Vol. 94, No. 2, 1981

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Invariant subspace lattices for a class of operators

Boon-Hua Ong

Vol. 94 (1981), No. 2, 385–405

We study the invariant subspace lattices for a one parameter family of operators {Tα}α on Lp(0,1), α a complex number, where

Tαf(x) = xf(x)+ α 0 f(t)dt,

and their adjoints Tα,

                 ∫ 1
T∗αf(x) = xf(x)+ α   f(t)dt.

The closed invariant subspaces for Tα are in one-to-one correspondence with certain closed ideals of α, where α is a Silov algebra with unit and in which the range α of the Riemann Liouville operator Jα

           1  ∫ x
(Jαf(x) =----    (x − t)α−1f(t)dt)
Γ (α)  0

is embedded as a closed ideal. When n is a positive integer, there is a complete lattice isomorphism between the closed ideals of n and the n-tuples (E0,E1,,En1) of closed subsets of [0,1] where E0 E1 En1 derived set of E0. Every closed ideal of n is the intersection of closed primary ideals. Similar results carry over to α where the real part of α is an integer and also to the adjoint operators.

Mathematical Subject Classification 2000
Primary: 47A15
Secondary: 46J10
Received: 9 August 1980
Published: 1 June 1981
Boon-Hua Ong