Vol. 48, No. 2, 1973

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Invariant subspaces, similarity and isometric equivalence of certain commuting operators in Lp

Robert E. Waterman

Vol. 48 (1973), No. 2, 593–613

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.

Mathematical Subject Classification 2000
Primary: 47A15
Secondary: 47B37
Received: 15 June 1972
Published: 1 October 1973
Robert E. Waterman