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Outer preserving linear operators |
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| Authors: | PC Gibson GF Margrave |
| Institution: | a Dept. Math. & Stat., York University, ON, Canada b Dept. Math. & Stat., University of Calgary, AB, Canada c Dept. Geoscience, University of Calgary, AB, Canada |
| Abstract: | A natural question about linear operators on the Hilbert-Hardy space is answered, motivated by work in geophysical imaging. Namely, which bounded linear operators on the Hardy space preserve the set of all shifted outer functions? A complete characterization is determined, which allows an explicit construction of all such operators. Every operator that preserves the set of shifted outer functions is necessarily a product-composition operator, consisting of composition with a shifted outer function followed by multiplication with a (possibly different) shifted outer function. Such operators represent important physical processes, including the propagation of seismic wave energy through the earth. Applications to seismic imaging are briefly discussed. |
| Keywords: | Hardy space Analytic function Outer function Bounded linear operator Composition operator Product-composition operator Semigroup Minimum-phase filter |
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