Bounded Linear Operators - Key Properties

The adjoint operator is a fundamental construction that extends the notion of transpose from finite-dimensional linear algebra to infinite-dimensional Banach and Hilbert spaces.

DefinitionAdjoint Operator (Banach Space)

Let $X, Y$ be Banach spaces and $T \in \mathcal{B}(X, Y)$ . The adjoint $T^* : Y^* \to X^*$ is defined by $\langle T^*(\phi), x \rangle = \langle \phi, T(x) \rangle$ for all $\phi \in Y^*$ and $x \in X$ .

The adjoint $T^*$ is a bounded linear operator with $\|T^*\| = \|T\|$ .

DefinitionAdjoint Operator (Hilbert Space)

Let $H_1, H_2$ be Hilbert spaces and $T \in \mathcal{B}(H_1, H_2)$ . The Hilbert space adjoint $T^* : H_2 \to H_1$ is the unique operator satisfying $\langle T(x), y \rangle_{H_2} = \langle x, T^*(y) \rangle_{H_1}$ for all $x \in H_1$ and $y \in H_2$ .

The existence of $T^*$ follows from the Riesz Representation Theorem.

ExampleComputing Adjoints

Matrix Operators: For finite-dimensional spaces, $T^*$ corresponds to the conjugate transpose of the matrix representing $T$
Multiplication Operator: If $(M_\phi f)(x) = \phi(x) f(x)$ on $L^2$ , then $(M_\phi^* f)(x) = \overline{\phi(x)} f(x)$
Integral Operator: If $(Kf)(x) = \int_a^b k(x,y) f(y) \, dy$ on $L^2[a,b]$ , then $(K^* f)(x) = \int_a^b \overline{k(y,x)} f(y) \, dy$
Shift Operator: On $\ell^2$ , if $(Sx)_n = x_{n+1}$ , then $(S^*x)_n = x_{n-1}$ (with $x_0 = 0$ )

DefinitionSpecial Classes of Operators

Let $T \in \mathcal{B}(H)$ where $H$ is a Hilbert space:

Self-adjoint: $T = T^*$
Normal: $TT^* = T^*T$
Unitary: $T^*T = TT^* = I$ (and $T$ is surjective)
Positive: $\langle Tx, x \rangle \geq 0$ for all $x$
Projection: $T^2 = T = T^*$

TheoremProperties of the Adjoint

Let $S, T \in \mathcal{B}(H_1, H_2)$ and $R \in \mathcal{B}(H_2, H_3)$ . Then:

$(S + T)^* = S^* + T^*$
$(\alpha T)^* = \overline{\alpha} T^*$
$(RT)^* = T^*R^*$
$(T^*)^* = T$
$\|T^*T\| = \|T\|^2$
$\ker(T^*) = (\text{Range}(T))^\perp$

Remark

Self-adjoint operators play a role analogous to symmetric matrices in finite dimensions. They have real spectra and admit spectral decompositions, making them essential in quantum mechanics where observables are represented by self-adjoint operators.

The adjoint operation provides a rich algebraic structure on $\mathcal{B}(H)$ , turning it into a $C^*$ -algebra. This structure is fundamental to operator theory and has applications ranging from quantum mechanics to signal processing.