Why Is the Spectral Family Defined This Way in Functional Analysis?

[thegreenlaser]

I've been reading Kreyszig's functional analysis book, and I'm a little confused why he defines the spectral family of a self-adjoint operator the way he does. For an operator $T$ he defines $T_{\lambda} = T - \lambda I$. Then he defines $T_{\lambda}^+ = 1/2\left(\left(T_{\lambda}^2\right)^{1/2} + T_{\lambda}\right)$. Finally, he defines the spectral family $E_{\lambda}$ for $\lambda \in \mathbb{R}$ so that $E_{\lambda}$ projects onto the null space of $T_{\lambda}^+$.

I realize that the definition works, but what motivates it? In the finite dimensional case, $E_{\lambda}$ was basically defined as projection onto all the eigenspaces corresponding to eigenvalues less than $\lambda$. Presumably this definition is some sort of generalization of the finite dimensional case, but I'm having a tough time seeing how.

 

[micromass]

Take a diagonal hermitian matrix and work out the different concepts for that. Or take a diagonal operator on an infinite dimensional space. This is just an operator of the form

$$T(x) = \sum_{i\in I} \lambda_i <x,e_i>e_i$$

where $e_i$ are an orthonormal basis of the (let's take a separable) Hilbert space. This is a generalization of diagonal matrices. Work out the different concepts for that. You will see that they agree with your intuition. That is already one reason to define the concepts like this.