What is a spectral projection?

Question

Let $T$ be an operator on some Hilbert space $H$, say $T\colon D(T)\subset H\to H$. How do we define a spectral projection in general?

Answer 1

I am not sure there is a universally accepted notion of spectral projection that applies in the generality asked by the OP.

For the case of a self-adjoint operator, most people would call a spectral projection any one of the projections $P(E)$, where $P$ is the spectral measure mentioned in the spectral Theorem, and $E$ is a Borel subset of $\sigma(T)$.

Very briefly, $P$ is map defined on $\mathscr B(\sigma(T))$ (Borel subsets of the spectrum of $T$), taking values in the set of all projections on $H$, such that $$T=\int_{\sigma(T)}\lambda \, dP(\lambda).$$

Most Functional Analysis books will have some version of the spectral Theorem. I highly recommend you study it!