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Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral Dirac operator phase. The compressed map has parametrix $P{V}^{\ast }P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their homogeneous line-bundle representations on the two-sphere are respectively $E_0\oplus E_2\oplus E_4\oplus \cdots$ and $E_2\oplus E_4\oplus \cdots$, so the equivariant Fredholm index is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.

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 byFor $G=SU(2)$ write $H={\widehat{⨁}}_n{E}_n\otimes {\mathbb{C}}^{m_n}$ and $H^{\prime }={\widehat{⨁}}_n{E}_n\otimes {\mathbb{C}}^{m_n^{\prime }}$, with finite multiplicities. By Schur lemma an intertwiner has blocks $I_{E_n}\otimes s_n$, where $s_n:{\mathbb{C}}^{m_n}\to {\mathbb{C}}^{m_n^{\prime }}$. If its rank is $r_n$, the kernel and cokernel multiplicities are $m_n-r_n$ and $m_n^{\prime }-r_n$. ConsequentlyOnly 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, $\mathrm{ind}S={\sum }_n(n+1)(m_n-m_n^{\prime })$. This calculation applies to any compact group with its finite-dimensional irreducibles in place of the $E_n$.