the spectral family of the operator $T:\ell^2\rightarrow\ell^2$ defined by $$T(\xi_1,\xi_2,\xi_3,....)=(\xi_1/1,\xi_2/2,\xi_3/3,....)$$
I am trying to get the application of Spectral theorem of bounded self-adjoint linear operator. All I know is that the set of eigenvalues is $\{1/n:n\in\mathbb Z_+\}$ and corresponding eigenvectors are $\{e_n:n\in\mathbb Z_+\}$ having $1$ at $n^{th}$ position and zero otherwise.
Solution:
$T_{\lambda}(\xi_1,\xi_2,\xi_3,....)=T(\xi_1,\xi_2,\xi_3,....)-\lambda(\xi_1,\xi_2,\xi_3,....)=(\xi_1/1,\xi_2/2,\xi_3/3,....)-(\lambda\xi_1,\lambda\xi_2,\lambda\xi_3,....)=(\xi_1(1/1-\lambda),\xi_2(1/2-\lambda),\xi_3(1/3-\lambda),....)$
Hence,$T_{\lambda}^2(\xi_1,\xi_2,\xi_3,....)=T_{\lambda}(\xi_1(1/1-\lambda),\xi_2(1/2-\lambda),\xi_3(1/3-\lambda),....)=(\xi_1(1/1-\lambda)^2,\xi_2(1/2-\lambda)^2,\xi_3(1/3-\lambda)^2,..)=(1/1-\lambda)^2,(1/2-\lambda)^2,(1/3-\lambda)^2,...)(\xi_1,\xi_2,\xi_3,....).$
So,$T_\lambda^2=((1/1-\lambda)^2,(1/2-\lambda)^2,(1/3-\lambda)^2,...)\implies (T_\lambda^2)^{1/2}=((1/1-\lambda),(1/2-\lambda),(1/3-\lambda),...)$
Now,$T_{\lambda}^+=\frac{1}{2}((T_\lambda+(T_\lambda^2)^{1/2})=\frac{1}{2}(2(1/1-\lambda),2(1/2-\lambda),2(1/3-\lambda),..)=((1/1-\lambda),(1/2-\lambda),(1/3-\lambda),...)$
Since, the eigenvalues are $1,1/2,1/3,1/4,...$
Computing the Null spaces
$\mathcal N(T_{1}^+)=(\xi_1,0,0,0,...)$
$\mathcal N(T_{1/2}^+)=(0,\xi_2,0,0,...)$
$\mathcal N(T_{1/3}^+)=(0,0,\xi_3,0,...)$ and so on
Computing the projections $E_\lambda=\mathcal H\rightarrow\ \mathcal N(T_{\lambda}^+)$,we get
$E_1(\xi_1,\xi_2,\xi_3,...)=(\xi_1,0,0,0,...)$
$E_{1/2}(\xi_1,\xi_2,\xi_3,...)=(0,\xi_2,0,0,...)$
$E_{1/3}(\xi_1,\xi_2,\xi_3,...)=(0,0,\xi_3,0,0,0,...)$ and so on is the spectral family.
above I've found the Spectral family of $T$.I want to know How can T be represented in the form as given in the following theorem?
Spectral theorem for Bounded Self-Adjoint Linear operators
Let $T:\mathcal H\rightarrow \mathcal H$ be a bounded self-adjoint linear operator on a complex Hilbert space $\mathcal H$.Then $T$ has the spectral representation $$T=\int_{m-0}^M\lambda dE_{\lambda}$$ Where $\mathcal E=(E_\lambda)$ is the spectral family associated with $T,$the integral is to be understood in the sense of uniform operator convergence,and for all $x,y\in \mathcal H$,$$\langle Tx,y\rangle=\int_{m-0}^M\lambda dw(\lambda)$$
Where the integral is an ordinary Riemann Stieltjes integral.