Let A be a weak-*Dirichlet
algebra of L∞(m). For 0 < p ≦∞, a closed subspace M of Lp(m) is called invariant
if f ∈ M and g ∈ A imply that fg ∈ M. Let B∞ be a weak-*closed subalgebra of
L∞(m) which contains A such that B∞M ⊆ M for an invariant subspace M. The
main result of this paper is a characterization of the left continuous invariant
subspaces for B∞, which is a natural generalization of simply invariant subspaces.
Applying this result with B∞= H∞(m) (or B∞= L∞(m)), the simply (or
doubly) invariant subspace theorem follows. Moreover this result characterizes
also the invariant subspaces which are neither simply nor doubly invariant.
Merrill and Lal characterized some special invariant subspaces of this kind.
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