Spectral Family:A real spectral family (or real decomposition of unity) is a one parameter family $\mathcal E=(E_{\lambda})_{\lambda\in \mathbb R}$ of projections $E_{\lambda}$ defined on a hilbert space $\mathcal H$ (of any dimension) which depends on a real parameter $\lambda$ and is such that
- $E_{\lambda}E_{\mu}=E_{\mu}E_{\lambda}=E_{\lambda}$[it is given that $\lambda<\mu$]
- $\lim_{\lambda\rightarrow-\infty}E_{\lambda}x=0$
- $\lim_{\lambda\rightarrow\infty}E_{\lambda}x=x$
- $E_{\lambda+0}x=\lim_{\mu\rightarrow\lambda+0}E_{\mu}x=E_\lambda x$
If an operator $T:\mathbb R^3\rightarrow \mathbb R^3$ is represented with respect to an orthonormal basis,by a matrix $$ \left[ \begin{array}{cc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array} \right] $$ What is the corresponding spectral family?
Since the spectral family includes eigenvalues so I computed the eigenvalues of the above matrix which are $\lambda=+1,+1,-1$
I'm not getting how to get the projections corresponding to the eigenvalues{1,1,-1}
need help in getting the spectral family
thank you!