Vol. 22, No. 1, 1967

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Operators and inner functions

Malcolm Jay Sherman

Vol. 22 (1967), No. 1, 159–170

Let L2 denote the Hilbert space of weakly measurable functions on the unit circle of the complex plane with values in a separable Hilbert space , and whose pointwise norms are square integrable with respect to Lebesgue measure. We are concerned with invariant subspaces of L2 , 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 , 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 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 is infinite dimensional, by exhibiting invariant subspaces ,𝒩 (which are also Rota subspaces) such that ℳ∩𝒩 = (0)-a result of independent interest.

Mathematical Subject Classification
Primary: 47.35
Received: 4 April 1965
Published: 1 July 1967
Malcolm Jay Sherman