Applying functions to operators
If a self-adjoint operator T diagonalises as Tx = Σ λₙ⟨x,eₙ⟩eₙ, then for any function f defined on σ(T) we simply define f(T)x = Σ f(λₙ)⟨x,eₙ⟩eₙ. This is the [[functional-calculus|functional calculus]]: apply f eigenvalue-by-eigenvalue. It is a genuine algebra homomorphism — (f+g)(T) = f(T)+g(T), (fg)(T) = f(T)g(T), and ‖f(T)‖ = sup_{λ∈σ(T)}|f(λ)| — so the messy algebra of operators becomes the easy algebra of functions on the spectrum.
Square root of a positive operator, via functional calculus.
Let T be self-adjoint with sigma(T) contained in [0, inf): a positive operator,
Tx = sum_n lambda_n <x, e_n> e_n, with each lambda_n >= 0.
Define S = T^{1/2} by applying f(t) = sqrt(t):
S x = sum_n sqrt(lambda_n) <x, e_n> e_n.
Check S^2 = T:
S^2 x = sum_n ( sqrt(lambda_n) )^2 <x, e_n> e_n
= sum_n lambda_n <x, e_n> e_n = T x. good
S is self-adjoint (real eigenvalues sqrt(lambda_n)) and positive (sqrt(lambda_n) >= 0).
So every positive operator has a unique positive square root --
the operator analogue of sqrt of a non-negative number.When the operator is unbounded
The operators of physics — position, momentum, energy — are not bounded, so they cannot be defined on all of H. An unbounded operator comes with a dense domain D(T) ⊊ H on which it acts, and good behaviour requires it to be closed (its graph is closed) rather than continuous. The differentiation operator T = i d/dx on L²(ℝ) is the model example: it is unbounded because i d/dx of eⁱᵏˣ is −k·eⁱᵏˣ, and k can be arbitrarily large, so no single constant bounds ‖Tf‖/‖f‖.
The spectral theorem in full, and quantum mechanics
When σ(T) is not a discrete set of eigenvalues, the sum Σ λₙ⟨x,eₙ⟩eₙ becomes an integral against a projection-valued measure E: Tx = ∫_{σ(T)} λ dE(λ). This is the general spectral theorem, valid for any (possibly unbounded) self-adjoint operator, and it covers the continuous spectrum case where there are no eigenvectors in H at all — the position operator on L²(ℝ) has purely continuous spectrum equal to all of ℝ. The functional calculus extends too: f(T) = ∫ f(λ) dE(λ).
This is the mathematical heart of quantum mechanics. An observable is a self-adjoint operator T; its spectrum σ(T) is the set of possible measured values, and the projection-valued measure E gives the probability of landing in a range of values. Time evolution is the unitary operator e^{−itH} built from the Hamiltonian H by the functional calculus with f(λ) = e^{−itλ}. The whole framework rests on the spectral theorem for unbounded self-adjoint operators — the destination this track has been climbing toward. The Fourier transform, not coincidentally, is exactly the unitary that diagonalises i d/dx.