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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 $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V] \qquad (1)$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?

Edit: We can shrink $W$ or augment $V$ in order to have $\dim H/W = \dim V$. Say $H/W \cong V \cong \mathbb{C}^N$, so that $X\times H/W \cong X\times V \cong X\times\mathbb{C}^N$, and therefore $$\mbox{Ind}_1(T) = [X\times\mathbb{C}^N] - [H/T(W)]\ ,$$ $$\mbox{Ind}_2(T) = [\ker T^V] - [X\times\mathbb{C}^N]\ .$$

Equation $(1)$ becomes $$[X\times\mathbb{C}^N] - [H/T(W)] = [\ker T^V] - [X\times\mathbb{C}^N]$$ and it holds iff there exists $k\geq0$ such that

$$\ker T^V \oplus H/T(W) \oplus (X\times\mathbb{C}^k) \cong X\times\mathbb{C}^{2N+k}$$ But why does there exists such $k$?

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 $x\in X$, so we can 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 $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V] \qquad (1)$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?

Edit: We can shrink $W$ or augment $V$ in order to have $\dim H/W = \dim V$. Say $H/W \cong V \cong \mathbb{C}^N$, so that $X\times H/W \cong X\times V \cong X\times\mathbb{C}^N$, and therefore $$\mbox{Ind}_1(T) = [X\times\mathbb{C}^N] - [H/T(W)]\ ,$$ $$\mbox{Ind}_2(T) = [\ker T^V] - [X\times\mathbb{C}^N]\ .$$

Equation $(1)$ becomes $$[X\times\mathbb{C}^N] - [H/T(W)] = [\ker T^V] - [X\times\mathbb{C}^N]$$ and it holds iff there exists $k\geq0$ such that

$$\ker T^V \oplus H/T(W) \oplus (X\times\mathbb{C}^k) \cong X\times\mathbb{C}^{2N+k}$$ But why does there exists such $k$?

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 $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V] \qquad (1)$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?

Edit: We can shrink $W$ or augment $V$ in order to have $\dim H/W = \dim V$. Say $H/W \cong V \cong \mathbb{C}^N$, so that $X\times H/W \cong X\times V \cong X\times\mathbb{C}^N$, and therefore $$\mbox{Ind}_1(T) = [X\times\mathbb{C}^N] - [H/T(W)]\ ,$$ $$\mbox{Ind}_2(T) = [\ker T^V] - [X\times\mathbb{C}^N]\ .$$

Equation $(1)$ becomes $$[X\times\mathbb{C}^N] - [H/T(W)] = [\ker T^V] - [X\times\mathbb{C}^N]$$ and it holds iff there exists $k\geq0$ such that

$$\ker T^V \oplus H/T(W) \oplus (X\times\mathbb{C}^k) \cong X\times\mathbb{C}^{2N+k} \qquad (2)$$ But why does this last statement hold?

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 $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V] \qquad (1)$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?

Edit: We can shrink $W$ or augment $V$ in order to have $\dim H/W = \dim V$. Say $H/W \cong V \cong \mathbb{C}^N$, so that $X\times H/W \cong X\times V \cong X\times\mathbb{C}^N$, and therefore $$\mbox{Ind}_1(T) = [X\times\mathbb{C}^N] - [H/T(W)]\ ,$$ $$\mbox{Ind}_2(T) = [\ker T^V] - [X\times\mathbb{C}^N]\ .$$

Equation $(1)$ becomes $$[X\times\mathbb{C}^N] - [H/T(W)] = [\ker T^V] - [X\times\mathbb{C}^N]$$ and it holds iff there exists $k\geq0$ such that

$$\ker T^V \oplus H/T(W) \oplus (X\times\mathbb{C}^k) \cong X\times\mathbb{C}^{2N+k}$$ But why does there exists such $k$?

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?

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 $$[X\times H/W] - [H/T(W)] = [\ker T^V] - [X\times V] \qquad (1)$$ in $K(X)$ ? Is there any reference that proves the equivalence of these indexes?

Edit: We can shrink $W$ or augment $V$ in order to have $\dim H/W = \dim V$. Say $H/W \cong V \cong \mathbb{C}^N$, so that $X\times H/W \cong X\times V \cong X\times\mathbb{C}^N$, and therefore $$\mbox{Ind}_1(T) = [X\times\mathbb{C}^N] - [H/T(W)]\ ,$$ $$\mbox{Ind}_2(T) = [\ker T^V] - [X\times\mathbb{C}^N]\ .$$

Equation $(1)$ becomes $$[X\times\mathbb{C}^N] - [H/T(W)] = [\ker T^V] - [X\times\mathbb{C}^N]$$ and it holds iff there exists $k\geq0$ such that

$$\ker T^V \oplus H/T(W) \oplus (X\times\mathbb{C}^k) \cong X\times\mathbb{C}^{2N+k} \qquad (2)$$ But why does this last statement hold?