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The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

Here are a few details. Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a connection $\nabla$ on $E$ compatible with the metric on $E$ such that

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.

The unique continuation is valid for generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

The operators you mentioned are such generalized Laplacians. Here are a few details. 

Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

A second order partial differential operator $L$ is called a generalized Laplacian if its principal symbol satisfies

$$ \sigma_L(\xi)=-|\xi|_g^2\cdot \mathbf{1}_{E_x},\;\;\forall x\in M,\;\;\forall \xi\in T^*_xM. $$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ has the form

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

for some a connection $\nabla$ on $E$ compatible with the metric on $E$ and an endomorphism $T$ of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.

The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

Here are a few details. Suppose that $E$ is a metric vector bundle ver the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatibvle with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a conection $\nabla$ on $E$ compatible with the metric on $E$ such that

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.

The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

Here are a few details. Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a connection $\nabla$ on $E$ compatible with the metric on $E$ such that

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.

The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

Suppose that $E$ is a metric vector bundle ver the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatibvle with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a conection $\nabla$ on $E$ compatible with the metric on $E$ such that

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$ then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtian the unique continuation.

The unique continuation is valid for generalized Laplacians. The operators you mentioned are such generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

Here are a few details. Suppose that $E$ is a metric vector bundle ver the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatibvle with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ then there exists a conection $\nabla$ on $E$ compatible with the metric on $E$ such that

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

where $T$ is an endomorphism of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.