The Left and Right Multiplication Operators on a Hilbert Space

Introduction

A Hilbert space carries bounded operators, and those operators act on one another by composition: for $A\in B(H)$ the left multiplication $L_A$ sends $T$ to $AT$, and the right multiplication $R_B$ sends $T$ to $TB$. In order that these be operators of a Hilbert space in their own right, the space on which they act must be given a Hilbert structure, and the natural one is the Hilbert–Schmidt inner product $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$ on $S_2(H)$. With that structure the multiplications are bounded, their adjoints are the multiplications by the adjoints, and the whole theory is carried by the rank-one operators: every Hilbert–Schmidt operator is a sum of rank-one operators, and on a rank-one operator the two multiplications act by shifting the vectors of the two sides.

This article fixes the two one-sided multiplications, computes their adjoints with respect to the Hilbert–Schmidt form, develops the rank-one decomposition that makes the computation transparent, and identifies the algebra they generate together with its commutant. The algebra $B(H)$ and the Hilbert–Schmidt class are Bounded Operators on a Hilbert Space and Compact Operators; the multiplication operators of an abstract algebra are The Left and Right Multiplication Operators on a Banach Algebra (Part II) and their Hilbert-algebra form is The Adjoint of the Left Multiplication on a Hilbert Algebra. The signed versions with a grade involution are The Signed Left Multiplication on a Hilbert Space and The Adjoint of the Left Multiplication on a Hilbert Space below.

Throughout, $H$ is a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$ linear in the first argument, $S_2(H)$ is the Hilbert–Schmidt class with the inner product $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$, and for $\xi,\eta\in H$ the rank-one operator is

$$ \xi\otimes\bar\eta:H\longrightarrow H,\qquad (\xi\otimes\bar\eta)(x)=\langle x,\eta\rangle\,\xi . $$

The left and right multiplications are $L_A(T)=AT$ and $R_B(T)=TB$ on $S_2(H)$, and the adjoint with respect to $\langle\cdot,\cdot\rangle_{\mathrm{HS}}$ is written $L_A^*$, $R_B^*$.

The One-Sided Multiplications

Definition. For $A,B\in B(H)$ the left multiplication and the right multiplication are

$$ L_A:S_2(H)\longrightarrow S_2(H),\quad L_A(T)=AT, \qquad R_B:S_2(H)\longrightarrow S_2(H),\quad R_B(T)=TB . $$

Proposition (boundedness and norms). $L_A$ and $R_B$ are bounded linear operators on $S_2(H)$ with

$$ \|L_A\|=\|A\|,\qquad \|R_B\|=\|B\|, $$

and the maps $A\mapsto L_A$ and $B\mapsto R_B$ are linear isometries of $B(H)$ into $B(S_2(H))$ that preserve products and the identity:

$$ L_AL_C=L_{AC},\quad R_BR_D=R_{BD},\quad L_I=R_I=I . $$

Proof. For a rank-one $\xi\otimes\bar\eta$ one has $\|\xi\otimes\bar\eta\|_{\mathrm{HS}}=\|\xi\|\|\eta\|$ and $L_A(\xi\otimes\bar\eta)=A\xi\otimes\bar\eta$, $R_B(\xi\otimes\bar\eta)=\xi\otimes\bar{B^*\eta}$; hence $\|L_A(\xi\otimes\bar\eta)\|_{\mathrm{HS}}=\|A\xi\|\|\eta\|\le\|A\|\|\xi\|\|\eta\|=\|A\|\|\xi\otimes\bar\eta\|_{\mathrm{HS}}$. Since the rank-one operators span a dense subspace of $S_2(H)$, $L_A$ extends with norm at most $\|A\|$, and the value $\|A\|$ is attained by choosing $\eta$ and $\xi$ of norm one with $\|A\xi\|=\|A\|$. Products and the identity follow from associativity, and faithfulness from the density of the rank-one operators.

Proposition (the commutant relation). The two one-sided families commute:

$$ L_AR_B=R_BL_A\qquad(A,B\in B(H)), $$

and this is the algebra statement $A(TB)=(AT)B$ read as an identity of operators on $S_2(H)$.

Proof. Both sides send $T$ to $ATB$, by associativity of the composition of the operators of $H$.

Adjoints with Respect to the Hilbert–Schmidt Form

Theorem (the adjoint of a one-sided multiplication). For all $A,B\in B(H)$,

$$ L_A^*=L_{A^*},\qquad R_B^*=R_{B^*} . $$

So the adjoint of a left multiplication is again a left multiplication, the adjoint of a right multiplication is again a right multiplication, and each one-sided family is closed under the adjunction.

Proof. Using $\langle L_AT,S\rangle_{\mathrm{HS}}=\operatorname{tr}(ATS^*) = \operatorname{tr}(T S^*A)=\operatorname{tr}(T(A^*S)^*)=\langle T,L_{A^*}S\rangle_{\mathrm{HS}}$, the first identity follows from the uniqueness of the Hilbert adjoint; the second is $\operatorname{tr}(TBS^*)=\operatorname{tr}(T(SB^*)^*)$.

Corollary (self-adjointness, normality and unitarity of the multiplications). For $A\in B(H)$:

$L_A$ is self-adjoint exactly when $A$ is self-adjoint, and skew-adjoint exactly when $A$ is skew-adjoint;
$L_A$ is normal exactly when $A$ is normal;
$L_A$ is unitary exactly when $A$ is unitary, and then $L_A^{-1}=L_{A^*}$;
$L_A$ is positive exactly when $A$ is positive.

The same four statements hold for $R_B$ with $B$ in place of $A$.

Proof. Each is the identity of the theorem read through the isometric representation: $L_A^*=L_{A^*}$ gives $L_A=L_A^*$ iff $A=A^*$ by faithfulness, $L_AL_A^*=L_{AA^*}$ against $L_A^*L_A=L_{A^*A}$ gives normality, unitarity adds $L_{AA^*}=L_{A^*A}=I$, and positivity is $\langle L_AT,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ATT^*)=\operatorname{tr}(T^*AT)\ge0$ exactly when $A$ is positive.

The Rank-One Decomposition

Definition. The rank-one decomposition of $S_2(H)$ is the expression of an operator as a norm-convergent sum of rank-one operators $T=\sum_n\xi_n\otimes\bar\eta_n$; the coefficient vectors are the singular vectors of $T$ and the sum is the singular-value expansion.

Proposition (the multiplications on rank-one operators). For $\xi,\eta,x\in H$,

$$ L_A(\xi\otimes\bar\eta)=A\xi\otimes\bar\eta,\qquad R_B(\xi\otimes\bar\eta)=\xi\otimes\bar{B^*\eta},\qquad \xi\otimes\bar\eta = \text{the operator } x\mapsto\langle x,\eta\rangle\xi . $$

Consequently the left multiplication acts on the first index and the right multiplication acts on the second, and $L_A$ and $R_B$ leave the set of rank-one operators invariant.

Proof. $L_A(\xi\otimes\bar\eta)(x)=A(\langle x,\eta\rangle\xi)=\langle x,\eta\rangle A\xi$, which is $A\xi\otimes\bar\eta$; $R_B(\xi\otimes\bar\eta)(x)=B(\langle x,\eta\rangle\xi)=\langle x,\eta\rangle B\xi$, and one computes $\langle x,\eta\rangle B\xi=\langle x,B^*\eta\rangle\xi$, so the second factor receives $B^*$.

Theorem (the span and the inner product). The rank-one operators span a dense subspace of $S_2(H)$, and the Hilbert–Schmidt inner product of two of them is

$$ \langle \xi\otimes\bar\eta,\ \xi'\otimes\bar\eta'\rangle_{\mathrm{HS}}=\langle \xi,\xi'\rangle\,\langle\eta',\eta\rangle . $$

Hence every $T\in S_2(H)$ has a norm-convergent expansion $T=\sum_n s_n\,\xi_n\otimes\bar\eta_n$ with orthonormal families $(\xi_n)$, $(\eta_n)$ and singular values $s_n$, and the multiplications act termwise on the expansion.

Proof. The pairings follow from $(\xi\otimes\bar\eta)(\xi'\otimes\bar\eta')^*=\xi\otimes\bar\eta\circ\eta'\otimes\bar\xi'$, whose trace is $\langle\xi,\xi'\rangle\langle\eta',\eta\rangle$; the expansion is the singular-value decomposition of the compact operator $T$ recalled from Compact Operators, and the density of the span is the density of the finite-rank operators in $S_2$.

The Generated Algebra and Its Commutant

Definition. The multiplication algebra of $H$ is the algebra generated in $B(S_2(H))$ by the left and right multiplications:

$$ \mathcal{M}(H)=\Bigl\{\sum_i L_{A_i}R_{B_i}:A_i,B_i\in B(H),\ \text{finite sums}\Bigr\}, \qquad \sum_iL_{A_i}R_{B_i}(T)=\sum_iA_iTB_i. $$

Proposition (the algebras generated by the two sides). The algebra generated by the left multiplications alone is $\{L_A:A\in B(H)\}\cong B(H)$, the algebra generated by the right multiplications alone is $\{R_B:B\in B(H)\}\cong B(H)$, and

$$ \mathcal{M}(H)=\{L_A:A\in B(H)\}'=\{R_B:B\in B(H)\}', $$

so the multiplication algebra is the commutant of each one-sided family. In finite dimension $H=\mathbb{K}^n$ the multiplication algebra is all of $B(M_n(\mathbb{K}))$.

Proof. A multiplication $\sum_iL_{A_i}R_{B_i}$ commutes with every $L_C$, and conversely an operator on $S_2(H)$ commuting with every left multiplication is a right multiplication in the following sense: it is determined by its value on a single rank-one operator and is the sum of finitely many two-sided multiplications. In finite dimension $n$ one has $\dim B(M_n)=n^4=\dim(M_n\otimes M_n^{\mathrm{op}})$, and the identification is an isomorphism, so the multiplication algebra is everything.

Example (finite dimension). 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 $L_A$ is the operator $T\mapsto AT$ and $R_B$ is $T\mapsto TB$, the left multiplications are the matrices $A\otimes I$ under the identification $M_n\cong\mathbb{K}^n\otimes\mathbb{K}^n$, the right multiplications are $I\otimes B^{\mathrm t}$, and the multiplications satisfy $L_A^*=L_{A^*}$, $R_B^*=R_{B^*}$ with the conjugate transpose.

Example (diagonal operators). For $H=\ell^2$ and $A=B=D_a$ diagonal with $a\in\ell^\infty$, the operator $L_{D_a}R_{D_a}$ acts on the matrix units $e_m\otimes\bar e_n$ by $e_m\otimes\bar e_n\mapsto a_ma_n\,e_m\otimes\bar e_n$, so the two-sided multiplication by a single diagonal element is diagonal in the matrix-unit basis with the product symbol; the same computation with $a\in c_0$ restricts to the compact operators and exhibits the multiplication algebra of the compact ideal.

Summary

On the Hilbert–Schmidt space $S_2(H)$ of a Hilbert space the bounded operators of $H$ act by the left multiplication $L_A(T)=AT$ and the right multiplication $R_B(T)=TB$, both bounded with $\|L_A\|=\|A\|$ and $\|R_B\|=\|B\|$, and the two representations $A\mapsto L_A$, $B\mapsto R_B$ are isometric and multiplicative. The adjoints are $L_A^*=L_{A^*}$ and $R_B^*=R_{B^*}$, so each one-sided family is self-adjoint, and self-adjointness, normality, unitarity and positivity of a multiplication are exactly the corresponding properties of the element. The two families commute, and they act on the rank-one operators $\xi\otimes\bar\eta$ by shifting the two vectors, $L_A(\xi\otimes\bar\eta)=A\xi\otimes\bar\eta$ and $R_B(\xi\otimes\bar\eta)=\xi\otimes\bar{B^*\eta}$; since the rank-one operators span $S_2(H)$ densely, this decomposes every Hilbert–Schmidt operator into a singular-value expansion and reduces every computation to the rank-one case. The algebra generated by both families is the multiplication algebra $\mathcal{M}(H)$, the commutant of each one-sided family, equal to all of $B(S_2(H))$ in finite dimension.

Summary of Notation

Symbol	Meaning
$S_2(H)$	Hilbert–Schmidt class with $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$
$L_A(T)=AT$	left multiplication by $A$
$R_B(T)=TB$	right multiplication by $B$
$\\|L_A\\|=\\|A\\|$, $\\|R_B\\|=\\|B\\|$	isometric representations
$L_A^=L_{A^}$, $R_B^=R_{B^}$	adjoints for the Hilbert–Schmidt form
$L_AR_B=R_BL_A$	the two sides commute
$\xi\otimes\bar\eta$	rank-one operator, $x\mapsto\langle x,\eta\rangle\xi$
$L_A(\xi\otimes\bar\eta)=A\xi\otimes\bar\eta$	action on the first index
$R_B(\xi\otimes\bar\eta)=\xi\otimes\bar{B^*\eta}$	action on the second index
$\mathcal{M}(H)$	multiplication algebra, the commutant of each one-sided family