g. Finally conditions are given for the similarity of Tf and
Tg.
