The Signed Adjoint of the Left Multiplication on a Hilbert Space

Introduction

The signed left multiplication of the graded Hilbert space is the one-sided operator $\ell_U(T)=U\,\alpha(T)=L_U\circ\alpha$, the left multiplication with the argument twisted by the grade involution. With respect to the Hilbert–Schmidt form of the category its adjoint is again a signed left multiplication,

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

so the adjoint of the signed left multiplication is the signed left multiplication by the $\alpha$-adjoint of the parameter. The formula is the working example of the general rule that the adjoint of a twisted one-sided operator is the same one-sided operator with the parameter adjoined and the twist re-applied to the parameter; it differs from the unsigned case, where $(L_U)^*=L_{U^*}$ with no twist, exactly by the occurrence of $\alpha$. Because the signed left multiplication is injective in its parameter — its value at the identity is the parameter itself — the formula turns into exact criteria: $\ell_U$ is self-adjoint exactly when $\alpha(U^*)=U$, it is an isometry exactly when $U$ is, and it is unitary exactly when $U$ is. The present article computes the adjoint, the criteria it produces, and the way the two one-sided adjoints assemble the adjoint of the signed sandwich; the unsigned counterpart is The Adjoint of the Left Multiplication on a Hilbert Space, and the two-sided case is The Signed Adjoint Sandwich on a Hilbert Space.

This article fixes the Hilbert–Schmidt form, the adjoint of the signed left multiplication and its explicit value, the self-adjointness, isometry and unitarity criteria, the relation of the one-sided adjoint to the adjoint of the signed sandwich, and the comparison with the unsigned and the graded-module cases. The signed left multiplication and the signed right multiplication are The Signed Left Multiplication on a Hilbert Space; the unsigned adjoint is The Adjoint of the Left Multiplication on a Hilbert Space; the two-sided signed adjoint is The Signed Adjoint Sandwich on a Hilbert Space, and the reflection case is The Signed Adjoint of the Reflection on a Hilbert Space; the module-level statement is The Graded Adjoint Action on a Module over a Hilbert Space; the algebraic form is The Signed Adjoint of the Left Multiplication on a Graded Algebra (Part II).

Throughout, $H=H^0\oplus H^1$ is a graded Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with parity operator $\Gamma$, grade involution $\alpha(T)=\Gamma T\Gamma$, and Hilbert–Schmidt form $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$ on $S_2(H)$. The signed left and right multiplications are $\ell_U=L_U\circ\alpha$, $\ell_U(T)=U\alpha(T)$, and $\varrho_V=R_V\circ\alpha$, $\varrho_V(T)=\alpha(T)V$; the unsigned ones are $L_U(T)=UT$ and $R_V(T)=TV$.

The Adjoint of the Signed Left Multiplication

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

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

so the adjoint of a signed left multiplication is the signed left multiplication by the $\alpha$-adjoint parameter; the adjoint of the signed right multiplication is, by the same computation, $(\varrho_V)^*=\varrho_{\alpha(V^*)}$.

Proof. Write $\ell_U=L_U\alpha$; the adjoint of a composite reverses the order, $(\ell_U)^*=\alpha^*L_U^*$, the grade involution is self-adjoint for the Hilbert–Schmidt form, $\alpha^*=\alpha$, and $L_U^*=L_{U^*}$ by The Adjoint of the Left Multiplication on a Hilbert Space; hence $(\ell_U)^*=\alpha L_{U^*}$, which acts by $T\mapsto\alpha(U^*T)=\alpha(U^*)\alpha(T)=\ell_{\alpha(U^*)}T$. The signed right case is $\varrho_V=R_V\alpha$, with the same three ingredients.

Proposition (the explicit value and the involution). The involution of the parameter that appears in the adjoint is the composite of the Hilbert adjoint and the grade involution,

$$ U\ \longmapsto\ \alpha(U^*)=\alpha(U)^* , $$

which is an involutive antiautomorphism of $B(H)$, $\alpha((UV)^*)=\alpha(U^*)\alpha(V^*)$ reversed as $\alpha(V^*)\alpha(U^*)$; the map $\ell_U\mapsto(\ell_U)^*$ is therefore an involution on the family of signed left multiplications.

Proof. $\alpha$ is multiplicative and the adjoint reverses the product, so $\alpha((UV)^*)=\alpha(V^*U^*)=\alpha(V^*)\alpha(U^*)$; the square of $U\mapsto\alpha(U^*)$ is the identity because both $\alpha$ and the adjoint are involutions and they commute, $\alpha(U^*)^*=\alpha(U)$.

Corollary (contrast with the unsigned case). For the unsigned left multiplication the adjoint is $(L_U)^*=L_{U^*}$; for the signed one it is $(\ell_U)^*=\ell_{\alpha(U^*)}$. The two agree exactly when $\alpha(U^*)=U^*$, that is, on the parameters fixed by the grade involution, and the discrepancy is the twist of the parameter by $\alpha$.

Proof. The unsigned identity is the previous article; the signed one is the theorem; comparing the two, $\ell_{\alpha(U^*)}=\ell_{U^*}$ iff $\alpha(U^*)=U^*$ by the injectivity of the parameter, which is the fixed-point condition.

Self-Adjointness, Isometry and Unitarity

Theorem (self-adjointness). The signed left multiplication is self-adjoint,

$$ (\ell_U)^*=\ell_U, $$

if and only if

$$ \alpha(U^*)=U . $$

In particular $\ell_U$ is self-adjoint when $U$ is self-adjoint and even, and it is never self-adjoint for a nonzero odd $U$.

Proof. The family is faithful: $\ell_A(I)=A\alpha(I)=A$, so $\ell_A=\ell_C$ iff $A=C$; the criterion is therefore the equality $\ell_{\alpha(U^*)}=\ell_U$, that is $\alpha(U^*)=U$. If $U$ is self-adjoint and even, $\alpha(U^*)=\alpha(U)=U$; if $U$ is odd and nonzero then $\alpha(U^*)=\alpha(U)^*=-U^*$, and $-U^*=U$ would make $U$ skew-adjoint and odd, contradicting $U^*=-\alpha(U)=\alpha(-U)$, so no nonzero odd $U$ satisfies the criterion.

Theorem (isometry and unitarity). The signed left multiplication is an isometry of $S_2(H)$ if and only if $U$ is an isometry, and it is unitary if and only if $U$ is unitary; more precisely

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

so $\ell_U$ is isometric exactly when $\alpha(U^*U)=I$, which is $U^*U=I$ because $\alpha$ is injective, and it is unitary exactly when in addition $UU^*=I$.

Proof. By the composition rules of the signed left multiplication, $\ell_{\alpha(U^*)}\ell_U=L_{\alpha(U^*)\alpha(U)}=L_{\alpha(U^*U)}$, and $\ell_U\ell_{\alpha(U^*)}=L_{U\alpha(\alpha(U^*))}=L_{UU^*}$ because $\alpha^2=\mathrm{id}$; the identity of $S_2(H)$ is $L_I$, and $L_{A}=L_I$ iff $A=I$; the injectivity of $\alpha$ identifies $\alpha(U^*U)=I$ with $U^*U=I$.

Corollary (partial isometries and the absence of projections). The signed left multiplication is a partial isometry exactly when $U$ is a partial isometry, its initial and final projections being $L_{\alpha(U^*U)}$ and $L_{UU^*}$; it is never an orthogonal projection, since $\ell_U^2=L_{U\alpha(U)}$ equals $\ell_U$ only for $U=I$ and $\ell_I=\alpha$ is not idempotent. The range of $\ell_U$ is $U\,B(H)$ and its kernel is $\alpha(\ker U)$.

Proof. The two factors $(\ell_U)^*\ell_U$ and $\ell_U(\ell_U)^*$ are $L_{\alpha(U^*U)}$ and $L_{UU^*}$, which are projections exactly when $UU^*$ and $\alpha(U^*U)$ are, and this is the partial-isometry condition for $\ell_U$ and, $\alpha$ preserving projections, for $U$. For the idempotence, $\ell_U^2=\ell_{U\alpha(U)}=L_{U\alpha(U)}$ and $L_{U\alpha(U)}=L_U$ iff $\alpha(U)=I$ iff $U=I$, while $\ell_I=\alpha$ has $\alpha^2=\mathrm{id}\neq\alpha$; the kernel and range statements are those of The Signed Left Multiplication on a Hilbert Space.

The One-Sided Adjoints and the Sandwich

Proposition (the adjoint of the sandwich from the one-sided adjoints). The signed sandwich factors as $S_{A,B}=\ell_AR_{\alpha(B)}=L_A\varrho_B$, and its adjoint is the composite of the two one-sided adjoints in the reverse order,

$$ (S_{A,B})^*=R_{\alpha(B)}^*\ell_A^*=R_{\alpha(B^*)}\ell_{\alpha(A^*)}, $$

which evaluates to $\alpha(A^*)\,\alpha(\cdot)\,\alpha(B^*)$ and is the closed form $S_{\alpha(A^*),\alpha(B^*)}$; the one-sided adjoint of the present article is therefore exactly the left factor of the sandwich adjoint.

Proof. The adjoint of a product reverses the order, giving $R_{\alpha(B)}^*\ell_A^*$; the one-sided adjoints are $R_{\alpha(B)}^*=R_{\alpha(B^*)}$ and $\ell_A^*=\ell_{\alpha(A^*)}$; the evaluation is $X\mapsto\alpha(A^*)\alpha(X)\alpha(B^*)$, which is the signed sandwich with parameters $\alpha(A^*)$ and $\alpha(B^*)$, in agreement with The Signed Adjoint Sandwich on a Hilbert Space.

Corollary (the two-sided family is closed and generated). The adjunction maps the signed left multiplications to signed left multiplications, the signed right multiplications to signed right multiplications, and the signed sandwiches to signed sandwiches; the algebra generated by the signed left and right multiplications is a $*$-algebra of operators on the Hilbert–Schmidt space, and its adjoint closure is the whole of it.

Proof. The three closure statements are the three adjoint formulas; an algebra closed under adjunction and under products is a $*$-algebra, and the closure statement is the definition read on the generating families.

Example (the rank-one computation). With $\alpha(\xi\otimes\bar\eta)=\Gamma\xi\otimes\overline{\Gamma\eta}$, the signed left multiplication acts on the rank-one operator by $\ell_U(\xi\otimes\bar\eta)=U\Gamma\xi\otimes\overline{\Gamma\eta}$ and the adjoint acts by $\ell_{\alpha(U^*)}(\xi\otimes\bar\eta)=\alpha(U^*)\Gamma\xi\otimes\overline{\Gamma\eta}$; the pairing of two rank-one operators reproduces the identity $\langle\ell_UT,S\rangle_{\mathrm{HS}}=\langle T,\ell_{\alpha(U^*)}S\rangle_{\mathrm{HS}}$, which is the theorem on the rank-one generators.

Examples and Degenerate Cases

Proposition (the degenerate case $\alpha=\mathrm{id}$). If the grading is trivial then $\ell_U=L_U$ and the adjoint theorem reduces to $(L_U)^*=L_{U^*}$; the self-adjointness criterion reduces to $U^*=U$, the isometry criterion to $U^*U=I$, and the signed theory is the unsigned theory of The Adjoint of the Left Multiplication on a Hilbert Space.

Proof. With $\alpha=\mathrm{id}$ the map $\ell_U$ is $L_U$ by definition, and every formula of the article reduces to the corresponding unsigned one; the reduction of the criteria is the substitution $\alpha=\mathrm{id}$.

Example (the diagonal graded case). When $H$ has a homogeneous orthonormal basis and $U$ is diagonal with entries $u_m$, the signed left multiplication has the eigenvalues $\varepsilon_m\varepsilon_nu_m$ on the matrix units $e_m\otimes\bar e_n$, with $\varepsilon_m=(-1)^{|e_m|}$; the adjoint has the conjugate eigenvalues $\varepsilon_m\varepsilon_n\bar u_m$, and the self-adjointness criterion $\alpha(U^*)=U$ reads $\bar u_m=u_m$ for a diagonal parameter, that is, the reality of the diagonal entries.

Example (the finite-dimensional sign count). For $H=\mathbb{K}^{p+q}$ with the diagonal grading, an even self-adjoint $U=\operatorname{diag}(A_0,A_0')$ satisfies $\alpha(U^*)=U$ and gives a self-adjoint $\ell_U$, while an odd self-adjoint $U$ gives $\alpha(U^*)=\alpha(U)=-U\neq U$ and never does; the exact set of self-adjoint signed left multiplications is the set of parameters with $\alpha(U^*)=U$, of which the even self-adjoint ones are the standard members.

Summary

With respect to the Hilbert–Schmidt form, the signed left multiplication $\ell_U=L_U\alpha$ has adjoint $(\ell_U)^*=\ell_{\alpha(U^*)}$, where $\alpha(U^*)=\alpha(U)^*$ is the composite of the Hilbert adjoint with the grade involution; the signed right multiplication has adjoint $(\varrho_V)^*=\varrho_{\alpha(V^*)}$, so each signed one-sided family is closed under adjunction and stays on its side. Because the family is faithful — $\ell_U$ determines $U$ as its value at the identity — the adjoint formula gives exact criteria: $\ell_U$ is self-adjoint exactly when $\alpha(U^*)=U$, an isometry exactly when $U$ is, unitary exactly when $U$ is, and a partial isometry exactly when $U$ is; in particular a nonzero odd parameter never gives a self-adjoint signed left multiplication. The signed sandwich factors as $S_{A,B}=\ell_AR_{\alpha(B)}=L_A\varrho_B$, and its adjoint is assembled from the two one-sided adjoints in the reverse order, giving the closed form $S_{\alpha(A^*),\alpha(B^*)}$; the algebra generated by the signed left and right multiplications is therefore a $*$-algebra on the Hilbert–Schmidt space. When the grading is trivial the article collapses to The Adjoint of the Left Multiplication on a Hilbert Space, and the module-level form of the same computation is The Graded Adjoint Action on a Module over a Hilbert Space.

Summary of Notation

Symbol	Meaning
$\ell_U(T)=U\alpha(T)=L_U\circ\alpha$	the signed left multiplication
$\varrho_V(T)=\alpha(T)V=R_V\circ\alpha$	the signed right multiplication
$(\ell_U)^=\ell_{\alpha(U^)}$	the adjoint of the signed left multiplication
$(\varrho_V)^=\varrho_{\alpha(V^)}$	the adjoint of the signed right multiplication
$\alpha(U^)=\alpha(U)^$	the adjoint parameter
$\alpha(U^*)=U$	self-adjointness criterion
$(\ell_U)^\ell_U=L_{\alpha(U^U)}$	the isometry computation
$(S_{A,B})^=R_{\alpha(B^)}\ell_{\alpha(A^*)}$	the sandwich adjoint from the one-sided adjoints
$\ell_A(I)=A$	faithfulness of the parameter
$\alpha=\mathrm{id}$	the degenerate case, signed equals unsigned