invariant subspaces of a direct sum of weighted shifts - Project Euclid
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invariant subspaces of a direct sum of weighted shifts - Project Euclid
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PACIFIC JOURNAL OF MATHEMATICS Vol. 27, No. 3, 1968
INVARIANT SUBSPACES OF A DIRECT SUM OF WEIGHTED SHIFTS ERIC A.
NORDGREN
The invariant subspaces of a direct sum of finitely many copies of the adjoint of a monotone I 2 shift are shown to be spanned by the finite dimensional invariant subspaces that they include. For the case of two copies of such a shift, the invariant subspaces are characterized in terms of a spanning set of vectors, and all infinite dimensional invariant subspaces are shown to be cyclic.
It was shown by Donoghue [2] that if an operator A on H2 is defined by Af(z) = zf(z/2), then A has a lattice of invariant subspaces anti-isomorphic to ω + 1. (This result has been generalized to a wider class of operators by Nikolskii [4].) Crimmins and Rαsenthal [1] have shown that the direct product of two (or even countably many) lattices of invariant subspaces is attainable as the lattice of invariant subspaces of some operator. If B and C are operators on a separable Hubert space such that their spectra are disjoint and no part of the spectrum of one is surrounded by spectrum of the other, then the lattice of invariant subspaces of B φ C is the direct product of the lattice of B by that of C. Thus, for example, their result gives the lattice of (A + 1) φ A. This prompts the question: What is the lattice of A φ A? One answer is given by Nikolskii [5] in terms of operators in the commutant of A φ A. Actually, his results are valid for any operator that is a direct sum of a finite number of copies of a monotone lp shift (see [3], p. 97). Adjoints of such operators will be studied in this paper, and the following results will be derived. The invariant subspaces of such an adjoint are spanned by the finite dimensional invariant subspaces that they include, and these are invariant subspaces of finite dimensional nilpotent operators (Theorem 1 and Theorem 2, Corollary 2). The infinite dimensional invariant subspaces are cyclic except possibly for a finite dimensional summand (Theorem 2, Corollary 3 and Theorem 3, Corollary 3). For a sum of two copies of the adjoint of a monotone I 2 shift, the invariant subspaces can be completely characterized in terms of a spanning set of vectors (Theorem 1, Corollary 1). We begin by establishing some notation. Although the natural setting for discussing shift operators is a sequence space, it will be somewhat more convenient to deal with functions on the unit circle X in the complex plane C Let ^ be a finite dimensional Hubert 587
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ERIC A. NORDGREN
space, μ normalized Lebesgue measure on X, and ξ> the Hubert space of measurable norm square integrable functions from I to ^ that are analytic. Thus, to say ί 7 is in § means F is a measurable function from X to
