and (4), $P$ is taken to be even-even with $m$ even. In all the cases (1)–(4), $\operatorname{res}A=0$, there is no contribution to $(-\lambda)^{-N}$ from the first sum in (1.1), and $$\tilde c''_0=\int\slint \tr a(x,\xi )\,\d\xi dx,\tag1.3$$ where $\tslint \tr a(x,\xi )\,\d\xi $ is a finite part integral (explained in detail in Section 3 below).

When $\nu \in\Bbb Z $, we say that $A$ or $a$ has [*even-even*]{} alternating parity (in short: is even-even), when the symbols with even (resp. odd) degree $\nu -j$ are even (resp. odd) in $\xi $: $$a_{\nu -j}(x,-\xi )=(-1)^{\nu -j}a_{\nu -j}(x,\xi ) \text{ for }|\xi
|\ge 1,\tag1.4$$ and the derivatives in $x$ and $\xi $ likewise have this property. $A$ and $a$ are said to have [*even-odd*]{} alternating parity in the reversed situation, where the symbols with even (resp. odd) degree $\nu -j$ are odd (resp. even) in $\xi $: $$a_{\nu -j}(x,-\xi )=(-1)^{\nu -j-1}a_{\nu -j}(x,\xi ) \text{ for
}|\xi |\ge 1,\tag1.5$$ etc. These properties are preserved under coordinate changes. For brevity, we shall say that $A$ (or $a$) [*has a parity that fits with the dimension $n$*]{}, when (3) or (4) holds.

[@KV] only considered the cases (1)–(3); the operators satisfying (3) were called odd-class operators. 