We discuss a certain geometric property X$\_{ of dual algebras generated by a polynomially bounded operator and property ($mathbb{A}$, $aleph$, =$aleph$) these are central to the study of $aleph$$\_{0}$$ imes$$aleph$$\_{0}$-systems of simultaneous equations of weak$^{*}$-continuous linear functionals on a dual algebra. In particular, we prove that if T $in$ $mathbb{A}$$^{M}$ satisfies a certain sequential property, then T $in$ $mathbb{A}$$^{M}$$\_{(H) if and only if the algebra A$\_{T}$ has property X$\_{0, 1/M}$, which is an improvement of Li-Pearcy theorem in [8].