There are different models of odd $K$-theory. In one case, one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a separable Hilbert space, one can consider the subspace $$\mathcal U_0=\bigl\{\,g\in\mathcal U\bigm|g+\mathrm{id}\text{ is Fredholm}\,\bigr\}\;.$$

In another model, one considers instead the space $\mathcal F$ of selfadjoint Fredholm operators on a Hilbert space that have an infinite number of positive as well as of negative eigenvalues. Both spaces are related by a Cayley transformation $$\mathcal F\owns D\mapsto \frac{D-i}{D+i}\in\mathcal U_0\subset\mathcal U_0\;.$$

Characteristic classes are cohomology classes on these classifying spaces. It is well-known that the cohomology ring is the exterior algebra over $\mathbb Z$ with one generator $\gamma_k$ in each odd degree $2k-1$. There is a de Rham type description for maps $u\colon M\to U$ of these generators of the form $$\gamma_k(u)=c_k\,\mathrm{tr}\bigl((u^{-1}\,du)^{2k-1}\bigr)$$ with appropriate constants $c_k\in\mathbb C$.

On the other hand, for a map $D\colon X\to\mathcal F$, the characteristic classes of $[D]\in K^1(X)$ should be determined entirely by the way that $\mathrm{spec}(D)\cap(-\varepsilon,\epsilon)$ (and maybe the corresponding eigenspaces) behave for some small $\varepsilon>0$.

My question is, how are these descriptions related? In there some book/article that makes characteristic classes of selfadjoint Fredholm operators more explicit and relates them to those of maps to $U$?

There are different models of odd $K$-theory. In one case, one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a separable Hilbert space, one can consider the subspace $$\mathcal U_0=\bigl\{\,g\in\mathcal U\bigm|g+\mathrm{id}\text{ is Fredholm}\,\bigr\}\;.$$

In another model, one considers instead the space $\mathcal F$ of selfadjoint Fredholm operators on a Hilbert space that have an infinite number of positive as well as of negative eigenvalues. Both spaces are related by a Cayley transformation $$\mathcal F\owns D\mapsto \frac{D-i}{D+i}\in\mathcal U_0\subset\mathcal U_0\;.$$

Characteristic classes are cohomology classes on these classifying spaces. It is well-known that the cohomology ring is the exterior algebra over $\mathbb Z$ with one generator $\gamma_k$ in each odd degree $2k-1$. There is a de Rham type description for maps $u\colon M\to U$ of these generators of the form $$\gamma_k(u)=c_k\,\mathrm{tr}\bigl((u^{-1}\,du)^{2k-1}\bigr)$$ with appropriate constants $c_k\in\mathbb C$.

On the other hand, for a map $D\colon X\to\mathcal F$, the characteristic classes of $[D]\in K^1(X)$ should be determined entirely by the way that $\mathrm{spec}(D)\cap(-\varepsilon,\varepsilon)$ (and maybe the corresponding eigenspaces) behave for some small $\varepsilon>0$.

My question is, how are these descriptions related? In there some book/article that makes characteristic classes of selfadjoint Fredholm operators more explicit and relates them to those of maps to $U$?

There are different models of odd $K$-theory. In one case, one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a separable Hilbert space, one can consider the subgroup $$\mathcal U_0=\bigl\{\,g\in\mathcal U\bigm|g+\mathrm{id}\text{ is Fredholm}\,\bigr\}\;.$$

In another model, one considers instead the space $\mathcal F$ of selfadjoint Fredholm operators on a Hilbert space that have an infinite number of positive as well as of negative eigenvalues. Both spaces are related by a Cayley transformation $$\mathcal F\owns D\mapsto \frac{D-i}{D+i}\in\mathcal U_0\subset\mathcal U_0\;.$$

Characteristic classes are cohomology classes on these classifying spaces. It is well-known that the cohomology ring is the exterior algebra over $\mathbb Z$ with one generator $\gamma_k$ in each odd degree $2k-1$. There is a de Rham type description for maps $u\colon M\to U$ of these generators of the form $$\gamma_k(u)=c_k\,\mathrm{tr}\bigl((u^{-1}\,du)^{2k-1}\bigr)$$ with appropriate constants $c_k\in\mathbb C$.

On the other hand, for a map $D\colon X\to\mathcal F$, the characteristic classes of $[D]\in K^1(X)$ should be determined entirely by the way that $\mathrm{spec}(D)\cap(-\varepsilon,\epsilon)$ (and maybe the corresponding eigenspaces) behave for some small $\varepsilon>0$.

My question is, how are these descriptions related? In there some book/article that makes characteristic classes of selfadjoint Fredholm operators more explicit and relates them to those of maps to $U$?

There are different models of odd $K$-theory. In one case, one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a separable Hilbert space, one can consider the subspace $$\mathcal U_0=\bigl\{\,g\in\mathcal U\bigm|g+\mathrm{id}\text{ is Fredholm}\,\bigr\}\;.$$

In another model, one considers instead the space $\mathcal F$ of selfadjoint Fredholm operators on a Hilbert space that have an infinite number of positive as well as of negative eigenvalues. Both spaces are related by a Cayley transformation $$\mathcal F\owns D\mapsto \frac{D-i}{D+i}\in\mathcal U_0\subset\mathcal U_0\;.$$

Characteristic classes are cohomology classes on these classifying spaces. It is well-known that the cohomology ring is the exterior algebra over $\mathbb Z$ with one generator $\gamma_k$ in each odd degree $2k-1$. There is a de Rham type description for maps $u\colon M\to U$ of these generators of the form $$\gamma_k(u)=c_k\,\mathrm{tr}\bigl((u^{-1}\,du)^{2k-1}\bigr)$$ with appropriate constants $c_k\in\mathbb C$.

On the other hand, for a map $D\colon X\to\mathcal F$, the characteristic classes of $[D]\in K^1(X)$ should be determined entirely by the way that $\mathrm{spec}(D)\cap(-\varepsilon,\epsilon)$ (and maybe the corresponding eigenspaces) behave for some small $\varepsilon>0$.

My question is, how are these descriptions related? In there some book/article that makes characteristic classes of selfadjoint Fredholm operators more explicit and relates them to those of maps to $U$?