Annales Academię Scientiarum Fennicę
Mathematica
Volumen 34, 2009, 109-129
Helsinki University of Technology, Institute of Mathematics
P.O. Box 1100, FI-02015 HUT, Finland; Kalle.Mikkola 'at' iki.fi
Abstract. We show that every operator that acts between two nonseparable Hilbert spaces can be ``block diagonalized'', where each diagonal block acts between two separable Hilbert spaces. Analogous results hold for operator-valued H\infty functions and others. Using these results, several theorems about representation, invertibility, factorization etc., which have previously been known only for separable Hilbert spaces, can now be generalized to arbitrary Hilbert spaces. We generalize several results often needed in systems and control theory, including the Lax-Halmos Theorem, Tolokonnikov's Lemma and the inner-outer factorization. We present our results both for the unit disc and for the half-plane.
2000 Mathematics Subject Classification: Primary 47B35, 46C99, 46E40.
Key words: Orthogonal subspaces, strong Hardy spaces of operator-valued functions, strongly essentially bounded functions, shift-invariant subspaces, translation-invariant operators, inner functions, left invertibility.
Reference to this article: K.M. Mikkola: Hankel and Toeplitz operators on nonseparable Hilbert spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 109-129.
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