Check the spectral theorem for this matrix

[LCSphysicist]

Homework Statement

.

Relevant Equations

.

I found three projection operators
$$P_{1}=
\begin{pmatrix}
1/2 & & \\
& -\sqrt{2}/2 & \\
& & 1/2
\end{pmatrix}$$
$$P_{2}=
\begin{pmatrix}
1/2 & & \\
& \sqrt{2}/2 & \\
& & 1/2
\end{pmatrix}$$
$$P_{3}=
\begin{pmatrix}
-1/\sqrt{2} & & \\
& & \\
& & 1/\sqrt{2}
\end{pmatrix}$$

From this five properties

, i am having trouble to prove the ii, iii, and iv. I mean, i could understand this conditions in the continuous situation, in which we can make the "eigenvalue tends to" something. But how to interpret it when we are dealing with discrete conditions? In which we have really just three eigenvalues.

 

[Office_Shredder]

What is the definition of E in those statements?

 

[Keith_McClary]

$E(\lambda)$ is an projection (operator) valued function. In the case of discrete eigenvalues it is a step function which has a step at each eigenvalue . $dE(\lambda)/d\lambda$ has a $\delta$ function spike at each eigenvalue, which gives you the discrete eigenvalue when you integrate.