The Adjoint of a Hermitian Operator

Introduction

Let $V$ be a complex vector space with a positive-definite Hermitian form $\langle\cdot,\cdot\rangle$ and let $H$ be an operator on $V$. The adjoint of $H$ with respect to the form is the unique operator $H^{*}$ with $$ \langle Hx, y\rangle = \langle x, H^{*}y\rangle \qquad (x, y \in V), $$ and $H$ is Hermitian (or self-adjoint) when $H^{*} = H$, that is when $\langle Hx,y\rangle = \langle x,Hy\rangle$ for all $x,y$. The adjoint is the operator that the Hermitian form of the category produces out of $H$, and its explicit form is transparent in a unitary frame: if $H$ has matrix $(H_{ij})$ there, then the matrix of $H^{*}$ is the conjugate transpose, $$ (H^{*})_{ij} = \overline{H_{ji}}, $$ so that a Hermitian operator is exactly an operator whose matrix is a Hermitian matrix, and $H$ is anti-Hermitian when $H^{*} = -H$, equivalently when $iH$ is Hermitian. The self-adjoint operators are the ones that a Hermitian form detects: the assignment $H \mapsto \langle H\cdot,\cdot\rangle$ is a bijection between the Hermitian operators and the Hermitian forms on $V$, the quadratic form $x\mapsto\langle Hx,x\rangle$ is real for Hermitian $H$, and the spectral theorem makes the Hermitian operators exactly the orthogonally diagonalisable ones with real eigenvalues.

The article has three sections: the adjoint and its explicit form; the Hermitian operators and the correspondence with the Hermitian forms; and the spectral statements. The Hermitian forms and the unitary group are Hermitian Geometry and the Unitary Group, the preceding article of this category; the sesquilinear forms and the polarisation are Sesquilinear Forms and the Lax–Milgram Theorem and Quadratic Forms and Polarisation; the spectral theorem is Self-Adjoint Operators and the Spectral Theorem; the adjoint operation in the general operator layer is Adjoints in a Banach Algebra and The L2 Adjoint of a Differential Operator; the real structure that conjugates an operator is The Involution on a Complex Vector Space. None of that is re-derived.

Throughout, $V$ is a complex vector space of dimension $m$ with the positive-definite Hermitian form $\langle\cdot,\cdot\rangle$, $H$ is an operator on $V$, $H^{*}$ is its adjoint, and $q_H(x) = \langle Hx,x\rangle$ is its quadratic form.

The Adjoint and Its Explicit Form

Proposition (existence and uniqueness of the adjoint). For every operator $H$ there is a unique operator $H^{*}$ with $\langle Hx,y\rangle = \langle x,H^{*}y\rangle$ for all $x,y$. In a unitary frame $e_1,\dots,e_m$ the matrix of $H^{*}$ is the conjugate transpose of the matrix of $H$, $(H^{*})_{ij} = \overline{H_{ji}}$, and the assignment $H\mapsto H^{*}$ is conjugate-linear, involutive and an anti-automorphism, $$ (aH+bK)^{*} = \bar aH^{*}+\bar bK^{*}, \qquad (H^{*})^{*} = H, \qquad (HK)^{*} = K^{*}H^{*}. $$

Proof. For fixed $y$ the map $x\mapsto\langle Hx,y\rangle$ is conjugate-linear, hence of the form $x\mapsto \langle x, z\rangle$ for a unique $z$, by the representation of conjugate-linear functionals through the positive-definite form; set $H^{*}y = z$. In a unitary frame the entries satisfy $\langle He_j,e_i\rangle = H_{ij}$ and $\langle e_j,H^{*}e_i\rangle = (H^{*})_{ij}$, so the defining identity gives $H_{ij} = \overline{(H^{*})_{ij}}$, that is $(H^{*})_{ij} = \overline{H_{ji}}$. The three properties are immediate from the definition and the antilinearity of the form in its second argument in the appropriate slot. Hermitian forms and the representation of functionals are Hermitian Geometry and the Unitary Group and Hermitian Forms and Unitary Geometry.

Proposition (the Hermitian operator). $H$ is Hermitian, $H^{*} = H$, if and only if its quadratic form is real, $\langle Hx,x\rangle \in \mathbb{R}$ for all $x$, if and only if its matrix in some unitary frame is a Hermitian matrix; $H$ is anti-Hermitian if and only if $iH$ is Hermitian, and every operator decomposes uniquely as $$ H = \tfrac12(H+H^{*}) + \tfrac12(H-H^{*}), $$ the sum of a Hermitian and an anti-Hermitian operator, the real and imaginary parts of $H$.

Proof. If $H^{*}=H$ then $\langle Hx,x\rangle = \overline{\langle x,Hx\rangle} = \overline{\langle Hx,x\rangle}$ is real; conversely if the quadratic form is real, the polarisation of $\langle Hx,y\rangle$ from $q_H$ shows $\langle Hx,y\rangle = \langle x,Hy\rangle$, so $H^{*}=H$. The matrix statement is the previous proposition; the decomposition is the projection onto the self-adjoint and skew-adjoint parts, using $(H+H^{*})^{*} = H+H^{*}$ and $(H-H^{*})^{*} = H^{*}-H$. This is Quadratic Forms and Polarisation and Hermitian Geometry and the Unitary Group.

The Hermitian Operators and the Hermitian Forms

Proposition (the correspondence). The assignment $$ H \longmapsto h_H, \qquad h_H(x,y) = \langle Hx, y\rangle , $$ is a bijection between the Hermitian operators on $V$ and the Hermitian forms on $V$; under it the definite operators correspond to the definite forms, $\langle Hx,x\rangle > 0$ for $x\neq0$, and the positive operators are exactly those of the form $H = K^{*}K$.

Proof. $h_H$ is sesquilinear and $h_H(y,x) = \overline{\langle Hy,x\rangle} = \overline{\langle y,H^{*}x\rangle} = \langle H^{*}x,y\rangle = h_{H^{*}}(x,y)$, so $h_H$ is Hermitian exactly when $H = H^{*}$; the converse is the representation of a sesquilinear form by an operator through the definite form. Finally $\langle K^{*}Kx,x\rangle = \langle Kx,Kx\rangle \ge 0$, and conversely a positive $H$ has a positive square root $K = \sqrt H$ with $H = K^2 = K^{*}K$. The positivity of operators and forms is Hermitian Geometry and the Unitary Group, and the square root is Self-Adjoint Operators and the Spectral Theorem.

Corollary (self-adjointness of the structural operators). The complex structure is anti-Hermitian, $J^{*} = -J$; a unitary operator $U$ satisfies $U^{*}U = UU^{*} = I$, so its adjoint is its inverse, $U^{*} = U^{-1}$. A conjugation $c$ compatible with the form, $h(cx,cy) = \overline{h(x,y)}$, is antiunitary: it is antilinear, so it has no ordinary adjoint, but the adjoint of an antilinear map defined by $h(cx,y) = \overline{h(x,c^{\dagger}y)}$ gives $c^{\dagger} = c$, so a compatible conjugation is self-adjoint in the antilinear sense.

Proof. $\langle Jx,y\rangle = -\langle x,Jy\rangle$ is the skew-adjointness of $J$ for a Hermitian form; $U^{*}U=I$ is the isometry condition, and the inverse of a unitary is its adjoint. For a compatible conjugation, $h(cx,y) = h(cx,c(cy)) = \overline{h(x,cy)}$ (using $c^2=\mathrm{id}$ and compatibility), which is exactly $c^{\dagger}=c$ under the antilinear adjoint convention. These are The Involution on a Complex Vector Space and Hermitian Geometry and the Unitary Group.

The Spectral Statements and the Positivity

Theorem (the spectral theorem, quoted). A Hermitian operator on a finite-dimensional Hermitian space has an orthonormal basis of eigenvectors and real eigenvalues; it is positive when all its eigenvalues are $> 0$ and positive semidefinite when they are $\ge 0$, and the eigenvalues of $H$ are the critical values of the quadratic form $q_H$ restricted to the unit sphere.

Proof. The statement and its proof are Self-Adjoint Operators and the Spectral Theorem; the reality of the eigenvalues follows from the reality of $q_H$, the orthogonality of the eigenvectors of distinct eigenvalues from the self-adjointness, and the variational description is the Rayleigh quotient $\lambda = \min_{x\neq0} q_H(x)/\langle x,x\rangle$ on the appropriate subspace. This is quoted and not re-derived.

Proposition (the operator norm and the numerical radius). The operator norm $\|H\| = \sup\{ \|Hx\| : \|x\| = 1\}$ equals the largest modulus of an eigenvalue of $H$ for normal operators (and in particular for Hermitian and unitary operators), and the numerical radius $w(H) = \sup\{ |\langle Hx,x\rangle| : \|x\|=1\}$ equals $\|H\|$ for Hermitian $H$.

Proof. For a normal operator there is an orthonormal eigenbasis and $\|Hx\|^2 = \sum_k|\lambda_k|^2|x_k|^2 \le (\max|\lambda_k|)^2\|x\|^2$ with equality on an eigenvector of the largest modulus; for Hermitian $H$ the eigenvalues are real and $w(H) = \max|\lambda_k| = \|H\|$ by the variational description. The spectral theorem and the norm are Self-Adjoint Operators and the Spectral Theorem and Normed and Banach Spaces.

Example (the projections and the Hermitian involutions). An orthogonal projection is a Hermitian operator with $P^2 = P$, its eigenvalues $0$ and $1$, its quadratic form $q_P(x) = \|Px\|^2 \ge 0$; it is the adjoint-invariant form of a direct-sum decomposition $V = \operatorname{im}P\oplus\ker P$ with the summands orthogonal. A Hermitian involution is a Hermitian operator with $H^2 = I$, that is a unitary self-adjoint operator: the unitary reflections of The Signed Adjoint of the Reflection on a Complex Vector Space are the standard examples, and the orthogonal projection $P$ is the Hermitian idempotent. The compatible conjugation of The Involution on a Complex Vector Space is the antilinear companion of these, and the Bergman projection is the infinite-dimensional instance, The Bergman Operator and Involutions of the Bergman Operator.

Summary

On a Hermitian space $(V,\langle\cdot,\cdot\rangle)$ the adjoint of an operator $H$ is the unique $H^{*}$ with $\langle Hx,y\rangle = \langle x,H^{*}y\rangle$; in a unitary frame it is the conjugate transpose, $(H^{*})_{ij} = \overline{H_{ji}}$, and the assignment is conjugate-linear, involutive and an anti-automorphism. $H$ is Hermitian exactly when its quadratic form is real, equivalently when its matrix is Hermitian; every operator splits into its Hermitian and anti-Hermitian parts; a Hermitian operator has real eigenvalues and an orthonormal eigenbasis by the spectral theorem, is positive exactly when $H = K^{*}K$, and its norm and numerical radius agree. The Hermitian operators are in bijection with the Hermitian forms by $h_H(x,y) = \langle Hx,y\rangle$, the definite forms corresponding to the positive operators; the structural operators satisfy $J^{*} = -J$ and $U^{*} = U^{-1}$, and a compatible conjugation is antiunitary, self-adjoint in the antilinear sense. The forms and the unitary group are Hermitian Geometry and the Unitary Group; the polarisation is Quadratic Forms and Polarisation; the spectral theorem is Self-Adjoint Operators and the Spectral Theorem; the linear conjugation is The Involution on a Complex Vector Space.

Summary of Notation

Symbol	Meaning
$H^{*}$	the adjoint, $\langle Hx,y\rangle=\langle x,H^{*}y\rangle$
$(H^{*})_{ij}=\overline{H_{ji}}$	the explicit form in a unitary frame
$H^{*}=H$	Hermitian operator; real quadratic form
$H=\frac12(H+H^{})+\frac12(H-H^{})$	Hermitian and anti-Hermitian parts
$h_H(x,y)=\langle Hx,y\rangle$	the Hermitian form of a Hermitian operator
$H=K^{*}K$	the positive operators
$J^{}=-J$, $U^{}=U^{-1}$	the complex structure and the unitary operators
$c^{\dagger}=c$	a compatible conjugation, antiunitary