the case where $\sigma$ is a cycle of consecutive letters. Finally, we prove that in this special case, $\bft_\sigma$ is in the image of the *Gelfand elements* $\mathrm{str}(\mathbf E^d)\in\bfZ(\gl(m|n))$ defined in Lemma \[gelfzhel\].

The idea behind the proof of Theorem \[MAINTHM\] is as follows. Let $D_\lambda$ be a Capelli operator, and let $z_\lambda\in\bfZ(\gl(m|2n))$ be an element in the inverse image of $D_\lambda$, whose existence follows from Theorem \[prpgw\]. We show that modulo the natural isomorphism induced by the trace form, the spherical polynomial $d_\lambda$ is the diagonal restriction of the symbol of the polynomial-coefficient differential operator corresponding to the radial part of $z_\lambda$. Furthermore, we show that the eigenvalue polynomial $c_\lambda$ is the image of the radial part of $z_\lambda$ (see Lemma \[cmulZa\]). We combine the latter two statements, as well as the refinement of the abstract Capelli problem obtained in Theorem \[prpgw\], to prove Theorem \[MAINTHM\].

Finally, the proof of Theorem \[thmconnSV\] goes as follows. By considering the action of $z_\lambda$ on the lowest weight vector of an irreducible $\gl(m|2n)$-module $V_\mu$, we prove in Proposition \[prpQlam\] that $c_\lambda(\mu)$ is a polynomial in the highest weight $\mu^*$ of the contragredient representation $V_\mu^*$. Denoting the latter polynomial $c_\lambda^*$, we verify that 