The [adjoint page](/page/The%20Adjoint%20of%20an%20Operator) developed the construction $T \mapsto T^*$ for bounded operators between [Hilbert spaces](/page/Hilbert%20Space) and established the defining identity $(Tx, y)_K = (x, T^*y)_H$. A natural question emerges: what happens when an operator equals its own adjoint? The condition $T = T^*$ — self-adjointness — is the infinite-dimensional analogue of the symmetry condition $A = A^\top$ for real matrices. In finite dimensions, the spectral theorem guarantees that every real symmetric matrix is diagonalisable by an orthogonal change of basis, with real eigenvalues. The central achievement of this page is the extension of that result to compact self-adjoint operators on separable Hilbert spaces: every such operator admits a complete orthonormal eigenbasis with real eigenvalues accumulating only at zero.

Self-adjointness is not merely a convenient technical condition. It is the *minimal algebraic requirement* that forces the spectrum to be real, the eigenspaces to be orthogonal, and the operator norm to be captured by the quadratic form $(Tx, x)_H$. These three properties — established on [the adjoint page](/page/The%20Adjoint%20of%20an%20Operator) — are the prerequisites for all that follows here. The present page develops the deeper structural theory: the spectral localisation result that pins the spectrum to the interval $[m, M]$ determined by the quadratic form, the full spectral theorem for compact self-adjoint operators, the Courant–Fischer variational characterisation of eigenvalues, and the consequences of positivity. We also introduce the operator order on self-adjoint operators and the notion of a positive square root.

## Motivation

[motivation] ### What Diagonalisability Means in Infinite Dimensions In $\mathbb{R}^n$, diagonalising a symmetric matrix $A$ means finding an orthonormal basis $\{e_1, \dots, e_n\}$ of eigenvectors: $Ae_k = \lambda_k e_k$. This decomposition is equivalent to writing \begin{align*} Ax = \sum_{k=1}^n \lambda_k (x \cdot e_k) e_k \quad \text{for all } x \in \mathbb{R}^n, \end{align*} which represents $A$ as a sum of rank-one projections scaled by eigenvalues. In an infinite-dimensional Hilbert space $H$, the analogous statement would be: a self-adjoint operator $T$ admits an orthonormal basis $\{e_k\}$ of eigenvectors such that \begin{align*} Tx = \sum_{k=1}^\infty \mu_k (x, e_k)_H e_k, \end{align*} with convergence in the norm of $H$. This is precisely what the spectral theorem for compact self-adjoint operators achieves — but two new phenomena appear that have no finite-dimensional counterpart. ### Two Obstacles Absent in Finite Dimensions First, a bounded self-adjoint operator on an infinite-dimensional space need not have *any* eigenvalues. The multiplication operator $M_t: L^2([0,1]) \to L^2([0,1])$ defined by $(M_t f)(x) = xf(x)$ is self-adjoint with $\sigma(M_t) = [0,1]$, but the equation $xf(x) = \lambda f(x)$ forces $f = 0$ a.e. for every $\lambda$ — there are no eigenvectors. The spectrum is entirely "[continuous](/page/Continuity)." Self-adjointness alone does not guarantee a diagonalisation. Second, even when eigenvalues exist and span a dense subspace, the convergence of the [series](/page/Series) $\sum \mu_k (x, e_k)_H e_k$ to $Tx$ is not automatic — it requires a norm estimate on the partial sums, which in turn requires control on how fast the eigenvalues decay. Compactness provides this control: if $T$ is compact, the eigenvalues must converge to zero, and the tail of the series is bounded by the largest remaining eigenvalue. ### The Role of Compactness Compactness resolves both obstacles simultaneously. A compact self-adjoint operator on an infinite-dimensional space always has eigenvalues (the operator norm is attained, and the extremal value is an eigenvalue), the eigenspaces are finite-dimensional (by compactness of the unit ball in finite dimensions), and the eigenvalues converge to zero (since the images of the orthonormal eigenvectors must have a convergent subsequence). These three facts combine to produce the full spectral decomposition. The spectral theorem for compact self-adjoint operators is the workhorse of elliptic PDE theory: the eigenvalue problems $-\Delta u = \lambda u$ on bounded domains, Sturm–Liouville problems, and the spectral theory of [integral](/page/Integral) operators with symmetric kernels all reduce to this result. [/motivation]

## Recap: Definition and Basic Properties

The [adjoint page](/page/The%20Adjoint%20of%20an%20Operator) established the following, which we use freely throughout this page.

[definition:Self-Adjoint Operator] Let $H$ be a Hilbert space with inner product $(\cdot, \cdot)_H$. A bounded linear operator $T \in \mathcal{L}(H)$ is **self-adjoint** if $T^* = T$, that is, \begin{align*} (Tx, y)_H = (x, Ty)_H \quad \text{for all } x, y \in H. \end{align*} [/definition]

The basic properties proved on [the adjoint page](/page/The%20Adjoint%20of%20an%20Operator) are: (a) every eigenvalue of a self-adjoint operator is real; (b) eigenvectors for distinct eigenvalues are orthogonal; (c) the operator norm satisfies $\|T\| = \sup_{\|x\|=1} |(Tx, x)_H|$. Property (c) is the key that connects the analytic quantity $\|T\|$ to the quadratic form, and it is used repeatedly in the spectral theory below.

## Positive Operators and the Operator Order

The quadratic form $x \mapsto (Tx, x)_H$ is real-valued for any self-adjoint operator $T$ (since $(Tx, x)_H = (x, Tx)_H = \overline{(Tx, x)_H}$). This opens the door to comparing self-adjoint operators through their quadratic forms, in the same way that real numbers are ordered.

The simplest and most important class of self-adjoint operators are those whose quadratic form is nonnegative. These are the operators for which "all eigenvalues are nonnegative" — the infinite-dimensional analogues of positive semidefinite matrices.

[definition:Positive Operator] Let $H$ be a Hilbert space. A self-adjoint operator $T \in \mathcal{L}(H)$ is **positive** (or **positive semidefinite**), written $T \ge 0$, if \begin{align*} (Tx, x)_H \ge 0 \quad \text{for all } x \in H. \end{align*} More generally, for self-adjoint $S, T \in \mathcal{L}(H)$, we write $S \le T$ if $T - S \ge 0$, i.e., if \begin{align*} (Sx, x)_H \le (Tx, x)_H \quad \text{for all } x \in H. \end{align*} This defines the **operator order** on the set of self-adjoint operators. [/definition]

The operator order is a partial order (reflexive, antisymmetric, transitive), but it is *not* a total order: there exist self-adjoint operators $S$ and $T$ such that neither $S \le T$ nor $T \le S$. This is already visible in $\mathbb{R}^2$: the diagonal matrices $\operatorname{diag}(1, 0)$ and $\operatorname{diag}(0, 1)$ are both positive, but neither dominates the other.

The operator order interacts well with algebraic operations: if $S \le T$, then $S + R \le T + R$ and $\lambda S \le \lambda T$ for $\lambda \ge 0$. However, the product of positive operators need not be positive (or even self-adjoint): if $S, T \ge 0$, the product $ST$ is self-adjoint only if $S$ and $T$ commute.

[example:The Laplacian As A Positive Operator] Let $U \subset \mathbb{R}^n$ be a bounded [open set](/page/Open%20Set) with smooth boundary, and consider the operator $A := -\Delta$ acting on $H^1_0(U) \cap H^2(U) \subset L^2(U)$. For $u \in H^1_0(U) \cap H^2(U)$, [integration by parts](/theorems/210) gives \begin{align*} (-\Delta u, u)_{L^2} = \int_U |\nabla u|^2 \, d\mathcal{L}^n \ge 0, \end{align*} where the [boundary](/page/Boundary) term vanishes because $u \in H^1_0(U)$. So $-\Delta$ is positive (in the sense of quadratic forms on its domain). This is the prototypical example: the positivity of $-\Delta$ is the reason the Dirichlet eigenvalue problem $-\Delta u = \lambda u$, $u|_{\partial U} = 0$ has only nonnegative eigenvalues. [/example]

### Why the Operator Order Matters

The operator order is not just an algebraic convenience — it gives a direct comparison tool for quadratic forms and hence for eigenvalues. If $S \le T$ are self-adjoint and compact, then the $k$-th eigenvalue of $S$ is at most the $k$-th eigenvalue of $T$ (this is a consequence of the [Courant–Fischer min-max principle](/theorems/553), which we state below). This monotonicity is the basis of eigenvalue comparison theorems in PDE theory: if one elliptic operator has "larger" coefficients than another, its eigenvalues are correspondingly larger.

## The Spectrum of a Self-Adjoint Operator

The basic properties from the adjoint page tell us that eigenvalues of a self-adjoint operator are real and that $\|T\| = \sup_{\|x\|=1} |(Tx, x)_H|$. The following theorem strengthens both statements: the *entire spectrum* (not just the eigenvalues) is real and confined to the interval determined by the quadratic form. This is a much stronger assertion, because the spectrum of a non-compact self-adjoint operator can contain points that are not eigenvalues — the continuous spectrum and residual spectrum also lie in $[m, M]$.

The proof exploits the fact that self-adjointness gives a lower bound on $\|(T - \lambda I)x\|_H$ whenever $\lambda$ has nonzero imaginary part or lies outside $[m, M]$. Such a lower bound shows $T - \lambda I$ is bounded below, and the [kernel-range duality](/theorems/551) then implies it is invertible.