determined uniquely by its radial component follows from the $KAK$ decomposition, which does not have an analogue in the context of supergroups. These issues complicate the formulation and proof of Theorem \[prpgw\] and Theorem \[MAINTHM\].

To achieve our goal in this paper, we have to overcome the above difficulties. Our first main result is Theorem \[prpgw\], where we give an affirmative answer to the abstract Capelli problem of Howe and Umeda for the Lie superalgebra $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$. In fact in Theorem \[prpgw\] we prove a slightly more precise statement that a $\gl(m|n)$-invariant differential operator of order $d$ is in the image of an element of $\bfZ(\gl(m|n))$ which has order $d$ with respect to the standard filtration of $\bfU(\gl(m|n))$. The latter refinement is needed in the proofs of Theorem \[MAINTHM\] and Theorem \[thmconnSV\], which we will elaborate on below. For the superpair $(\gl(m|n)\times\gl(m|n),\gl(m|n))$, the corresponding abstract Capelli theorem indeed follows directly from the work of Molev [@Molev], as explained in Appendix \[appxB\] (see Theorem \[thabsscap\]).

Our second main goal concerns the Capelli eigenvalue problem for the Lie superalgebra $\gl(m|2n)$ acting on $\sS^2(\C^{m|2n})$. Extending [@Sahi Theorem 1], we show that the eigenvalues of the Capelli operators are given by polynomials 