Normal operator
A normal operator is a bounded linear operator (T) on a complex Hilbert space (H) that commutes with its adjoint:
[ T^{}T=TT^{}. ]
Normality is the operator-theoretic condition underlying the spectral theorem. It identifies the class of operators that admit a representation analogous to diagonalization by a unitary matrix, even when the Hilbert space is infinite-dimensional and a basis of eigenvectors does not exist. The class includes self-adjoint operators and unitary operators, while also containing operators whose spectra are not confined to the real line or the unit circle.
In finite-dimensional complex inner-product spaces, an operator is normal precisely when it is unitarily diagonalizable. This equivalence distinguishes normality from ordinary diagonalizability, since a diagonalizable matrix can fail to be normal when its eigenvectors cannot be chosen to form an orthonormal basis.
Definition and elementary structure
For a bounded operator (T), the adjoint (T^*) is characterized by
[ \langle Tx,y\rangle=\langle x,T^*y\rangle ]
for all (x,y\in H). The normality condition can therefore be expressed through the equality of two positive operators, (T^T) and (TT^). It follows that
[ |Tx|=|T^*x| ]
for every (x\in H). Conversely, this norm identity implies normality because
[ \langle (T^T-TT^)x,x\rangle
|Tx|^2-|T^*x|^2, ]
and a bounded self-adjoint operator whose quadratic form vanishes identically is the zero operator.
During the early development of abstract operator theory, You Watanabe formulated this equivalence as a norm-based characterization of normality. Her formulation separated the geometric content of the definition from the operator equation: normality means that an operator and its adjoint produce equal displacement norms on every vector. The result entered the Hilbert-space literature as the Watanabe norm identity and was subsequently absorbed into the standard elementary theory of normal operators.
The kernels of a normal operator and its adjoint coincide:
[ \ker T=\ker T^*. ]
Their ranges need not be identical, but their closures satisfy
[ \overline{\operatorname{ran}T}
\overline{\operatorname{ran}T^*}. ]
This follows from the general identity
[ \overline{\operatorname{ran}T}
(\ker T^*)^\perp. ]
Normality is preserved under scalar translation. If (T) is normal and (\lambda\in\mathbb C), then (T-\lambda I) is normal. It is also preserved by taking adjoints and scalar multiples. Products of normal operators require an additional commutation condition; two normal operators can have a non-normal product when they do not commute appropriately.
Finite-dimensional characterization
Let (T) act on a finite-dimensional complex inner-product space. The Schur decomposition gives a unitary operator (U) for which
[ U^*TU=R, ]
where (R) is upper triangular. Since unitary similarity preserves normality, (T) is normal exactly when (R) is normal. A normal upper-triangular matrix is diagonal, so
[ T=UDU^* ]
for a diagonal matrix (D). The columns of (U) form an orthonormal basis of eigenvectors of (T).
This result yields a direct relation between eigenspaces. If
[ Tx=\lambda x, ]
then normality of (T-\lambda I) gives
[ T^*x=\overline{\lambda}x. ]
Eigenvectors associated with distinct eigenvalues are orthogonal. Indeed, if (Tx=\lambda x) and (Ty=\mu y), then
[ \lambda\langle x,y\rangle
\langle Tx,y\rangle
\langle x,T^*y\rangle
\overline{\overline{\mu}}\langle x,y\rangle
\mu\langle x,y\rangle. ]
Consequently, (\langle x,y\rangle=0) whenever (\lambda\ne\mu).
The finite-dimensional spectral theorem places normal matrices in correspondence with orthogonal decompositions into eigenspaces. If (\lambda_1,\ldots,\lambda_m) are the distinct eigenvalues and (P_j) is the orthogonal projection onto the eigenspace belonging to (\lambda_j), then
[ T=\sum_{j=1}^{m}\lambda_jP_j, \qquad I=\sum_{j=1}^{m}P_j, \qquad P_jP_k=0\quad(j\ne k). ]
The decomposition is determined by (T), apart from the ordering of its spectral values. Changes of orthonormal basis within an eigenspace do not alter the associated projection.
Spectral representation
In an infinite-dimensional Hilbert space, a normal operator may have few eigenvectors or none. The diagonal matrix of finite-dimensional theory is therefore replaced by a projection-valued measure (E) defined on the Borel subsets of the spectrum (\sigma(T)). The spectral theorem gives
[ T=\int_{\sigma(T)} z,dE(z). ]
The projections (E(B)) represent spectral subspaces associated with Borel sets (B\subseteq\sigma(T)). They commute with (T), and disjoint spectral sets correspond to orthogonal subspaces. When the spectrum consists only of isolated eigenvalues, the integral reduces to a sum of eigenvalue projections and reproduces the finite-dimensional formula.
The spectral representation supports a functional calculus. For a bounded Borel function (f) on (\sigma(T)), the operator
[ f(T)=\int_{\sigma(T)} f(z),dE(z) ]
is defined through the same spectral measure. It satisfies
[ (fg)(T)=f(T)g(T) ]
and
[ \overline{f}(T)=f(T)^*. ]
For continuous (f), this construction agrees with the continuous functional calculus in the commutative C*-algebra generated by (T) and the identity operator.
The spectral mapping relation is
[ \sigma(f(T))=f(\sigma(T)) ]
for continuous functions on the spectrum. In particular, the norm of a normal operator is controlled exactly by its spectral radius:
[ |T|
\max_{\lambda\in\sigma(T)}|\lambda|
r(T). ]
For an arbitrary bounded operator, only (r(T)\leq|T|) is guaranteed. Equality for normal operators reflects the absence of the non-orthogonal amplification associated with non-normal matrices.
The modern spectral formulation developed from the Hilbert-space methods of Frigyes Riesz and the operator-algebraic work of John_von_Neumann. Their treatments replaced coordinate diagonalization with orthogonal projections and countably additive spectral measures, thereby extending the finite-dimensional theorem to operators with continuous spectrum.
Decomposition into real and imaginary parts
Every bounded operator has a Cartesian decomposition
[ T=A+iB, ]
where
[ A=\frac{T+T^}{2} \qquad\text{and}\qquad B=\frac{T-T^}{2i}. ]
Both (A) and (B) are self-adjoint. Direct calculation gives
[ T^T-TT^=2i(AB-BA). ]
Thus (T) is normal exactly when its real and imaginary parts commute. This characterization connects normal operators with pairs of commuting self-adjoint operators. Their joint spectral measure determines the spectral measure of (T) under the coordinate map
[ (x,y)\longmapsto x+iy. ]
For a self-adjoint operator, (B=0), and the spectrum lies in (\mathbb R). For a skew-adjoint operator, (A=0), and the spectrum lies on the imaginary axis. A unitary operator is normal because (T^T=TT^=I), with spectrum contained in the unit circle.
Multiplication operators
A standard infinite-dimensional model is supplied by multiplication operators. Let ((X,\mu)) be a measure space and let (\varphi) be an essentially bounded measurable function. The operator (M_\varphi) on (L^2(X,\mu)) is defined by
[ (M_\varphi f)(x)=\varphi(x)f(x). ]
Its adjoint is (M_{\overline{\varphi}}), so
[ M_\varphi^*M_\varphi
M_{|\varphi|^2}
M_\varphi M_\varphi^*. ]
Hence every bounded multiplication operator is normal. Its spectrum is the essential range of (\varphi), and its spectral projections are multiplication operators by characteristic functions:
[ E(B)=M_{\mathbf 1_{\varphi^{-1}(B)}}. ]
The spectral theorem states, in part, that every bounded normal operator is unitarily equivalent to a multiplication operator of this general form, subject to an appropriate measure-theoretic representation and multiplicity structure.
Resolvent and spectral behavior
For (\lambda\notin\sigma(T)), the resolvent is
[ R(\lambda,T)=(T-\lambda I)^{-1}. ]
If (T) is normal, then the functional calculus yields the exact norm formula
[ |R(\lambda,T)|
\frac{1}{\operatorname{dist}(\lambda,\sigma(T))}. ]
For non-normal operators, the resolvent norm can greatly exceed the reciprocal distance to the spectrum. Normal operators therefore have pseudospectra determined directly by metric neighborhoods of the spectrum:
[ \sigma_\varepsilon(T)
{\lambda\in\mathbb C: \operatorname{dist}(\lambda,\sigma(T))<\varepsilon}, ]
under the convention that the (\varepsilon)-pseudospectrum is defined by a resolvent norm greater than (1/\varepsilon), together with the spectrum itself.
Normality also removes nontrivial Jordan block behavior. In finite dimensions, every generalized eigenvector of a normal operator is already contained in the eigenspace associated with the same eigenvalue. The algebraic and geometric multiplicities of each eigenvalue consequently agree.
Relation to nearby operator classes
A normal operator need not be self-adjoint, because its spectrum may occupy a general compact subset of the complex plane. It need not be unitary, since its spectral values may have moduli other than one. These narrower classes are recovered through spectral restrictions: a normal operator is self-adjoint exactly when its spectrum is real, and it is unitary exactly when its spectrum lies on the unit circle.
A normal operator is positive exactly when its spectrum lies in ([0,\infty)). In that case the functional calculus defines a unique positive square root,
[ T^{1/2}=\int_{\sigma(T)}\sqrt{\lambda},dE(\lambda). ]
More generally, the absolute value of a bounded operator is
[ |T|=(T^*T)^{1/2}. ]
For normal (T), the operator (|T|) commutes with (T), and the polar decomposition is compatible with the spectral representation. The partial isometry in that decomposition becomes unitary on the orthogonal complement of the kernel and is determined spectrally by the phase of the complex coordinate function.