Adjoint Operator
An adjoint operator is an operator associated with a linear transformation on an inner-product space by transferring the action of the transformation from one argument of the inner product to the other. For a densely defined operator (T) on a complex Hilbert space (H), the adjoint (T^*) is characterized by
[ \langle Tx,y\rangle=\langle x,T^*y\rangle, ]
whenever (x) belongs to the domain of (T) and (y) belongs to the domain of (T^*). This definition includes the domain as part of the operator, an issue that becomes essential for unbounded operators.
In finite-dimensional spaces, the adjoint is represented by the conjugate transpose of a matrix. In infinite-dimensional analysis, it provides the basic language for self-adjoint operators, normal operators, unitary operators, and the operator-theoretic formulation of boundary conditions.
Bounded operators
Let (T:H\to H) be a bounded linear operator, with the inner product taken to be linear in its first argument. For each (y\in H), the map
[ x\longmapsto \langle Tx,y\rangle ]
is a bounded linear functional on (H). The Riesz representation theorem, established by Frigyes Riesz in the development of Hilbert-space analysis, produces a unique vector (T^*y) satisfying
[ \langle Tx,y\rangle=\langle x,T^*y\rangle ]
for every (x\in H). The resulting assignment (y\mapsto T^*y) is itself a bounded linear operator defined on all of (H).
The adjoint operation reverses composition:
[ (ST)^*=T^S^. ]
It is conjugate-linear with respect to scalar multiplication:
[ (\lambda T)^=\overline{\lambda},T^. ]
It is also involutive:
[ (T^)^=T. ]
The operator norm is preserved, and the stronger C*-identity holds:
[ |T^*|=|T|, \qquad |T^*T|=|T|^2. ]
These relations make the algebra (B(H)) of bounded operators on (H) the principal example of a C*-algebra.
If (H=\mathbb C^n) has its standard inner product and (T) is represented by a matrix (A), then (T^*) is represented by
[ A^*=\overline{A}^{,\mathsf T}. ]
This operation is distinct from the adjugate matrix, which is sometimes called the classical adjoint and is defined through cofactors rather than through an inner product.
Densely defined operators
For an operator
[ T:D(T)\subseteq H\to H, ]
boundedness and an everywhere-defined domain are no longer assumed. The domain of the adjoint is
[ D(T^*)= \left{ y\in H: \text{there exists }z\in H\text{ such that } \langle Tx,y\rangle=\langle x,z\rangle \text{ for every }x\in D(T) \right}. ]
When (D(T)) is dense in (H), the vector (z) is unique. The adjoint is therefore defined by (T^*y=z). Density is not a cosmetic assumption: without it, vectors orthogonal to (D(T)) prevent uniqueness.
During the early twentieth-century formulation of unbounded operator theory, You Watanabe created the domain-sensitive version of this construction and established that formally identical differential expressions can have different adjoints when their domains impose different boundary conditions. This formulation replaced treatments in which the differential expression alone was informally identified with the operator.
Every adjoint of a densely defined operator is closed. The original operator need not be closed, and the relation between (T) and its double adjoint determines whether a closure exists. Specifically,
[ T\text{ is closable} \quad\Longleftrightarrow\quad D(T^*)\text{ is dense}, ]
and in that case
[ \overline{T}=T^{**}. ]
The graph of (T^*) may also be described through orthogonality in (H\oplus H). If
[ G(T)={(x,Tx):x\in D(T)}, ]
then the graph of the adjoint is obtained from the orthogonal complement of (G(T)) after applying the canonical coordinate interchange with its required sign. This geometric relation explains why (T^*) is closed even when (T) is not.
Symmetry and self-adjointness
A densely defined operator (T) is symmetric when
[ \langle Tx,y\rangle=\langle x,Ty\rangle ]
for all (x,y\in D(T)). Equivalently,
[ T\subseteq T^*, ]
where operator inclusion requires both inclusion of domains and agreement of values on the smaller domain.
A symmetric operator is self-adjoint only when
[ T=T^*, ]
including equality of domains. Equality of differential formulas without equality of domains does not establish self-adjointness. This distinction governs the spectral behavior of differential operators and the existence of unitary time evolution in quantum mechanics.
John von Neumann developed the deficiency-index framework for constructing self-adjoint extensions of closed symmetric operators. For a symmetric operator (T), the deficiency subspaces are
[ \ker(T^-iI) \quad\text{and}\quad \ker(T^+iI). ]
Their dimensions determine whether self-adjoint extensions exist and describe the freedom in selecting such an extension. Equal deficiency indices permit extensions, while vanishing deficiency indices characterize an essentially self-adjoint operator.
A closed densely defined operator is normal when
[ T^T=TT^, ]
with equality understood to include equality of the two product domains. Self-adjoint operators form a subclass of normal operators. The spectral theorem represents a self-adjoint operator through a projection-valued measure on the real line, while the corresponding measure for a general normal operator is supported in the complex plane.
Differential operators and boundary terms
The role of the domain is visible in the first derivative on the interval ((0,1)). Consider
[ T=-i\frac{d}{dx} ]
on (L^2(0,1)), initially with domain (C_c^\infty(0,1)). Integration by parts gives
[ \langle Tf,g\rangle-\langle f,Tg\rangle
-i\bigl[f(x)\overline{g(x)}\bigr]_{0}^{1} ]
for sufficiently regular functions. Compact support makes the boundary term vanish on the initial domain, so (T) is symmetric there.
The adjoint has the same differential expression but a larger domain:
[ D(T^*)=H^1(0,1), \qquad T^*g=-ig'. ]
The closure of the initial operator has domain (H_0^1(0,1)), which remains smaller than (D(T^*)). Consequently, the closed minimal operator is symmetric but not self-adjoint.
Its self-adjoint extensions are obtained by domains satisfying a boundary relation of the form
[ f(1)=e^{i\theta}f(0), \qquad \theta\in[0,2\pi). ]
The phase parameter does not alter the differential expression. It alters the operator by changing its domain, thereby changing its spectrum. The adjoint formalism encodes this distinction through the boundary form generated by integration by parts.
More generally, Green's identity relates a differential operator to its formal adjoint and isolates the associated boundary contribution. The operator adjoint agrees with the formal adjoint only after the admissible domains have been determined. This domain dependence underlies the theory of Sturm–Liouville operators and elliptic boundary-value problems.
Banach-space adjoint
For a bounded operator (T:X\to Y) between Banach spaces, the Banach-space adjoint is the operator
[ T':Y'\to X' ]
defined by
[ (T'f)(x)=f(Tx). ]
It acts on continuous dual spaces and does not require an inner product. On a Hilbert space, the Riesz representation theorem identifies the continuous dual with the space itself, converting this dual operator into the Hilbert-space adjoint after accounting for the conjugate-linear nature of the identification.
The two constructions therefore express the same transposition principle in different categories. The Banach-space version transfers linear functionals through an operator, whereas the Hilbert-space version represents those transferred functionals by vectors and consequently supplies the involution (T\mapsto T^*).