The Adjoint of a Bounded Operator

Introduction

The adjoint of a bounded operator on a Hilbert space is defined by the identity $\langle Tx, y\rangle = \langle x, T^{*}y\rangle$, and the assignment $T \mapsto T^{*}$ is an isometric involutive anti-automorphism of the algebra $B(H)$ of bounded operators: it is conjugate-linear, reverses the order of a product, is its own inverse, and preserves the norm, so $B(H)$ is an involutive Banach algebra and the identity $\lVert T^{*}T\rVert = \lVert T\rVert^{2}$ holds. Topologically, the adjoint is continuous for the norm, the weak, the strong and the weak-star topologies, and it is the transpose of $T$ read through the Riesz identification of $H$ with its dual, $T^{*} = R^{-1}T'R$; the spectrum transforms by $\sigma(T^{*}) = \overline{\sigma(T)}$, and on a finite-dimensional space the adjoint is the conjugate transpose of the matrix.

The existence of the adjoint and its basic identities are Banach and Hilbert Spaces, which supply the inner product, the Riesz representation theorem, the projection theorem and the bounded operators; the present article develops the operator $T^{*}$ as an element of the involutive algebra $B(H)$, its topological continuity and its identification with the transpose. The transposition of a continuous operator is The Dual Operator; the adjoint defined by an abstract pairing is The Dual Pairing and the Adjoint; the involutions of $B(H)$ and the classical groups they define are Involutions of the Bounded Operators; the abstract involutive Banach algebras are Involutive Banach Algebras and the Gelfand–Naimark Theorem. The forms and the indefinite adjoints are Part III, and no analysis is used beyond the inner product.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$ and $H$ is a Hilbert space over $\mathbb{K}$ with inner product $\langle\cdot,\cdot\rangle$ linear in the first argument, norm $\lVert\cdot\rVert$ and dual $H'$; the Riesz map is $R : H \to H'$, $(Ry)(x) = \langle x, y\rangle$, a conjugate-linear isometric isomorphism; $B(H)$ is the algebra of bounded operators with the operator norm; and $T' : H' \to H'$ is the transpose of $T$.

The Adjoint as an Involution of the Operator Algebra

Theorem (existence and the defining properties). For every $T \in B(H)$ there is a unique $T^{*} \in B(H)$ with

$$ \langle Tx, y\rangle = \langle x, T^{*}y\rangle \qquad (x, y \in H), $$

and $\lVert T^{*}\rVert = \lVert T\rVert$, $(S+T)^{*} = S^{*}+T^{*}$, $(\lambda T)^{*} = \overline{\lambda}T^{*}$, $(\mathrm{id})^{*} = \mathrm{id}$ and $T^{**} = T$.

Proof. The existence, the uniqueness, the norm equality and the algebraic identities are established in Banach and Hilbert Spaces from the Riesz representation theorem; they are quoted here and not reproved.

Theorem (the anti-automorphism). For all $S, T \in B(H)$,

$$ (ST)^{*} = T^{*}S^{*} , $$

so $T \mapsto T^{*}$ is an involutive anti-automorphism of $B(H)$: it reverses products, is conjugate-linear, is its own inverse, and fixes the identity. Consequently $B(H)$ is an involutive Banach algebra, and the adjoint is isometric.

Proof. For $x, y \in H$, $\langle STx, y\rangle = \langle Tx, S^{*}y\rangle = \langle x, T^{*}S^{*}y\rangle$, and the uniqueness of the adjoint gives $(ST)^{*} = T^{*}S^{*}$; the remaining statements are the theorem above.

Proposition (the $C^{*}$-identity). For every $T \in B(H)$,

$$ \lVert T^{*}T\rVert = \lVert T\rVert^{2} , $$

and this identity exhibits $B(H)$ as a $C^{*}$-algebra for the adjoint involution.

Proof. For $\lVert x\rVert \leq 1$, $\lVert Tx\rVert^{2} = \langle Tx,Tx\rangle = \langle T^{*}Tx,x\rangle \leq \lVert T^{*}T\rVert\lVert x\rVert^{2}$, so $\lVert T\rVert^{2} \leq \lVert T^{*}T\rVert$; the reverse inequality is the submultiplicativity $\lVert T^{*}T\rVert \leq \lVert T^{*}\rVert\lVert T\rVert = \lVert T\rVert^{2}$. The abstract theory of the $C^{*}$-algebras that the identity defines is Involutive Banach Algebras and the Gelfand–Naimark Theorem and Operator Algebras, named here and not developed.

Theorem (the adjoint is the transpose through Riesz). With $R$ the Riesz map,

$$ T^{*} = R^{-1}\,T'\,R , $$

so the Hilbert adjoint is the transpose of The Dual Operator conjugated by the Riesz identification; consequently $T \mapsto T^{*}$ is the transpose read on $H$ and inherits the reversal of composition and, in the reflexive case, the symmetry $T^{**} = T$.

Proof. For $y \in H$ and $x \in H$, $(R^{-1}T'R\,y)(x) = R^{-1}(T'Ry)(x) = (T'Ry)(x) = (Ry)(Tx) = \langle Tx, y\rangle = \langle x, T^{*}y\rangle = (R T^{*} y)(x)$, so $R^{-1}T'Ry = T^{*}y$. The reversal and the symmetry are the corresponding properties of the transpose.

Topological Continuity of the Adjoint

Theorem (continuity for the operator topologies). The involution $T \mapsto T^{*}$ of $B(H)$ is continuous, and in fact a homeomorphism, for the norm topology, the weak operator topology, the strong operator topology and the weak-star (ultraweak) topology; it is isometric for the norm.

Proof. Norm continuity and isometry are the theorem above. For the weak operator topology, the seminorms are $T \mapsto \lvert\langle Tx,y\rangle\rvert$, and $\lvert\langle T^{*}x,y\rangle\rvert = \lvert\langle x,Ty\rangle\rvert$ shows that the seminorms of $T^{*}$ are those of $T$ with the arguments exchanged; hence the involution is a homeomorphism. The strong operator topology is handled by $\lVert T^{*}x\rVert = \sup_{\lVert y\rVert\leq1}\lvert\langle T^{*}x,y\rangle\rvert = \sup_{\lVert y\rVert\leq1}\lvert\langle x,Ty\rangle\rvert$; the ultraweak topology is the weak-star topology on $B(H)$ as the dual of the trace-class operators, and the involution on the predual is the transpose of The Dual Operator and the Weak Topology, so it is continuous.

Corollary (commutants and bicommutants). The adjoint preserves the commutant, $T \in \{S\}' \Rightarrow T^{*} \in \{S\}'$ when $S$ is self-adjoint, and in general $\{S\}^{*} = \{S^{*}\}'$; hence a self-adjoint set of operators has a self-adjoint commutant, and the von Neumann bicommutant theorem applies to the self-adjoint algebras, which are owned by Operator Algebras.

Proof. If $TS = ST$ then $S^{*}T^{*} = (TS)^{*} = (ST)^{*} = T^{*}S^{*}$, so $T^{*}$ commutes with $S^{*}$; for self-adjoint $S$ this gives $T^{*} \in \{S\}'$, and the general identity follows by applying it to $S^{*}$.

Spectral and Finite-Dimensional Properties

Proposition (the spectrum transforms by conjugation). For $T \in B(H)$,

$$ \sigma(T^{*}) = \overline{\sigma(T)} , $$

and $T$ is invertible exactly when $T^{*}$ is, with $(T^{-1})^{*} = (T^{*})^{-1}$; the involution therefore preserves invertibility and the unitary group.

Proof. $T - \lambda \mathrm{id}$ is invertible with bounded inverse exactly when its adjoint $T^{*} - \overline{\lambda}\,\mathrm{id}$ is, because the adjoint of an invertible operator is invertible with adjoint inverse; hence $\lambda \in \rho(T)$ iff $\overline{\lambda} \in \rho(T^{*})$. The inverse formula follows by taking adjoints in $T T^{-1} = \mathrm{id}$.

Proposition (self-adjoint, skew, normal, unitary). The involution splits $B(H)$ into the self-adjoint operators $T = T^{*}$ and the skew-adjoint operators $T = -T^{*}$, with

$$ B(H) = B(H)^{\mathrm{sa}} \oplus B(H)^{\mathrm{skew}} , \qquad T = \tfrac12(T+T^{*}) + \tfrac12(T-T^{*}) , $$

when $2$ is invertible; $T$ is normal when $TT^{*} = T^{*}T$ and unitary when $T^{*}T = TT^{*} = \mathrm{id}$. The self-adjoint and skew-adjoint parts are real vector subspaces, and the unitary operators form a group.

Proof. The averaging maps $\tfrac12(\mathrm{id}\pm\ast)$ are real-linear idempotents with the stated images, and the direct sum is the abstract decomposition of an involutive algebra; the unitary operators are closed under multiplication and inversion by the anti-automorphism property.

Example (the finite-dimensional case). For $H = \mathbb{K}^{n}$ with the standard inner product and $A$ the matrix of $T$ in the standard basis, the matrix of $T^{*}$ is the conjugate transpose $A^{*} = \overline{A}^{\mathsf{T}}$, the self-adjoint operators are the Hermitian matrices (symmetric in the real case), the unitary operators are the unitary matrices (orthogonal in the real case), and the $C^{*}$-identity is $\lVert A^{*}A\rVert = \lVert A\rVert^{2}$ for the spectral norm. The transpose through Riesz is the identity of the finite-dimensional duality $(\mathbb{K}^{n})' = \mathbb{K}^{n}$ up to the conjugate structure.

Example (the sequence space). On $H = \ell^{2}$ the adjoint of a bounded operator with matrix $(a_{ij})$ has matrix $(a_{ji}^{*})$; the unilateral shift has the backward shift as its adjoint, the shift being an isometry with $S^{*}S = \mathrm{id}$ and $SS^{*}$ the projection onto the closed span of the coordinate vectors $e_{2}, e_{3}, \dots$, a proper projection, so the shift is not unitary.

Example (the multiplication operators). On $H = L^{2}(\mu)$ the multiplication $M_{f}$ by an essentially bounded function $f$ has adjoint $M_{\bar f}$, the involution acting by conjugation of the symbol; $M_{f}$ is self-adjoint exactly when $f$ is real almost everywhere, unitary exactly when $\lvert f\rvert = 1$ almost everywhere, and normal always, the multiplication operators being the model commutative self-adjoint algebra.

Summary

For a bounded operator $T$ on a Hilbert space there is a unique $T^{*}$ with $\langle Tx,y\rangle = \langle x,T^{*}y\rangle$, of the same norm, and the assignment $T \mapsto T^{*}$ is an isometric involutive anti-automorphism of $B(H)$, satisfying $(ST)^{*} = T^{*}S^{*}$, $T^{**} = T$ and the $C^{*}$-identity $\lVert T^{*}T\rVert = \lVert T\rVert^{2}$; it is the transpose of $T$ conjugated by the Riesz map, $T^{*} = R^{-1}T'R$. The involution is a homeomorphism for the norm, weak, strong and weak-star topologies of $B(H)$, it preserves commutants of self-adjoint sets, it sends the spectrum to the conjugate spectrum, $\sigma(T^{*}) = \overline{\sigma(T)}$, and it implements the splitting of $B(H)$ into the self-adjoint and the skew-adjoint operators, with the normal and the unitary operators defined by the involutive identities. In finite dimension the involution is the conjugate transpose of the matrix. The abstract involutions of $B(H)$ and the classical groups they define are Involutions of the Bounded Operators, and the adjoint defined by an abstract pairing is The Dual Pairing and the Adjoint.

Summary of Notation

Further Reading

• Paul R. Halmos, A Hilbert Space Problem Book (Springer, second edition, 1982), for the adjoint, the operator topologies and the unilateral shift.
• John B. Conway, A Course in Functional Analysis (Springer, second edition, 1990), for the Hilbert-space adjoint, the weak and strong operator topologies and the spectral theory.
• Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras I (Academic Press, 1983), for the involutive algebra $B(H)$, its topologies and the commutant.
• Walter Rudin, Functional Analysis (McGraw-Hill, second edition, 1991), for the Riesz representation theorem, the adjoint and the weak-star topology.
• Gert K. Pedersen, Analysis Now (Springer, 1989), for the $C^{*}$-identity, the involutive Banach algebras and the adjoint.