The bounded spectral theorem replaces diagonalization by integration against spectral projections. It is the correct form of functional calculus when an operator need not have a basis of eigenvectors.
Continuous functional calculus
Functional calculus lets one apply scalar functions to operators. The guiding principle is that an operator with real spectral behavior should support the same continuous operations as its spectrum.
Theorem. Let \(T=T^*\in\mathcal L(H)\) and let \(K=\sigma(T)\subset\mathbb R\). There is a unique isometric unital \(*\)-homomorphism \(\Phi:C(K)\to\mathcal L(H)\) such that \(\Phi(p)=p(T)\) for every polynomial \(p\). We write \(\Phi(f)=f(T)\).
Proof. For real polynomials \(p\), the spectral mapping theorem for polynomials gives \(\|p(T)\|=\sup_{\lambda\in K}|p(\lambda)|\). The map \(p\mapsto p(T)\) is therefore norm-continuous from polynomials in \(C(K)\) to \(\mathcal L(H)\). By Weierstrass approximation, polynomials are dense in \(C(K)\), so the map extends uniquely to an isometric homomorphism on \(C(K)\).
The algebraic properties pass to the limit from polynomials. The \(*\)-property follows because \(\overline p(T)=p(T)^*\) for polynomials and then by uniform approximation. Uniqueness follows from density of polynomials in \(C(K)\).
Spectral measures from Riesz representation
Positive linear functionals on \(C(\sigma(T))\) become measures by Riesz representation. This is the bridge from algebraic calculus to spectral integration.
Theorem. For each \(x,y\in H\), there is a unique complex regular Borel measure \(\mu_{x,y}\) on \(\sigma(T)\) such that \(\langle f(T)x,y\rangle=\int_{\sigma(T)} f(\lambda)\,\mathrm{d}\mu_{x,y}(\lambda)\) for all \(f\in C(\sigma(T))\).
Proof. For fixed \(x\), the map \(f\mapsto\langle f(T)x,x\rangle\) is a positive bounded linear functional on \(C(K)\), with norm \(\le\|x\|^2\|f\|_\infty\). The Riesz-Markov representation theorem gives a positive finite measure \(\mu_{x,x}\). For \(x,y\), define \(\mu_{x,y}\) by polarization from the four positive measures associated with \(x+y\), \(x-y\), \(x+iy\), and \(x-iy\). The polarization identity gives the displayed formula and uniqueness.
Projection-valued measure
A spectral measure is projection-valued rather than scalar-valued because it must remember how different spectral regions act on the Hilbert space.
Theorem. There exists a unique projection-valued measure \(E\) on \(\sigma(T)\) such that \(T=\int\lambda\,\mathrm{d}E(\lambda)\) and \(f(T)=\int f(\lambda)\,\mathrm{d}E(\lambda)\) for bounded Borel \(f\).
Proof. First define \(E(B)\) weakly by \(\langle E(B)x,y\rangle=\mu_{x,y}(B)\). Countable additivity follows from countable additivity of scalar measures. Positivity gives \(E(B)\ge0\), and multiplicativity of the continuous functional calculus extends by monotone-class arguments from continuous functions to bounded Borel functions, yielding \(E(B\cap C)=E(B)E(C)\). Hence \(E(B)^2=E(B)=E(B)^*\), so \(E(B)\) is an orthogonal projection. The integral formula follows by approximation of bounded Borel functions by simple functions in the spectral measure calculus.
Spectral projections
Spectral projections decompose the space according to where the spectrum lies. They replace eigenspace projections when eigenvectors are not available.
Corollary. If \(B\subset\sigma(T)\) is Borel, then \(E(B)H\) is the spectral subspace on which \(T\) has spectrum contained in \(\overline B\), and \(E(B)\) commutes with every bounded operator commuting with \(T\).
Proof. The restriction formula follows from \(T E(B)=\int_B\lambda\,\mathrm{d}E(\lambda)\). If \(S\) commutes with \(T\), it commutes with every polynomial in \(T\), hence with every \(f(T)\) for \(f\in C(\sigma(T))\), and by bounded Borel approximation it commutes with \(E(B)\).
Compact case as diagonal calculus
In the compact self-adjoint case, the abstract calculus reduces to familiar diagonal formulas. This example should be kept as the mental model for the general theorem.
Example. For compact self-adjoint \(T\), the spectral measure is atomic away from \(0\): \(E(\{\lambda_j\})\) is projection onto the eigenspace of \(\lambda_j\). The continuous functional calculus reduces to \(f(T)x=f(0)P_{\ker T}x+\sum_j f(\lambda_j)\langle x,e_j\rangle e_j\).