Dual & Adjoint Operators

An operator acts forward on vectors. Its dual action asks how every scalar measurement changes when pulled back through that operator. In a Hilbert space, Riesz representation turns the pulled-back measurement into another vector; that represented operator is the Hilbert adjoint.

This chapter uses inner products linear in the first argument.

Banach-space dual operator #

For $T\in\mathcal B(X,Y)$, define

$$T':Y^*\to X^*,\qquad T'f=f\circ T.$$

Then $T'$ is bounded and

$$\|T'\|=\|T\|, \qquad (ST)'=T'S'.$$

The reversed composition order is the same contravariance seen in matrix transposes. The construction needs no inner product and is sometimes denoted $T^*$ in Banach-space texts; using $T'$ here avoids confusing it with the Hilbert adjoint.

Annihilator identities expose solvability:

$$\ker T'=(\mathop{\mathrm{ran}}T)^\perp, \qquad (\mathop{\mathrm{ran}}T')_\perp=\ker T.$$

In Hilbert notation these become orthogonal-complement relationships, with a closure where required:

$$\overline{\mathop{\mathrm{ran}}T}=(\ker T^*)^\perp, \qquad \overline{\mathop{\mathrm{ran}}T^*}=(\ker T)^\perp.$$

Hilbert adjoint #

For $T\in\mathcal B(H_1,H_2)$, the Hilbert adjoint $T^*\in\mathcal B(H_2,H_1)$ is uniquely defined by

$$\langle Tx,y\rangle_{H_2}=\langle x,T^*y\rangle_{H_1} \quad\text{for all }x,y.$$

Existence follows because $x\mapsto\langle Tx,y\rangle$ is a bounded functional on $H_1$, so Riesz representation supplies $T^*y$. It satisfies

$$\|T^*\|=\|T\|,qquad (ST)^*=T^*S^*,qquad (T^*)^*=T,qquad \|T^*T\|=\|T\|^2.$$

For matrices with the standard complex inner product, $T^*$ is the conjugate transpose. For real matrices it is the transpose.

Examples #

For multiplication on $L^2$,

$$(M_gf)(x)=g(x)f(x),\qquad M_g^*=M_{\overline g}.$$

For an integral operator

$$(Tf)(x)=\int_a^b k(x,t)f(t)dt,$$

Fubini's theorem gives

$$(T^*g)(t)=\int_a^b\overline{k(x,t)}g(x)dx$$

under the needed integrability hypotheses. A Hermitian kernel $k(x,t)=\overline{k(t,x)}$ produces a self-adjoint operator.

The unilateral right shift on $\ell^2$, $S(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$, has adjoint $S^*(x_1,x_2,\ldots)=(x_2,x_3,\ldots)$. Here $S^*S=I$ but $SS^*\ne I$: an isometry need not be unitary.

Important operator classes #

• Self-adjoint: $T=T^*$. Its spectrum is real.
• Unitary: $T^*T=TT^*=I$. It preserves inner products and norms.
• Normal: $T^*T=TT^*$. The spectral theorem applies in its appropriate form.
• Positive: $\langle Tx,x\rangle\ge0$ for every $x$. A bounded positive operator is self-adjoint and has a unique positive square root.

Symmetric and self-adjoint are not interchangeable for unbounded operators. Symmetry means $\langle Tx,y\rangle=\langle x,Ty\rangle$ on $D(T)$; self-adjointness also requires $D(T)=D(T^*)$.

Unbounded adjoints are domain constructions #

Let $T:D(T)\subset H_1\to H_2$ be densely defined. A vector $y\in H_2$ lies in $D(T^*)$ when there exists $z\in H_1$ such that

$$\langle Tx,y\rangle=\langle x,z\rangle \quad\text{for every }x\in D(T),$$

and then $T^*y=z$. Density makes $z$ unique. Integration by parts shows why boundary conditions enter $D(T^*)$: the boundary term must vanish for every primal-domain vector.

Diagnostic checklist #

• Decide whether the context is Banach dual $T'$ or Hilbert adjoint $T^*$.
• State the complex inner-product convention.
• Keep range closures in orthogonal-complement identities.
• For integral kernels, conjugate and swap variables.
• For unbounded operators, compute $D(T^*)$ rather than only a formal differential expression.