Classification of $H^2$-functions according to the degree of their cyclicity

The vector-valued functions $f$ in the Hardy space $H^2(E)$ are classified according to their approximation capabilities with respect to the backward shift operator $S^*$, $S^*f\overset{\operatorname{def}}=\frac{f-f(0)}z$, i.e., according to the “size” of the closed linear span $\operatorname{span}(S^{*k}f:k\geqslant0)$.
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