Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 121-130

# FOURIER MULTIPLIERS FOR L^{2} 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 L^{2} 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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