The Adjoint of the Left Multiplication on a Hilbert Space

Introduction

The left multiplication of a Hilbert space is the operator $L_U(T)=UT$ on the Hilbert–Schmidt space, and the form of the category with respect to which its adjoint is taken is the Hilbert–Schmidt inner product $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$. The adjoint of the left multiplication is again a left multiplication,

$$ (L_U)^*=L_{U^*}, $$

so the adjoint stays on the side it started from, and the map $U\mapsto L_U$ is a $*$-representation of the algebra on its own Hilbert–Schmidt space: it is multiplicative, isometric and compatible with the involution. This is the property that the present article computes and explores; it is the operator-theoretic counterpart of the identity $L_x^*=L_{x^{\dagger}}$ of the Hilbert algebra, and it is the reason the left multiplications form a self-adjoint family. The computation is carried by the rank-one operators, on which the left multiplication simply shifts the left vector, and the same computation gives the adjoint of the right multiplication, $R_V^*=R_{V^*}$, and the adjoints of the two-sided sandwich in the signed case, which is flagged below.

This article fixes the Hilbert–Schmidt form, the adjoint of the left multiplication and its explicit value on the rank-one operators, the self-adjointness, normality, unitarity and positivity criteria that the identity produces, the $*$-representation structure and the commutant, and the passage to the signed left multiplication. The two-sided multiplication algebra is The Left and Right Multiplication Operators on a Hilbert Space, where the rank-one decomposition and the Hilbert–Schmidt form are established; the bounded operators and their adjoints are Bounded Operators on a Hilbert Space; the Hilbert-Schmidt class is Compact Operators; the signed left multiplication is The Signed Left Multiplication on a Hilbert Space, and its adjoint is The Signed Adjoint of the Left Multiplication on a Hilbert Space below; the two-sided signed adjoint is The Signed Adjoint Sandwich on a Hilbert Space; the Hilbert-algebra form of the same identity, with the Tomita operator, is The Adjoint of the Left Multiplication on a Hilbert Algebra (Part II).

Throughout, $H$ is a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with inner product linear in the first argument, $S_2(H)$ is the Hilbert–Schmidt class with the inner product $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$, $B(H)$ acts on $S_2(H)$ by $L_U(T)=UT$ and $R_V(T)=TV$, and the adjoint with respect to $\langle\cdot,\cdot\rangle_{\mathrm{HS}}$ is written with a star. The rank-one operator is $\xi\otimes\bar\eta:\zeta\mapsto\langle\zeta,\eta\rangle\xi$, and the involution on the elements is $U\mapsto U^*$.

The Hilbert–Schmidt Form and the Adjoint

Definition. On the Hilbert–Schmidt class $S_2(H)$ the Hilbert–Schmidt form is

$$ \langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*), $$

linear in the first argument and conjugate-linear in the second; it is an inner product, and $S_2(H)$ is a Hilbert space in it. The adjoint of a bounded operator is taken with respect to this form.

Theorem (the adjoint of the left multiplication). For every $U\in B(H)$ the left multiplication $L_U$ is bounded on $S_2(H)$ and

$$ (L_U)^*=L_{U^*}, $$

so the adjoint of a left multiplication is the left multiplication by the adjoint; the same computation gives $R_V^*=R_{V^*}$ for the right multiplication, and each one-sided family is closed under the adjunction.

Proof. The adjoint identity follows from the trace identity: $\langle L_UT,S\rangle_{\mathrm{HS}}=\operatorname{tr}(UTS^*)=\operatorname{tr}(T(U^*S)^*)=\langle T,U^*S\rangle_{\mathrm{HS}}=\langle T,L_{U^*}S\rangle_{\mathrm{HS}}$; the right case is $\operatorname{tr}(TVS^*)=\operatorname{tr}(T(SV^*)^*)$; uniqueness of the Hilbert adjoint and the density of $S_2(H)$ in the operators of the form give the identities for all elements.

Proposition (boundedness and the rank-one expression). For all $U$ and $\xi,\eta\in H$,

$$ L_U(\xi\otimes\bar\eta)=(U\xi)\otimes\bar\eta,\qquad R_V(\xi\otimes\bar\eta)=\xi\otimes\bar{V^*\eta}, $$

and the adjoint is computed on rank-one operators by transposing the vector:

$$ \langle L_U(\xi\otimes\bar\eta),\zeta\otimes\bar\kappa\rangle_{\mathrm{HS}} =\langle U\xi,\zeta\rangle\langle\kappa,\eta\rangle =\langle \xi\otimes\bar\eta,L_{U^*}(\zeta\otimes\bar\kappa)\rangle_{\mathrm{HS}} . $$

Proof. The rank-one action is the computation $UT=U(\xi\otimes\bar\eta)$, and for the right multiplication $(\xi\otimes\bar\eta)V=\xi\otimes\bar\eta\circ V$ gives the transpose formula; the adjoint pairing is the trace of the product of two rank-one operators, $\operatorname{tr}((\xi\otimes\bar\eta)(\zeta\otimes\bar\kappa)^*)=\langle\xi,\zeta\rangle\langle\kappa,\eta\rangle$.

Corollary (the multiplications are isometric and multiplicative). The maps $U\mapsto L_U$ and $V\mapsto R_V$ are linear, multiplicative and isometric: $\|L_U\|=\|U\|$, $\|R_V\|=\|V\|$, and the two families commute, $L_UR_V=R_VL_U$.

Proof. The norm identity is $\|UT\|_{\mathrm{HS}}\le\|U\|\|T\|_{\mathrm{HS}}$ with equality at a rank-one operator for which the vector is aligned with $U$; multiplicativity is associativity, and the commutativity is $U(TV)=(UT)V$.

Self-Adjointness, Normality and the Order

Theorem (criteria for a left multiplication). For $U\in B(H)$:

(i) $L_U$ is self-adjoint exactly when $U$ is self-adjoint;

(ii) $L_U$ is normal exactly when $U$ is normal;

(iii) $L_U$ is unitary exactly when $U$ is unitary;

(iv) $L_U$ is positive exactly when $U$ is positive;

(v) $L_U$ is an orthogonal projection exactly when $U$ is an orthogonal projection.

Proof. (i) $L_U^*=L_{U^*}$ equals $L_U$ iff $U^*=U$, the map being injective. (ii) $L_UL_U^*=L_{UU^*}$ and $L_U^*L_U=L_{U^*U}$ coincide iff $UU^*=U^*U$. (iii) $(L_U)^*L_U=L_{U^*U}$ is the identity iff $U^*U=I$, and $L_UL_U^*=I$ iff $UU^*=I$; the identity of $S_2(H)$ is $L_I$. (iv) $L_U$ is positive iff it is self-adjoint, so $U$ is, and $\langle L_UT,T\rangle_{\mathrm{HS}}=\operatorname{tr}(UTT^*)=\operatorname{tr}(T^*UT)$ is nonnegative for all $T$ iff $U\ge0$. (v) An orthogonal projection is a positive idempotent; $L_U^2=L_{U^2}$ is $L_U$ iff $U^2=U$, and the positivity is (iv).

Proposition (the adjoint and the involution). The involution $U\mapsto U^*$ on the elements and the adjunction on the operators agree under the representation:

$$ (L_U)^*=L_{U^*},\qquad (R_V)^*=R_{V^*}, $$

so the two representations are $*$-representations of $B(H)$ (and of its opposite) on $S_2(H)$, and the compatibility of the involution with the multiplications, $(UV)^*=V^*U^*$, is the compatibility of the adjunction with the composition, $(L_UL_V)^*=(L_V)^*(L_U)^*$.

Proof. The two identities are the theorem; passing to adjoints reverses the order of a product, $(L_UL_V)^*=L_V^*L_U^*$, which is the operator form of the anti-multiplicativity of the involution.

Corollary (the commutation theorem for one side). The commutant of $\{L_U:U\in B(H)\}$ in $B(S_2(H))$ contains the right multiplications $R_V$, and in finite dimension it is exactly the algebra generated by them; consequently $\{L_U:U\in B(H)\}'=\{R_V\}$, and the double commutant is $\{L_U\}$.

Proof. The first containment is the commutativity of the two families; in finite dimension the multiplication algebra generated by the two families is the whole of $B(M_n(\mathbb{K}))$ via the identification $M_n\cong\mathbb{K}^n\otimes\mathbb{K}^n$, and the commutant of the left family is the right family by the standard double-commutant computation, which gives the second identity and, applying it twice, the third.

The Signed Case

Proposition (the signed left multiplication to be adjointed). With the same form, the signed left multiplication of the graded Hilbert space, $\ell_U=L_U\circ\alpha$, has adjoint

$$ (\ell_U)^*=\alpha\,L_{U^*}=\ell_{\alpha(U^*)}, $$

because the grade involution is a $*$-automorphism and is self-adjoint for the Hilbert–Schmidt form, $\alpha^*=\alpha$; so the adjoint of a signed left multiplication is again a signed left multiplication, and the explicit computation is The Signed Adjoint of the Left Multiplication on a Hilbert Space below.

Proof. $(L_U\alpha)^*=\alpha^*L_U^*=\alpha L_{U^*}$, and $\alpha L_{U^*}$ acts by $X\mapsto\alpha(U^*X)=\alpha(U^*)\alpha(X)$, which is $\ell_{\alpha(U^*)}$; the self-adjointness of $\alpha$ is $\langle\alpha X,Y\rangle_{\mathrm{HS}}=\operatorname{tr}(\Gamma X\Gamma Y^*)=\operatorname{tr}(X\Gamma Y^*\Gamma)=\langle X,\alpha Y\rangle_{\mathrm{HS}}$.

Forward reference. The two-sided adjoint, the sandwich $S_{A,B}=L_A\varrho_B=\ell_AR_{\alpha(B)}$ and its adjoint $S_{\alpha(A^*),\alpha(B^*)}$, the unitarity condition they define, and the reflection case are The Signed Adjoint Sandwich on a Hilbert Space and The Signed Adjoint of the Reflection on a Hilbert Space, written in parallel; the graded-module adjoint action is The Graded Adjoint Action on a Module over a Hilbert Space. Nothing of those is used here.

Example (the finite-dimensional picture). For $H=\mathbb{K}^n$ identify $S_2(H)$ with $M_n(\mathbb{K})$ and the Hilbert–Schmidt form with $\langle S,T\rangle=\operatorname{tr}(ST^*)$; then $L_U$ is the operator $T\mapsto UT$, the adjoint is $T\mapsto U^*T$, and the matrix of $L_U$ in the standard basis is $U\otimes I$. The self-adjointness of $L_U$ is the self-adjointness of $U$, and the eigenvalues of $L_U$ are the eigenvalues of $U$ each repeated $n$ times, so the spectral picture of the multiplication is read from the element.

Example (rank-one vectors and the shift). On a rank-one operator, $L_U$ shifts only the left vector and keeps the right one, $L_U(\xi\otimes\bar\eta)=(U\xi)\otimes\bar\eta$; the adjoint shifts the same vector back by $U^*$, and iterating gives $L_U^n(\xi\otimes\bar\eta)=(U^n\xi)\otimes\bar\eta$. The rank-one decomposition thus carries the adjoint, the powers and the whole functional calculus of $L_U$ back to the element $U$.

Summary

With respect to the Hilbert–Schmidt form $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$, the left multiplication $L_U(T)=UT$ has adjoint $(L_U)^*=L_{U^*}$ and the right multiplication $R_V(T)=TV$ has adjoint $R_V^*=R_{V^*}$, so both one-sided families are closed under adjunction and stay on their side. The representation $U\mapsto L_U$ is linear, multiplicative and isometric, it is a $*$-representation in the sense that it carries the involution of the elements, $U\mapsto U^*$, to the adjunction of the operators, and the two families $L(B(H))$ and $R(B(H))$ commute; the commutant of each is the other, and the double commutant is the family itself. Consequently self-adjointness, normality, unitarity, positivity and projectionness of $L_U$ are exactly the corresponding properties of $U$, and the rank-one operators carry the adjoint and the powers to the element by shifting one vector. The signed left multiplication $\ell_U=L_U\alpha$ has adjoint $(\ell_U)^*=\ell_{\alpha(U^*)}$, again on the left, which is the first step of the signed adjoint theory developed in The Signed Adjoint of the Left Multiplication on a Hilbert Space and its companions.

Summary of Notation

Symbol	Meaning
$S_2(H)$	the Hilbert–Schmidt class
$\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$	the Hilbert–Schmidt form
$L_U(T)=UT$, $R_V(T)=TV$	the left and right multiplications
$(L_U)^=L_{U^}$	adjoint of the left multiplication
$R_V^=R_{V^}$	adjoint of the right multiplication
$\alpha(T)=\Gamma T\Gamma$, $\alpha^*=\alpha$	the grade involution, self-adjoint for the form
$(\ell_U)^=\ell_{\alpha(U^)}$	adjoint of the signed left multiplication
$\xi\otimes\bar\eta$	the rank-one operator
$L_U(\xi\otimes\bar\eta)=(U\xi)\otimes\bar\eta$	the rank-one action
$\{L_U\}'=\{R_V\}$	the commutation theorem