Confusion on Theorem in Kato's book

Question

On page 432 (pdf-page: 455) of Kato's book perturbation theory of linear operators, I do not understand why in Theorem 1.15 $$H_n = \int dE_n(\lambda)$$

instead of the ususal thing

$$H_n=\int \lambda dE_n(\lambda).$$

Is this a misprint?

Answer 1

Looks like a misprint, as $\int dE_n(\lambda) = {\rm Id}$, and this would imply $H_n = {\rm Id}$ for all $n$. Just writing down the spectral decomposition of $H_n$ gives $H_n = \int \lambda\, dE_n(\lambda)$, as you write.