0$. The class $\mathcal{S}_q^{\gamma}=
\mathcal{S}_q^{\gamma}(\mathbb{R}^m\times \mathbb{R}^{m'},\mathfrak {K})$ consists of functions
$\mathbf{a}(y,{\eta}),\; (y,{\eta})\in \mathbb{R}^m\times \mathbb{R}^{m'}$,
such that for any $(y,{\eta})$, $\mathbf{a}(y,{\eta})$ is a bounded operator in
$\mathfrak{K}$ and, moreover,
\begin{gather}
\|D_{\eta}^\alpha D_y^{\beta} \mathbf{a}(y,{\eta})\|\le
C_{\alpha,{\beta}}(1+|{\eta}|)^{-|\alpha|+{\gamma}}, \label{e3.1} \\
|D_{\eta}^\alpha D_y^{\beta} \mathbf{a}(y,{\eta})|_{q\over
-{\gamma}+|\alpha|}\le C_{\alpha,{\beta}}.\label{e3.2}
\end{gather}
for $|\alpha|,|{\beta}|\le N$.
\end{definition}
Note here that for the case when $M$ is a $k$-dimensional
compact manifold and $a(y,z,{\eta},{{\zeta}})$ is a classical pseudodifferential
symbol of order less than ${\gamma}$ on $\mathbb{R}^m\times M$, the operator
valued symbol $\mathbf{a}(y,{\eta})=a(y,z,{\eta},D_z)$ acting in $\mathfrak{
K}=L_2(M)$ belongs to $\mathcal{S}_k^{\gamma}$ for any $N$. A more involved
example arises in the study of operators with discontinuous symbols.
Suppose that the symbol $a(y,z,{\eta},{{\zeta}})$ has compact support in $z$,
order ${\gamma}\le0$ positively homogeneous in $({\eta},{{\zeta}})$ (with a
certain smoothening near the point $({\eta},{{\zeta}})=0$), but near the
subspace $z=0$ it is positively homogeneous of order ${\gamma}$ in $z$ variable,
thus having a singularity at the subspace $z=0$. The operator symbol $
\mathbf{a}(y,{\eta})$ is a bounded operator in $\mathfrak{K}=L_2(\mathbb{R}^k)$,
differentiation in ${\eta}$ lowers the homogeneity order in
$({\eta},{{\zeta}})$, but the singularity in $z$ prevents it from acting into
usual Sobolev spaces (it is here the need for weighted Sobolev spaces arises).
However, in the terms of the Definition \ref{def3.1}, the properties of the operator
symbol are easily described: it belongs to $\mathcal{S}_q^{\gamma}$ for any $q>
k$. This example will be the basic one in considerations in Sect. 7.
The interpolation inequality $|\mathbf{a}|_q^q\le|\mathbf{a}|_p^p\|
\mathbf{a}\|^{q-p}$ for $p< q$ implies that for $-{\gamma}+|\alpha|-q>0$ the
derivatives in (\ref{e3.1}), (\ref{e3.2}) belong to trace class and for
$-{\gamma}+|\alpha|-q>m$ the integral of its trace class norm with respect to
${\eta}$ converges. The same holds for any $\mathfrak{s}_p$ - norm, provided
$|\alpha|$ is big enough. On the other hand, since
\begin{equation}
|\mathbf{a}{\mathbf{b}}|_{(p^{-1}+q^{-1})^{-1}}\le |\mathbf{a}|_p|
{\mathbf{b}}|_q,\label{e3.3}
\end{equation}
the product of symbols $\mathbf{a}\in
\mathcal{S}^{\gamma}_q$ and ${\mathbf{b}}\in\mathcal{S}^{\delta}_q$ belongs to $
\mathcal{S}^{{\gamma}+{\delta}}_q$.
For a symbol in $\mathbf{a}\in\mathcal{S}_q^{\gamma}$ and a function $f(\lambda)$
analytical in a sufficiently large domain in the complex plain, the symbol
$f(\mathbf{a})$ can be defined by means of the usual analytical functional
calculus for bounded operators. One can check directly that for any such $f$,
the symbol $f(\mathbf{a})$ belongs to $\mathcal{S}_q^0$; if, additionally,
$f(0)=0$, then $f(\mathbf{a})\in \mathcal{S}^{\gamma}_q$, moreover, if
$f(0)=f'(0)=\dots=f^{({\nu})}(0)=0$ then $f(\mathbf{a})\in
\mathcal{S}_q^{({\nu}+1){\gamma}}$. Thus, $\mathcal{S}^{{\gamma}}_q$ becomes a local
$*$-subalgebra in the algebra of bounded continuous operator-valued functions on
$\mathbb{R}^m\times \mathbb{R}^{m'}$.
We are going to sketch the operator-valued version of the usual
pseudodifferential calculus. The main difference of this calculus from the
usual one is the notion of 'negligible' operators. In the scalar case, one
considers as negligible the infinitely smoothing operators. In our case, we
take trace class operators as negligible, and it is up to a trace class error,
that the classical relations of the pseudodifferential calculus will be shown to
hold. This is sufficient for the needs of index theory.
Having a symbol $\mathbf{a}(y,y',{\eta})\in
\mathcal{S}_q^{\gamma}(\mathbb{R}^{2m}\times\mathbb{R}^m,\mathfrak{K})$, we define the
pseudodifferential operator with this symbol as
\begin{equation}
(OPS(\mathbf{a})u)(y)=(\mathbf{a}(y,y',D_y)u)(y)=(2\pi)^{-m}\int\int
e^{i(y-y'){\eta}}\mathbf{a}(y,y',{\eta}) u(y')d{\eta} dy',\label{e3.4}
\end{equation}
where
$u(y)$ is a function on $\mathbb{R}^m$ with values in $\mathfrak{K}$. In
particular, if $\mathbf{a}$ does not depend on $y'$, this is the usual formula
involving the Fourier transform:
\begin{equation}
\mathbf{a}(y,D_y)u=OPS(\mathbf{a})=
\mathcal{F}^{-1}\mathbf{a}(y,{\eta})\mathcal{F} u,\label{e3.5}
\end{equation}
Without any changes, on the
base of (\ref{e3.1}), the standard reasoning applied in the scalar case to give precise
meaning to (\ref{e3.4}), (\ref{e3.5}) defines the action of the operator $\mathbf{a}(y,D_y)$
on rapidly decaying smooth functions $u$ and establishes its boundedness in
$L_2$. We are going to show now is that the property (\ref{e3.2}) produces trace class
estimates.
The following proposition gives a sufficient condition for a pseudodifferential
operator to belong to trace class.
\begin{proposition} \label{prop3.2}
Let the operator-valued symbol $
\mathbf{a}(y,y',{\eta})$ in $\mathbb{R}^{2m}\times\mathbb{R}^m$ be smooth with respect
to $y,y'$, let all $y,y'$-derivatives $D_{y^{}}^{\beta} D_{y'}^{{\beta}'}
\mathbf{a}$ up to some (sufficiently large) order $N$ be trace class operators with
trace class norm bounded uniformly in $y,y'$. Suppose that
$g(y),h(y)=O((1+|y|)^{-2m})$. Then the operator $h\mathbf{a}(y,y',D_y)g$
belongs to $\mathfrak{s}_1(L_2(\mathbb{R}^m;\mathfrak{K}))$.
\end{proposition}
\paragraph{Proof} Suppose first that the functions $h,g$ have compact support in some
unit cubes $Q,Q'$. Take smooth functions $f,f'$ compactly supported in
concentric cubes with twice as large side such that $hf=h, gf'=g$. We can
represent our operator $hf\mathbf{a}(y,y',D_y)f'g=h\mathbf{a}(y,y',D_y)g$ in
the form
\begin{equation}
hf\mathbf{a}(y,y',D_y)f'g=(2\pi)^{-2m}\int\int
e^{iy{{\zeta}}+iy'{{\zeta}}'}h\mathbf{a}_{{{\zeta}},{{\zeta}}'}(D_y)g
d{{\zeta}} d{{\zeta}}' g,\label{e3.6}
\end{equation}
where $\mathbf{a}_{{{\zeta}},{{\zeta}}'}({\eta})=\int\int
e^{-i(y{{\zeta}}+y'{{\zeta}}')}f(y)\mathbf{a}(y,y',{\eta})f'(y')dydy'$. The
conditions imposed on the symbol $\mathbf{a}$ guarantee that the symbol
$\mathbf{a}_{{{\zeta}},{{\zeta}}'}({\eta})$ is a trace class operator for all
${\eta},{{\zeta}},{{\zeta}}'$, its trace class norm is in $L_1$ with respect to
${\eta}$ variable and decays rapidly at infinity in ${{\zeta}},{{\zeta}}'$. We
will use this to prove that for all ${{\zeta}},{{\zeta}}'$ the operator
$h\mathbf{a}_{{{\zeta}},{{\zeta}}'}(D_y)g$ belongs to the trace class and its
trace class norm decreases sufficiently fast as ${{\zeta}},{{\zeta}}'$ tend to
$\infty$. In order to do this, we factorize this operator into the product of
two Hilbert-Schmidt operators with rapidly decreasing Hilbert-Schmidt norm.
Recall that for a pseudodifferential operator with operator-valued symbol
$\mathbf{k}(y,{\eta})$, one has $|
{\mathbf{k}}(y,D_y)|_2^{2}=(2\pi)^{-m}\int\int|\mathbf{k}(y,{\eta})|_2^{2}dyd{\eta }$,
and, similarly for an operator with symbol $\mathbf{k}(y',{\eta})$. Represent
the symbol $\mathbf{a}_{{{\zeta}},{{\zeta}}'}({\eta})$ as the product $
{\mathbf{b}}_{{{\zeta}},{{\zeta}}'}({\eta}){\mathbf{c}}_{{{\zeta}},{{\zeta}}'}({\eta})$
where ${\mathbf{b}}_{{{\zeta}},{{\zeta}}'}({\eta})=|
\mathbf{a}_{{{\zeta}},{{\zeta}}'}({\eta})|^{1/2}$. The symbol $h(y)
{\mathbf{b}}_{{{\zeta}},{{\zeta}}'}({\eta})$ belongs to the Hilbert-Schmidt class $\mathfrak{
s}_2(\mathfrak{K})$ at any point $(y,{\eta})$, the Hilbert-Schmidt norm belongs to
$L_2$ in $(y,{\eta})$ variables and decays fast as as $({{\zeta}},{{\zeta}}')$
tend to infinity. Therefore, the operator $h(y)
{\mathbf{b}}_{{{\zeta}},{{\zeta}}'}(D_y)$ belongs to the Hilbert-Schmidt class, with norm
fast decaying in $({{\zeta}},{{\zeta}}')$. The same reasoning takes care of
${\mathbf{c}}_{{{\zeta}},{{\zeta}}'}g$. Thus the trace class norm of the
integrand on the right-hand side in (\ref{e3.6}) decays fast in
$({{\zeta}},{{\zeta}}')$, and this, after integration in ${{\zeta}},
{{\zeta}}'$, establishes the required property of $h\mathbf{a}(y,y',D_y)g$.
Note here, that the trace norm of the operator $h\mathbf{a}(y,y',D_y)g$ is estimated
by the $L_2$-norms of the functions $h,g$ over the cubes $Q,Q'$. To dispose of
the condition of $h,g$ to have compact support, we take a covering of the space
by a lattice of unit cubes $Q_j$ and define $h_j,g_j$ as restrictions of $h,g$
to the corresponding cube. Then the reasoning above can be applied to each of
the operators $h_j\mathbf{a}(y,y',D_y)g_{j'}$, and the series of trace class
norms of these operators converges. \hfill$\Box$\smallskip
\begin{remark} \label{rmk3.3} \rm
Note that we do not impose on the operator-valued symbol any smoothness
conditions in ${\eta}$ variable. This proves to be useful later, especially, in
Sect.7. A somewhat unusual presence of both functions $g,h$ (instead of just
one of them, as one might expect comparing with the scalar theory) is explained
by the fact that without smoothness conditions with respect to ${\eta}$, our
pseudodifferential operators are not necessarily pseudo-local in any reasonable
sense.
\end{remark}
\begin{remark} \label{rmk3.4} A special case where Proposition \ref{prop3.2} can be
used for establishing trace class properties is the one of the symbol $
\mathbf{a}$ decaying sufficiently fast in $y,y'$, together with derivatives, without
factors $g,h$. In fact, consider $\mathbf{a}=(1+|y|^2)^{-N}
{\mathbf{b}}(1+{|y'|}^2)^{-N}$, with $N$ large enough, and apply Proposition \ref{prop3.2} to the
symbol ${\mathbf{b}}$.
\end{remark}
If symbols belong to the classes $\mathcal{S}_q^{\gamma}$, the usual properties
and formulas in the pseudodifferential calculus hold, with our modification of
the notion of negligible operators.
\begin{theorem}[Pseudo-locality] \label{thm3.5}
Let the symbol $\mathbf{a}(y,y',{\eta})$ belong to $\mathcal{S}_q^{\gamma}(\mathbb{R}^{2n}\times\mathbb{R}^n)$ for some $q>0,{\gamma}\le0$,
let $h,g$ be bounded functions with disjoint supports, at least one of them
being compactly supported. Then (for $N$ large enough) the operator $h
{\mathbf{ a}}(y,y',D)g$ belongs to $\mathfrak{s}_1(L_2(\mathbb{R}^m;\mathfrak{K}))$, moreover,
$$
|h\mathbf{a}(y,y',D)g|_{\mathfrak{s}_1}\le C||g||_\infty ||h||_\infty
(1+d^{-N})\max\{C_{\alpha,\beta}; |\alpha|,|\beta|\le N\},
$$
where
$C_{\alpha,{\beta}}$ are constants in (\ref{e3.1}), (\ref{e3.2}) and $d={\operatorname{dist}}
({\operatorname{supp}}(g),{\operatorname{supp}}(h))$.
\end{theorem}
\paragraph{Proof}
First, let $h$ have compact support. Take two more bounded functions $h',g'\in
C^{\infty}$ with disjoint supports such that $\supp h'$ is compact,
$hh'=h,\;gg'=g$. Again represent the operator in question in the form
\begin{equation}
h\mathbf{a}(y,y',D_y)g=(2\pi)^{-m}\int e^{iy{{\zeta}}}h(y)
{\mathbf{ a}}_{{{\zeta}}}(y',D)g(y') d{{\zeta}} ,\label{e3.7}
\end{equation}
where $
{\mathbf{ a}}_{{{\zeta}}}(y',{\eta})=\int e^{iy{{\zeta}}}h'(y)
{\mathbf{ a}}(y,y',{\eta})g'(y')dy.$ We will show that the integrand in (\ref{e3.7}) belongs to
the trace class and its trace norm is integrable with respect to ${{\zeta}}$.
We have
\begin{equation}
(\mathbf{a}_{{{\zeta}}}(y',D) u)(y)=(2\pi)^{-m}\int\int
e^{i{\eta}(y-y')}h'(y)
{\mathbf{ a}}_{{\zeta}}(y',{\eta})g'(y')u(y')dy'd{\eta}.\label{e3.8}
\end{equation}
The first order
partial differential operator $L=L(D_{\eta})=-i|y-y'|^{-2}(y-y')D_{\eta} $ has
the property $Le^{i{\eta}(y-y')}=e^{i{\eta}(y-y')}$, so we can insert $L^N$
into (\ref{e3.8}) for any $N$. After integration by parts (first formal, but then
justified in the usual way), we obtain that (\ref{e3.7}) equals
$$
(2\pi)^{-m}\int\int
e^{i{\eta}(y-y')}h'(y)|y-y'|^{-2N}((y-y')D_{\eta})^N
{\mathbf{ a}}_{{\zeta}}(y',{\eta})g'(y')u(y')dy'd{\eta}.
$$
Since the supports of $h',g'$
are disjoint, the function
$$
h'(y)|y-y'|^{-2N}((y-y')D_{\eta})^N
{\mathbf{ a}}_{{\zeta}}(y',{\eta})g'(y')
$$
is smooth with respect to $y,y'$. By choosing
$N$ large enough, we can, using (\ref{e3.1}), (\ref{e3.2}), arrange it to belong to trace
class and have trace class norm decaying fast in $y',{\eta},{{\zeta}}$,
together with as many derivatives as we wish. Now, according to
Proposition \ref{prop3.2} (see Remark \ref{rmk3.4}), this implies that
the trace class norm of the operator
(\ref{e3.8}) decays fast in ${{\zeta}}$, and the result follows, together with the
estimate.
The same reasoning works if not $h$ but $g$ has a compact support, one just
makes the representation similar to (\ref{e3.7}), making Fourier transform in $y'$
variable. \hfill$\Box$\smallskip
The usual formula expressing the symbol of the composition of operators via the
symbols of the factors also holds in the operator-valued situation.
\begin{theorem} \label{thm3.6}
Let the symbols $\mathbf{a}(y,{\eta}),{\mathbf{b}}(y,{\eta})$ belong to $
\mathcal{S}_q^{\gamma}(\mathbb{R}^{2m}\times\mathbb{R}^m)$ for some $q>0,{\gamma}\le0$
and $h(y)=O((1+|y|)^{-m-1})$. Then, for $N$ large enough, the operator
$hOPS(\mathbf{a})OPS({\mathbf{b}})$ $ -hOPS({\mathbf{c}}_N)$ belongs to trace
class, where, as usual,
\begin{equation}
{\mathbf{c}}_N=\mathbf{a}\circ_N
{\mathbf{b}}=\sum_{|\alpha|< N}(\alpha!)^{-1}\partial_{\eta}^{\alpha}\mathbf{a}
D_y^{\alpha}{\mathbf{b}}.\label{e3.9}
\end{equation}
\end{theorem}
\paragraph{Proof} We follow the
standard way of proving the composition formula, however the remainder term will
be estimated by means of Proposition \ref{prop3.2}.
Suppose first that $h$ has a compact support in a unit cube $Q$. Take a
function $g\in C_0^\infty$ which is equal to $1$ in the concentric cube with
side 2 and vanishes outside the concentric cube with side 3. Set $
{\mathbf{b}}=g{\mathbf{b}}+(1-g){\mathbf{b}}={\mathbf{b}}'+{\mathbf{b}}''$. For the symbol
${\mathbf{b}}''$, we have $h\mathbf{a}\circ_N{\mathbf{b}}''=0$, at the same time,
$hOPS(\mathbf{a})OPS({\mathbf{b}}'')$ is trace class due to the pseudo-locality
property. Thus, $hOPS(\mathbf{a})OPS({\mathbf{b}}'')-hOPS(
\mathbf{a}\circ_N{\mathbf{b}}'')$ belongs to $\mathfrak{s}_1$, with trace class norm
controlled by the $L_\infty$ norm of $h$ in $Q$. Next, since $h
\mathbf{a}=hg\mathbf{a}$, we can assume that $\mathbf{a}$ has a compact support in
$y$.
We represent the operator ${\mathbf{b}}'(y,D)$ as the integral similar to (\ref{e3.6}):
$$
{\mathbf{b}}'(y',D)=\int e^{izy'}{\mathbf{b}}_{{\zeta}}'(D)dz,
$$
where
\begin{equation}
{\mathbf{b}}_{{\zeta}}'({\eta})=(2\pi)^{-m}\int e^{-iy'{{\zeta}}}
{\mathbf{b}}'(y',{\eta})dy'.\label{e3.10}
\end{equation}
Then the difference $hOPS(
\mathbf{a})OPS({\mathbf{b}}')-hOPS({\mathbf{c}}_N)$ can be written as
\begin{equation}
\int h(
\mathbf{a}(y,D) e^{izy'}
{\mathbf{b}}_{{\zeta}}'(D)-\sum_{|\alpha|