Adjoint of a bounded operator

Let $H$ and $K$ be complex Hilbert spaces whose inner products are linear in the first variable, and let $T:H\to K$ be a bounded linear operator. The adjoint of $T$ is the unique bounded operator $T^*:K\to H$ satisfying

\langle Tx,y\rangle_K=\langle x,T^*y\rangle_H \qquad (x\in H,\ y\in K).

Existence and uniqueness follow from the Riesz representation theorem applied to $x\mapsto\langle Tx,y\rangle_K$ . The adjoint reverses the direction of the map, so self-adjointness is defined only when source and target agree.

Norm and algebraic identities

The adjoint has the same operator norm as the original operator and satisfies

\lVert T^*\rVert=\lVert T\rVert,\qquad T^{**}=T,\qquad (ST)^*=T^*S^*.

It is conjugate-linear in the operator:

(\alpha S+\beta T)^* =\overline{\alpha}S^*+\overline{\beta}T^*.

When $T\in B(H)$ , the identity $\lVert T^*T\rVert=\lVert T\rVert^2$ makes the adjoint the involution in the $C^*$ -algebra $B(H)$ .

Kernels, ranges, and matrices

The orthogonality relations

\ker T^*=(\operatorname{Ran}T)^\perp,\qquad \ker T=(\operatorname{Ran}T^*)^\perp

connect algebraic failure of injectivity with density of the opposite range. For finite-dimensional Hilbert spaces with orthonormal bases, $T^*$ is represented by the conjugate-transpose matrix. For the unilateral shift on $\ell^2(\mathbb N)$ , the adjoint is the backward shift $(x_0,x_1,\ldots)\mapsto(x_1,x_2,\ldots)$ .

Relation to unbounded adjoints

The adjoint of a densely defined operator is defined by the same inner-product identity, but its domain may be a proper dense subspace. For bounded $T$ defined on all of $H$ , that construction yields an everywhere-defined bounded $T^*$ . Conflating the two settings can hide the domain questions responsible for symmetry, self-adjointness, and extension theory.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. DOI record. Relevant: Chapter II on Hilbert-space operators and adjoints.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. DOI record. Relevant: §2.5 on operators on Hilbert space.