We discuss a direct and very transparent connection between the poly‐Bergman type spaces on the upper half‐plane Π and the action of the following two‐dimensional singular integral operators
(SΠφ)(z) = −1π ∫ Πφ(zgr;)dv(ζ)(ζ−z)2 and (SΠ*φ)(z) = −1π ∫ Πφ(zgr;)dv(ζ)(ζ̄−z̄)2,
where dv is the standard Lebesgue plane measure.

We show that both SΠ and SΠ* admit a simple (functional) model: each of them is unitary equivalent to the direct sum of two unilateral shifts, forward and backward, both taken with infinite multiplicity.

This fact leads to an easy access to the majority of properties of these operators.

Topics
General topology
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© 2011 American Institute of Physics.
2011
American Institute of Physics
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