Annales Academię Scientiarum Fennicę
Mathematica
Volumen 33, 2008, 121-130

FOURIER MULTIPLIERS FOR L² FUNCTIONS WITH VALUES IN NONSEPARABLE HILBERT SPACES AND OPERATOR-VALUED H^p BOUNDARY FUNCTIONS

Kalle M. Mikkola

Helsinki University of Technology, Institute of Mathematics
P.O. Box 1100, FI-02015 HUT, Finland; Kalle.Mikkola 'at' hut.fi

Abstract. We extend the standard Fourier multiplier result to square integrable functions with values in (possibly nonseparable) Hilbert spaces. As a corollary, we extend the standard Hardy class boundary trace result to H^p (even Nevanlinna or bounded type) functions whose values are bounded linear operators between Hilbert spaces. Both results have been well-known in the case that the Hilbert spaces are separable. Naturally, the results apply to functions over the unit circle/disc or over the real-line/half-plane or over other similar domains, even multidimensional in the case of the multiplier result. We briefly treat some related results, generalizations to Banach spaces and counter-examples.

2000 Mathematics Subject Classification: Primary 42B15, 46E40, 42B30; Secondary 28B05, 47B35.

Key words: Fourier multipliers, translation-invariant operators, time-invariant, Toeplitz operators, Hardy spaces of operator-valued functions, vector-valued functions, strongly measurable functions, nontangential limits, boundary trace.

Reference to this article: K.M. Mikkola: Fourier multipliers for L² functions with values in nonseparable Hilbert spaces and operator valued H^p boundary values. Ann. Acad. Sci. Fenn. Math. 33 (2008), 121-130.

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