6.1 Adjoint operators in Hilbert spaces

Adjoint operators are crucial in Hilbert spaces, linking an operator's action to the inner product structure. They're defined by a unique property that swaps the operator's effect between inner product arguments, revealing deep connections in operator theory.

Adjoints exist for all bounded linear operators and have key properties like norm preservation. They're essential for studying self-adjoint, normal, and unitary operators, which are vital in functional analysis and quantum mechanics.

Adjoint Operators in Hilbert Spaces

Definition of adjoint operators

• Let $H$ be a Hilbert space and $T: H \to H$ be a bounded linear operator
• The adjoint of $T$, denoted by $T^*$, is a unique bounded linear operator satisfying the property $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x, y \in H$
  • This property relates the action of $T$ on the first argument of the inner product to the action of $T^*$ on the second argument
• The adjoint operator allows for the study of $T$ through the inner product structure of the Hilbert space
• Properties of the adjoint operator include $\|T^*\| = \|T\|$, $(T^*)^* = T$, $(S + T)^* = S^* + T^*$ for bounded linear operators $S$ and $T$, $(\alpha T)^* = \overline{\alpha} T^*$ for any scalar $\alpha$, and $(ST)^* = T^*S^*$ for bounded linear operators $S$ and $T$
  • These properties demonstrate the close relationship between an operator and its adjoint

Existence and uniqueness of adjoints

• Existence of the adjoint operator $T^*$ for a bounded linear operator $T: H \to H$ can be proven using the Riesz Representation Theorem
  1. Define a linear functional $\phi_y: H \to \mathbb{C}$ by $\phi_y(x) = \langle Tx, y \rangle$ for each $y \in H$
  2. By the Riesz Representation Theorem, there exists a unique $z \in H$ such that $\phi_y(x) = \langle x, z \rangle$ for all $x \in H$
  3. Define $T^*y = z$, then $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x, y \in H$
• Uniqueness of the adjoint operator can be proven by assuming $S$ and $T$ are bounded linear operators satisfying $\langle Sx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in H$
  • Then $\langle (S-T)x, y \rangle = 0$ for all $x, y \in H$
  • Choosing $y = (S-T)x$ yields $\|(S-T)x\|^2 = 0$ for all $x \in H$, implying $S = T$ and proving uniqueness

Calculation of specific adjoints

• Identity operator $I$ on a Hilbert space $H$ is self-adjoint, meaning $I^* = I$
  • This can be verified by observing $\langle Ix, y \rangle = \langle x, y \rangle = \langle x, Iy \rangle$ for all $x, y \in H$
• Multiplication operator $M_f$ on the Hilbert space $L^2(\mu)$, defined by $(M_f\varphi)(x) = f(x)\varphi(x)$, has adjoint $(M_f)^* = M_{\overline{f}}$
  • This can be shown by calculating $\langle M_f\varphi, \psi \rangle = \int f(x)\varphi(x)\overline{\psi(x)} d\mu = \int \varphi(x)\overline{f(x)\psi(x)} d\mu = \langle \varphi, M_{\overline{f}}\psi \rangle$ for $\varphi, \psi \in L^2(\mu)$
• Properties of adjoints can be verified for these specific operators, such as $\|I^*\| = \|I\| = 1$ and $(M_f)^{**} = (M_{\overline{f}})^* = M_f$

Operators vs adjoints in inner products

• The adjoint operator $T^*$ is defined by the property $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x, y \in H$, which relates the action of $T$ on the first argument of the inner product to the action of $T^*$ on the second argument
• This property allows for the study of an operator $T$ through the inner product structure of the Hilbert space
• Properties of the adjoint operator, such as $\|T^*\| = \|T\|$, demonstrate the close relationship between an operator and its adjoint
• The adjoint operator is a key concept in the study of self-adjoint $T = T^*$, normal $TT^* = T^*T$, and unitary $TT^* = T^*T = I$ operators in Hilbert spaces
  • These special classes of operators have important properties and applications in functional analysis and quantum mechanics