= Solution
The kernel and cokernel of an equivariant <Fredholm operator> are finite-dimensional representations. Define its <equivariant Fredholm index> in the <representation ring of a compact group> by
$$
\boxed{\operatorname{ind}_G S=[\ker S]-[\operatorname{coker}S].}
$$
For $G=SU(2)$ write $H=\widehat\bigoplus_n E_n\otimes\mathbb C^{m_n}$ and $H'=\widehat\bigoplus_n E_n\otimes\mathbb C^{m'_n}$, with finite multiplicities. By <Schur lemma> an intertwiner has blocks $I_{E_n}\otimes s_n$, where $s_n:\mathbb C^{m_n}\to\mathbb C^{m'_n}$. If its rank is $r_n$, the kernel and cokernel multiplicities are $m_n-r_n$ and $m'_n-r_n$. Consequently
$$
\boxed{\operatorname{ind}_G S=\sum_n(m_n-m'_n)[E_n].}
$$
Only finitely many coefficients can be nonzero, because the global kernel and cokernel are finite-dimensional. Closed range is part of the <Fredholm> hypothesis and supplies the uniform lower bound on the remaining blocks; multiplicity matching alone would not establish that hypothesis. The ordinary index is obtained by taking dimensions, $\operatorname{ind}S=\sum_n(n+1)(m_n-m'_n)$. This calculation applies to any compact group with its finite-dimensional irreducibles in place of the $E_n$.
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