Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.
For a continuous map $T\colon X\to F$, there exists a closed subspace $W\subseteq H$ with $\dim H/W<\infty$ such that $W\cap\ker T_x=0$ for all $x\in X$ and $H/T(W) =\bigcup_{x\in X} H/T_x(W)$ is a vector bundle over $X$ (See appendix of K-Theory, Anderson & Atiyah). Then one can show that $$\mbox{Ind}_1(T) = [X\times H/W] - [H/T(W)] \in K(X)$$ does not depend on $W$.
On the other hand, there exists a finite dimensional subspace $V\subseteq H$ such that $V+T_x(H) = H$ for all $x\in X$, and define $T^V\colon X\to F(H\oplus V, H)$ by $T^V_x(u,v) = T_x u + v$. Then $T^V_x$ is surjective and $\dim\ker T^V_x$ is constant on $x$. Thus $\ker T^V = \bigcup_{x\in X} \ker T_x$ is also a vector bundle over $X$. One can show that $$ \mbox{Ind}_2(T) = [\ker T^V] - [X\times V] \in K(X)$$ does not depend on $V$.
These index maps are called the family index of families of Fredholm operators in $H$, and it made me suspect that they are equal.
Question: Is it true that $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V]$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?