I am studying Yosida's Functional Analysis book and I am having a bit of trouble to understand the definitions of differential and elliptic operators on a manifold (Chap XIV, Sec 2). The relevant definitions are:
Definition 1: A 2nd order (linear) differential operator $A$ on a compact manifold $M$ is, in local coordinates, expressed by $$A = a^{ij}\frac{\partial^2}{\partial x_i\partial x_j} + b^j\frac{\partial}{\partial x_j},$$ where the coefficients are smooth, real-valued and satisfy the transformation rule $$\hat a^{kl} = a^{ij}\frac{\partial \hat x_k}{\partial x_i}\frac{\partial \hat x_l}{\partial x_j}\quad\text{ and }\quad \hat b^k = a^{ij}\frac{\partial^2 \hat x_k}{\partial x_i\partial x_j} + b^j\frac{\partial \hat x_k}{\partial x_j}$$ by the coordinate change $(x_1,\ldots,x_N)\mapsto(\hat x_1,\ldots,\hat x_N)$.
Definition 2: The operator $A$ above is said to be strictly elliptic if there is some constant $c > 0$ such that $$a^{ij}(x)\xi_i\xi_j\geq c|\xi|^2,\quad\forall x\in M,\,\forall \xi\in\mathbb{R}^N.$$
Then, my questions are:
I can see why the transformation rules make sense, but why is the definition of strictly elliptic operator independent of the choice of coordinate system? It seems to me that it should have something to do with the coordinate change preserving orientation, but I could not prove it.
When treating differential operators in $\mathbb{R}^N$, Def 1 contains an order 0 term but, curiously, in this case it does not. Why is that the case? It seems to me that the theory would work the same if there should be a constant term.
I think I now understand why (2) is necessary, it is way of making the main proof simpler. When studying these (strictly elliptic without constant coefficient) differential operators, Yosida's goal is to show that they generate a $C_0$ semigroup of contractions by means of the Lumer-Phillips theorem. This means that we have to check that $$\left\|I-\frac{1}{\mu}A\right\|\geq 1,\quad\forall \mu > 0.$$ The book then proceeds with the proof by considering $$ A:C^{\infty}(M)\subseteq C(M)\to C(M),$$ where $C(M)$ is equipped with the sup norm a usual, takes a function $f\in C^{\infty}(M)$ and lets $x_0$ be a maximum point of $f$. Then, by carefully choosing a system of coordinates at $x_0$ where $${(a^{ij}(x_0))} = \text{diag}\,\{\lambda_1,\ldots,\lambda_N\},\quad\lambda_j\geq 0,$$ (we can do this by using our strict ellipticity hypothesis), we compute $$\left(I-\frac 1\mu A\right)f (x_0) = f(x_0) -\frac 1\mu b^j(x_0)\frac{\partial f}{\partial x_j}(x_0) - \frac 1\mu\lambda_j\frac{\partial^2 f}{\partial x_j^2}(x_0)\geq f(x_0) = \|f\|,$$ for the first order terms vanish and the second order derivatives are all negative at a maximum point. Taking the supremum over $x\in M$ on the left hand side yields the desired estimate.
Finally, note how on the computation above, if $A$ had a constant term, we would not be able to estimate $$f(x_0) -c(x_0)f(x_0)- \frac 1\mu\lambda_j\frac{\partial^2 f}{\partial x_j^2}(x_0)$$ as easily as we did above.