2.4 Adjoint operators and their properties

Adjoint operators are key players in functional analysis, linking operators between Hilbert spaces. They're defined by a special property that connects inner products, making them crucial for understanding operator behavior.

Adjoints have unique properties like conjugate linearity and norm preservation. They're used to classify operators as self-adjoint, normal, or unitary, which has big implications in quantum mechanics and other fields using Hilbert spaces.

Adjoint Operators

Definition of adjoint operator

• Let $H_1$ and $H_2$ be Hilbert spaces and $T: H_1 \to H_2$ be a bounded linear operator
• The adjoint operator of $T$, denoted by $T^*$, is a bounded linear operator from $H_2$ to $H_1$
  • Satisfies the adjoint property $\langle Tx, y \rangle_{H_2} = \langle x, T^*y \rangle_{H_1}$ for all $x \in H_1$ and $y \in H_2$
  • Also known as the defining property of the adjoint operator
  • Allows for the study of the relationship between an operator and its adjoint ($T$ and $T^*$)
  • Plays a crucial role in the theory of bounded linear operators on Hilbert spaces ($L^2$, $\ell^2$)

Existence and uniqueness of adjoint

• Existence:
  • For each $y \in H_2$, define a linear functional $\phi_y: H_1 \to \mathbb{C}$ by $\phi_y(x) = \langle Tx, y \rangle_{H_2}$
  • By the Riesz Representation Theorem, there exists a unique $z \in H_1$ such that $\phi_y(x) = \langle x, z \rangle_{H_1}$ for all $x \in H_1$
    • This theorem guarantees the existence of a unique element in $H_1$ representing the linear functional $\phi_y$
  • Define $T^*y = z$, then $\langle Tx, y \rangle_{H_2} = \langle x, T^*y \rangle_{H_1}$ for all $x \in H_1$ and $y \in H_2$
• Uniqueness:
  • Suppose $S: H_2 \to H_1$ is another operator satisfying the adjoint property
  • Then, $\langle x, T^*y \rangle_{H_1} = \langle Tx, y \rangle_{H_2} = \langle x, Sy \rangle_{H_1}$ for all $x \in H_1$ and $y \in H_2$
  • By the uniqueness part of the Riesz Representation Theorem, $T^*y = Sy$ for all $y \in H_2$, implying $T^* = S$
• Properties:
  • Linearity: $(aT + bS)^* = \overline{a}T^* + \overline{b}S^*$ for all bounded linear operators $T, S$ and scalars $a, b$
    • The adjoint operator preserves linear combinations, with complex conjugates of the scalars
  • Boundedness: $\|T^*\| = \|T\|$
    • The adjoint operator has the same operator norm as the original operator
  • Double adjoint: $(T^*)^* = T$
    • Taking the adjoint of the adjoint operator yields the original operator

Calculation of adjoint operators

• Example 1: Let $T: L^2[0, 1] \to L^2[0, 1]$ be defined by $(Tf)(x) = \int_0^x f(t) dt$. Then, $(T^*g)(x) = \int_x^1 g(t) dt$
  • The adjoint of the integral operator from $0$ to $x$ is the integral operator from $x$ to $1$
• Example 2: Let $T: \ell^2 \to \ell^2$ be defined by $(Tx)_n = \frac{x_n}{n}$. Then, $(T^*y)_n = \frac{y_n}{n}$
  • The adjoint of the operator dividing each component by its index is the same operator
• Example 3: Let $T: \mathbb{C}^n \to \mathbb{C}^m$ be a matrix operator. Then, $T^*$ is the conjugate transpose of the matrix representing $T$
  • The adjoint of a matrix operator is the conjugate transpose of the matrix ($A^*$ or $A^H$)

Operator vs adjoint relationships

• Self-adjointness:
  • An operator $T$ is self-adjoint if $T = T^*$
    • The operator is equal to its adjoint
  • Equivalently, $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y$ in the Hilbert space
    • The inner product is symmetric with respect to the operator
  • Self-adjoint operators have real eigenvalues and orthogonal eigenvectors
    • Spectral properties are similar to real symmetric matrices
• Normality:
  • An operator $T$ is normal if $TT^* = T^*T$
    • The operator commutes with its adjoint
  • Self-adjoint operators are always normal
    • Self-adjointness is a stronger condition than normality
  • Normal operators can be diagonalized by a unitary operator
    • They have an orthonormal basis of eigenvectors ($U^*TU$ is diagonal, $U$ unitary)

Properties and Applications

Prove the existence and uniqueness of the adjoint operator and its properties

• Conjugate linearity: $\langle Tx, y \rangle = \overline{\langle x, T^*y \rangle}$
  • The inner product with the adjoint is the complex conjugate of the inner product with the original operator
• Adjoint of the adjoint: $(T^*)^* = T$
  • Taking the adjoint twice yields the original operator
• Adjoint of the inverse: $(T^{-1})^* = (T^*)^{-1}$, if $T$ is invertible
  • The adjoint of the inverse is the inverse of the adjoint
• Adjoint of the composition: $(ST)^* = T^*S^*$
  • The adjoint of a composition is the composition of the adjoints in reverse order

Analyze the relationship between an operator and its adjoint, such as self-adjointness and normality

• Positive operators:
  • An operator $T$ is positive if $\langle Tx, x \rangle \geq 0$ for all $x$ in the Hilbert space
    • The inner product with the operator is non-negative
  • Positive operators are always self-adjoint
    • Positivity implies self-adjointness
  • The spectrum of a positive operator is a subset of $[0, \infty)$
    • Eigenvalues of positive operators are non-negative real numbers
• Unitary operators:
  • An operator $U$ is unitary if $UU^* = U^*U = I$
    • The operator and its adjoint are inverses of each other
  • Unitary operators preserve inner products: $\langle Ux, Uy \rangle = \langle x, y \rangle$
    • The inner product is invariant under unitary transformations
  • The spectrum of a unitary operator is a subset of the unit circle in the complex plane
    • Eigenvalues of unitary operators have modulus 1
• Applications:
  • Quantum mechanics: observables are represented by self-adjoint operators
    • Ensures real eigenvalues (measurable quantities) and orthogonal eigenvectors (states)
  • Fourier analysis: the Fourier transform is a unitary operator on $L^2(\mathbb{R})$
    • Preserves the $L^2$ norm (Parseval's identity) and has an inverse transform
  • Sturm-Liouville theory: eigenvalue problems for self-adjoint differential operators
    • Arises in the study of vibrating strings, heat conduction, and quantum mechanics