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}^+##.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Spectral Family Definition

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