Self-Adjoint Operators and the Spectral Theorem

Adjoints, self-adjoint, and normal

On a Hilbert space H every bounded operator T has an adjoint T*, defined by ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x, y, using the inner product. T is self-adjoint if T = T*, normal if TT* = T*T, and unitary if T*T = TT* = I. Self-adjoint is the operator analogue of a real symmetric matrix; normal is the analogue of any matrix unitarily diagonalisable. The adjoint is the bridge that lets geometry — angles and orthogonality — speak to the algebra of T.

Two facts you can verify by hand for self-adjoint T (T = T*).

(1) <Tx, x> is real for every x.
    <Tx, x> = <x, T*x> = <x, Tx> = conj(<Tx, x>),
    and z = conj(z) means z is real.

(2) Eigenvalues are real and eigenvectors for distinct eigenvalues are orthogonal.
    Let Tx = lambda x with x != 0.  Then
       lambda <x,x> = <Tx, x> = <x, Tx> = conj(lambda) <x,x>,
    so lambda = conj(lambda): lambda is real.
    If also Ty = mu y with mu != lambda, then
       lambda <x,y> = <Tx, y> = <x, Ty> = mu <x,y>   (mu real)
       => (lambda - mu)<x,y> = 0 => <x,y> = 0.

The norm equals the spectral radius

For a self-adjoint operator the spectrum is real and lives in the interval [m, M] where m = inf‖x‖=1 ⟨Tx,x⟩ and M = sup‖x‖=1 ⟨Tx,x⟩, with both endpoints actually in σ(T). A striking consequence: ‖T‖ = r(T) = max(|m|, |M|). So for self-adjoint operators the norm — an analytic quantity — is read off directly from the spectrum, and ‖T‖ = sup{|λ| : λ ∈ σ(T)}. A positive operator (⟨Tx,x⟩ ≥ 0) is exactly a self-adjoint operator with σ(T) ⊆ [0, ∞).

The spectral theorem for compact self-adjoint operators

Combine compactness with self-adjointness and you get the cleanest infinite-dimensional diagonalisation there is. The [[ana-spectral-theorem|spectral theorem]] for compact self-adjoint operators states: there is an orthonormal basis (eₙ) of (the closure of the range of) T consisting of eigenvectors, with real eigenvalues λₙ → 0, and Tx = Σₙ λₙ ⟨x, eₙ⟩ eₙ for every x. Each eₙ diagonalises T; the whole operator is a weighted sum of rank-one projections onto its eigenlines. This is the exact infinite-dimensional twin of diagonalising a symmetric matrix by an orthonormal eigenbasis.

1. Show T has an eigenvalue λ₁ with |λ₁| = ‖T‖ by maximising ⟨Tx,x⟩ on the unit sphere; compactness makes the maximiser attainable.
2. Restrict T to the orthogonal complement of e₁; this subspace is T-invariant and T stays compact self-adjoint there.
3. Repeat to extract e₂, e₃, … with |λₙ| decreasing; compactness forces λₙ → 0.
4. Verify the eₙ span the closure of the range, so Tx = Σ λₙ⟨x,eₙ⟩eₙ holds for all x.