Please consider the following theorem extracted from "Fibre Bundles" by Dale Husemoller:

This theorem is formulated in the context of the Adams operations for K-theory:

Using Dirac notation we can write:

$$ {\Psi}^{k} | \beta_{{2\,m}} \rangle= k^{m} | \beta_{{2\,m}} \rangle $$

it is to say, $| \beta_{{2\,m}} \rangle $ is an eigenvector of eigenvalue $ k^{m} $ for the Adams operations $ {\Psi}^{k}$ on $ \tilde{K}(S^{2m})$ .

A simple formal proof is as follows

$$ {\Psi}^{k} | \beta_{{2\,m}} \rangle= {\Psi}^{k} [ | a_{1} \rangle \otimes | a_{2} \rangle \otimes... \otimes| a_{m} \rangle ]= ( {\Psi}^{k} | a_{1} \rangle )\otimes( {\Psi}^{k} | a_{2} \rangle) \otimes... \otimes| ( {\Psi}^{k}a_{m} \rangle )$$

which is reduced to

$$ {\Psi}^{k} | \beta_{{2\,m}} \rangle= ( k | a_{1} \rangle )\otimes( k| a_{2} \rangle) \otimes... \otimes(| ka_{m} \rangle )=k^{m} [ | a_{1} \rangle \otimes | a_{2} \rangle \otimes... \otimes| a_{m} \rangle ]$$

and then

$$ {\Psi}^{k} | \beta_{{2\,m}} \rangle= k^{m} | \beta_{{2\,m}} \rangle$$

Then, my question is : Do you know any physical application of this theorem? Many thanks.