The Signed Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions
Introduction
The signed left multiplication is the operator $T_a=L_a\alpha$, the composition of the left multiplication by $a$ with the grade involution; it is the basic odd operator of the algebra of arithmetic functions. Its adjoint for the coefficient form, the signed adjoint of the left multiplication, is $$ T_a^{\alpha\dagger}=T_a^*=\alpha\,\Theta_a=\Theta_{\alpha(a)}\,\alpha , $$ the composition of the transposed multiplication by $\alpha(a)$ with the grade involution. This article computes the signed adjoint, its products and its square, the self-adjointness and unitarity criteria, and the comparison with the left multiplication by $\delta(a)$, which fails exactly as in the non-graded case. The signed left multiplication itself is The Signed Left Multiplication on the Algebra of Arithmetic Functions; the adjoint of the plain left multiplication is The Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions, and the two-sided version is The Signed Adjoint Sandwich on the Algebra of Arithmetic Functions. Nothing here reads a distance as an object.
The Signed Adjoint
Definition and closed form
Definition. For $a\in\mathcal{A}$ the signed left multiplication is $T_a=L_a\alpha$, and its signed adjoint is $$ T_a^{\alpha\dagger}=T_a^*,\qquad T_a^{\alpha\dagger}=\alpha\,\Theta_a=\Theta_{\alpha(a)}\,\alpha . $$
Theorem. With respect to the coefficient form, $$ \langle T_af,g\rangle=\sum_n\Bigl(\sum_{mk=n}a(m)\lambda(k)f(k)\Bigr)\overline{g(n)}=\langle f,\alpha\Theta_ag\rangle , $$ so that $T_a^{\alpha\dagger}=\alpha\Theta_a=\Theta_{\alpha(a)}\alpha$, with the explicit form $$ \bigl(T_a^{\alpha\dagger}g\bigr)(k)=\lambda(k)\sum_{m\ge1}\overline{a(m)}\,g(mk)=\sum_{m\ge1}\lambda(m)\overline{a(m)}\,\lambda(k)\,g(mk). $$ The signed adjoint is anti-linear in $a$; it is the adjoint of the adjoint, $(T_a^{\alpha\dagger})^*=T_a$; and it is never of the form $T_b$ for a nondegenerate $a$.
Proof. Insert $T_a=L_a\alpha$, use $\alpha^*=\alpha$ and $L_a^*=\Theta_a$ from The Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions, and the identity $\alpha\Theta_a=\Theta_{\alpha(a)}\alpha$ of the transposed multiplication. The anti-linearity is the conjugation in $\Theta_a$.
Products and square
Theorem. The signed adjoints satisfy $$ T_a^{\alpha\dagger}T_b^{\alpha\dagger}=\Theta_{\alpha(a)}\Theta_{\alpha(b)}=\Theta_{\alpha(a)*\alpha(b)}=\Theta_{\alpha(a*b)},\qquad \bigl(T_a^{\alpha\dagger}\bigr)^2=\Theta_{\alpha(a)*\alpha(a)}, $$ and the products with the signed left multiplication are $$ T_a^{\alpha\dagger}T_b=\alpha\Theta_aL_b\alpha=\Theta_{\alpha(a)}L_b,\qquad T_aT_b^{\alpha\dagger}=L_a\Theta_{\alpha(b)} , $$ so the signed adjoints close under composition, forming the transposed algebra with the elements $\alpha(a)$.
Proof. The products are computed by inserting the definitions and cancelling $\alpha^2=\mathrm{id}$; the multiplicativity $\Theta_c\Theta_d=\Theta_{c*d}$ of the transposed multiplication gives the first display. This is the transposed-algebra computation of The Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions with the twist $\alpha$.
Corollary (self-adjointness and unitarity). The signed left multiplication is self-adjoint exactly for the real scalar multiples of the identity, $$ T_a=T_a^{\alpha\dagger}\iff L_{\alpha(a)}=\Theta_a\iff a=\gamma\varepsilon,\ \gamma\in\mathbb{R}, $$ and it is unitary exactly for the unitary scalar multiples of the identity, $$ T_a\ \text{unitary}\iff \alpha\Theta_aL_a\alpha=L_a\Theta_a=\mathrm{id}\iff a=\gamma\varepsilon,\ |\gamma|=1 . $$ It is an isometry exactly when $\Theta_aL_a=\mathrm{id}$, which holds for $a=\gamma\delta_p$, $|\gamma|=1$.
Proof. Test the self-adjointness on $\delta_1$: $T_a\delta_1=\alpha(a)$ and $T_a^{\alpha\dagger}\delta_1=\alpha\Theta_a\delta_1=\lambda(1)\overline{a(1)}\delta_1=\overline{a(1)}\delta_1$, so equality forces $a=\gamma\varepsilon$, and then $T_a=\gamma\alpha$, $T_a^{\alpha\dagger}=\overline\gamma\alpha$, equal exactly for $\gamma$ real; the unitarity is the product identity, as in The Signed Left Multiplication on the Algebra of Arithmetic Functions.
The Comparison with the Involution
Theorem (comparison). The expected formula in which the signed adjoint is the signed left multiplication of the transformed element, $$ T_a^{\alpha\dagger}=T_{\delta(a)}\quad\text{with}\quad\delta=\sigma\alpha, $$ does not hold over the algebra of arithmetic functions; the right side is $T_{\delta(a)}=L_{\delta(a)}\alpha=\alpha L_{\alpha(\delta(a))}\alpha$ and the left side is $\alpha\Theta_a$, and the two differ whenever $a$ is not a scalar multiple of the identity. The signed adjoint is a transposed operation on the elements, not a multiplication by a transformed element.
Proof. The right side is $T_{\delta(a)}=L_{\delta(a)}\alpha=\alpha L_{\alpha(\delta(a))}$, a left multiplication conjugated by $\alpha$, while the left side is $\alpha\Theta_a$; the two coincide exactly when $\Theta_a=L_{\alpha(\delta(a))}$, which by the criterion of the corollary forces $a=\gamma\varepsilon$. This is the graded analogue of the obstruction of The Sandwich on the Algebra of Arithmetic Functions.
Worked Examples
Example ($a=\varepsilon$). $T_\varepsilon=\alpha$ and $T_\varepsilon^{\alpha\dagger}=\alpha$; the grade involution, self-adjoint and unitary.
Example ($a=\delta_p$). $T_{\delta_p}^{\alpha\dagger}=\alpha\Theta_{\delta_p}=\alpha S_p$, the signed adjoint of the isometry $\alpha D_p$; the product $T^{\alpha\dagger}T=\alpha S_pD_p\alpha=P_p$ is the projection onto the multiples of $p$.
Example ($a=\mathbf 1$). $T_{\mathbf 1}^{\alpha\dagger}=\alpha\Theta_{\mathbf 1}$; the operator is unbounded on the finitely supported functions, and its square is $\Theta_{\alpha(\mathbf 1)*\alpha(\mathbf 1)}=\Theta_{\lambda*\lambda}$.
Failure of the Degenerate Cases
The signed adjoint of the left multiplication fails in four degenerate configurations. First, it is not a signed left multiplication, so the class of the signed left multiplications is not closed under adjunction; the adjoints form the transposed class. Second, the self-adjointness and the unitarity hold only at the scalar elements, so the graded operators that are self-adjoint are the degenerate ones. Third, the transposed multiplication is unbounded for a general element and the adjoint requires a weighted space for its domain. Fourth, the involution formula $T_a^{\alpha\dagger}=T_{\delta(a)}$ fails in the same way as the formula $L_a^*=L_{a^*}$ fails for the plain left multiplication; the only form restoring it is the rank-one augmentation form, on which the whole operator theory collapses. These are the boundary cases of the signed adjoint.
Summary
The signed left multiplication $T_a=L_a\alpha$ has the signed adjoint $T_a^{\alpha\dagger}=\alpha\Theta_a=\Theta_{\alpha(a)}\alpha$ for the coefficient form, anti-linear in $a$, with $(T_a^{\alpha\dagger})^*=T_a$, products $T_a^{\alpha\dagger}T_b^{\alpha\dagger}=\Theta_{\alpha(a*b)}$ and square $\Theta_{\alpha(a)*\alpha(a)}$. It is self-adjoint exactly for $a=\gamma\varepsilon$ with $\gamma$ real, unitary exactly for $|\gamma|=1$, and an isometry exactly when $\Theta_aL_a=\mathrm{id}$; the involution formula $T_a^{\alpha\dagger}=T_{\delta(a)}$ fails outside the scalars, because the adjoint is a transposed operation and not a multiplication. The degenerate cases are the non-closure of the class, the scalar self-adjointness, the unboundedness and the rank-one augmentation form.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $T_a=L_a\alpha$ | Signed left multiplication |
| $T_a^{\alpha\dagger}=\alpha\Theta_a=\Theta_{\alpha(a)}\alpha$ | Signed adjoint |
| $(T_a^{\alpha\dagger}g)(k)=\lambda(k)\sum_m\overline{a(m)}g(mk)$ | Explicit form |
| $(T_a^{\alpha\dagger})^*=T_a$ | Adjoint of the adjoint |
| $T_a^{\alpha\dagger}T_b^{\alpha\dagger}=\Theta_{\alpha(a*b)}$ | Product |
| $(T_a^{\alpha\dagger})^2=\Theta_{\alpha(a)*\alpha(a)}$ | The square |
| $a=\gamma\varepsilon$, $\gamma\in\mathbb{R}$ | Self-adjointness |
| $a=\gamma\varepsilon$, $|\gamma|=1$ | Unitarity |
| $\delta=\sigma\alpha$ | The composite involution |
Further Reading
- Sterling Berberian, Introduction to Hilbert Space (Oxford University Press, 1961), for the adjoints of the products of operators.
- Paul Halmos, A Hilbert Space Problem Book (Springer, 1982), for the self-adjoint and unitary operators.
- Israel Gohberg and Mark Krein, Theory of Volterra Operators in Hilbert Space (American Mathematical Society, 1970), for the transposed convolution operators.
- Tsit Yuen Lam, A First Course in Noncommutative Rings (Springer, 2001), for the involution and the transformed elements.
- Hugh Montgomery and Robert Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, 2007), for the arithmetic convolution.