Approximate representation of Bergman submodules

Abstract

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L _a ²(B _d) satisfies ${P_M} - \sum\limits_i {{M_{{z^i}}}} {P_M}M_{{z^i}}^* \leqslant \frac{c}{{N + 1}}{P_M}$ for some number c > 0, then there is a sequence {f _j} of multipliers and a positive number c′ such that $c'{P_M} \leqslant \sum\limits_j {{M_{{f_j}}}} M_{{f_j}}^* \leqslant {P_M}$, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

Keywords

Approximate representation / Essential normality / Bergman submodule

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