Summary: SIMILARITY OF NESTS IN Lp, 1 p 6= 2 < 1
A. ARIAS* and P. MULLER
ABSTRACT. In this note we prove that some aspect of the similarity theory for the
Volterra nest in Lp(0 1) for 1 < p 6= 2 < 1 is like that for p = 1 we thus answer a
question from ALWW].
A nest N in a Banach space X is a totally ordered family of closed subspaces closed
under intersection and closed unions and containing 0 and X. The nest algebra induced
by N is the set of all T 2 B(X) that leave invariant every element of N i.e., TN N for
every N 2 N.
The main example for our purpose is the Volterra nest in Lp(0 1), 1 p < 1 where
N = fNt : 0 t 1g and Nt is the set of those functions f 2 Lp(0 1) that have support
contained in 0 t]. For p = 2 the similaritytheory tells us that this is the canonical example
of a continuous nest (see L]).
Another consequence of the similarity theory (see D] and L]) says the following: If
: 0 1] ! 0 1] is strictly increasing and onto, then there exists T 2 B(L2(0 1)) invertible
such that for every t 2 0 1], TNt = N (t).
The last question was considered by Allen, Larson, Ward and Woodward ( ALWW])
for p = 1 where they proved that such a T exists if and only if both and ;1 are