Is $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ by itself a valid operator?

Question

If I simply consider just one of the combinations $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ both of them take from $\Omega^r(M)\to \Omega^r(M)$ for some manifold $M$. But do they make sense as operators individually or do they only make sense when they come together as the Laplacian $\Delta = \text{d}^{\dagger}\text{d} + \text{d}\text{d}^{\dagger}$? I have not yet seen them play any role individually in any literature as such, hence, my query. Can anyone point out any application of these individual operators in the literature?

Answer 1

Let $M$ be a Riemannian manifold, then the exterior differential $\mathrm d$ and its adjoint $\mathrm{d}^*$ are both $\mathbb R$-vector space endomorphisms on $\Omega(M)$ - i.e. elements of $\mathrm{End}_{\mathbb{R}}(\Omega(M))$ - and hence the compositions $\mathrm{d}^*\mathrm{d}$ and $\mathrm{d}\mathrm{d}^*$ are both well-defined linear operators. See below for the definition of the adjoint and some references.

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Let $M$ be a Riemannian manifold, then we can define $\mathrm{d^*}\in\mathrm{End}_{\mathbb{R}}(\Omega(M))$ to be the adjoint of $\mathrm{d}$, i.e. the unique operator such that $$\int_M\langle \mathrm d\alpha,\beta\rangle(x)|\mathrm dx|=\int_M\langle \alpha,\mathrm d^*\beta\rangle(x)|\mathrm dx|$$ for all $\alpha,\beta\in\Omega_0(M)$, where $\Omega_0(M)\subseteq\Omega(M)$ is the space of compactly supported differential forms and $x\mapsto|\mathrm dx|$ is the Riemannian density. See page 68 in Heat Kernels and Dirac Operators. Note that if $M$ is oriented, then $\mathrm d^*$ can be equivalently defined in terms of the Hodge star operator, see Definition 7.2.9 and Theorem 7.2.10 in Mathematical Gauge Theory. (In the second reference $\mathrm d^*$ is called the codifferential.)