Let Lℋ2 denote the Hilbert
space of weakly measurable functions on the unit circle of the complex plane with
values in a separable Hilbert space ℋ, and whose pointwise norms are square
integrable with respect to Lebesgue measure. We are concerned with invariant
subspaces of Lℋ2 , by which we mean closed subspaces invariant under the right
shift operator, and will be especially interested in those invariant subspaces
which arise from a bounded operator on ℋ, using a construction due to Rota
and Lowdenslager. We begin by relating the determinant of the “Rota inner
function” of an operator to the characteristic polynomial of the operator, along
with a similar interpretation of the minimal polynomial, when ℋ is finite
dimensional. We then consider some general questions about intersections and
unions of invariant subspaces, and use the results to establish a factorization
theorem for finite dimensional inner functions (the set of all 𝒰∗𝒱, where 𝒰,𝒱
are inner, is the same as the set of all 𝒰𝒱∗). We show this theorem false if
ℋ is infinite dimensional, by exhibiting invariant subspaces ℳ,𝒩 (which
are also Rota subspaces) such that ℳ∩𝒩 = (0)-a result of independent
interest.
