Spectral projection of a self-adjoint or normal operator

Definition

Let $T$ be a self-adjoint or normal operator with spectral measure $E_T$ , and let $B$ be a Borel subset of $\mathbb R$ in the self-adjoint case or $\mathbb C$ in the normal case. The spectral projection of $T$ for $B$ is

E_T(B)=1_B(T),

where $1_B(T)$ is defined by the Borel functional calculus. It is an orthogonal projection. Its range is the spectral subspace on which $T$ has spectrum concentrated in the closure of $B$ , and it reduces $T$ , including the domain when $T$ is unbounded.

Projection-valued measure laws

The assignment $B\mapsto E_T(B)$ is a projection-valued measure:

E_T(\varnothing)=0,\qquad E_T(\sigma(T))=I,\qquad E_T(B\cap C)=E_T(B)E_T(C).

For pairwise disjoint $B_n$ , the projections $E_T(B_n)$ are orthogonal and $E_T(\bigcup_nB_n)=\sum_nE_T(B_n)$ , where the sum converges strongly. These laws encode countable additivity at the operator level.

Eigenvalues, intervals, and reconstruction

For $\lambda\in\mathbb C$ , the atomic projection $E_T(\{\lambda\})$ has range $\ker(T-\lambda I)$ ; it is nonzero exactly when $\lambda$ is an eigenvalue. If $T$ is self-adjoint, the family

E_\lambda=E_T((-\infty,\lambda])

is an increasing right-continuous resolution of the identity. The operator is reconstructed as

T=\int_{\mathbb R}\lambda\,dE_T(\lambda),

with its domain determined by square integrability of the coordinate function.

Operator-algebraic role

Every spectral projection belongs to the von Neumann algebra generated by $T$ 's bounded resolvents, while it need not lie in the norm-closed $C^*$ -algebra generated by a bounded $T$ . This distinction explains why von Neumann algebras support measurable Borel operations. In quantum mechanics, spectral projections of a self-adjoint observable represent sharp yes-or-no events.

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised ed., Academic Press, 1980. DOI record. Relevant: Chapter VII on spectral resolutions and functional calculus.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III on spectral projections in von Neumann algebras.