On some analytical index formulas related to operator-valued symbols (Q2778483)

In this very well written paper the author considers a class of pseudodifferential operators with operator-valued symbols and obtains analytical index formulas for the elliptic operators in the class. The author gives a careful justification for the introduction of pseudodifferential operators of the form NEWLINE\[NEWLINE\bigl(Op(a) u\bigr)(y)= (2\pi)^{-m}\int_{\mathbb R^m} \int_{\mathbb R^m} e^{i(y-y') \eta}a (y,y',\eta) u(y')\,d\eta\, dy',\tag{1}NEWLINE\]NEWLINE where the amplitude \(a(y,y',\eta)\) belongs to the class \({\mathcal S}_q^\gamma (\mathbb R^{2m} \times\mathbb R^m,H)\) and \(u\) is a function defined on \(\mathbb R^m\) with values in the Hilbert space \(H\).NEWLINENEWLINENEWLINEThat \(a(y,y',\eta)\) belongs to the class \({\mathcal S}_q^\gamma (\mathbb R^{2m}\times \mathbb R^m,H)\) means that \(a(y,y',\eta)\) is a \(C^N\) function defined on \(\mathbb R^{2m} \times\mathbb R^m\) with values in the linear and continuous operators on the Hilbert space \(H\), satisfying the estimates NEWLINE\[NEWLINE\bigl \|\partial^\alpha_\eta \partial^\beta_{yy'} a(y,y',\eta) \bigr\|\leq C_{\alpha, \beta}\bigl(1+ |\eta|\bigr)^{-|\alpha |+\gamma}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\bigl|\partial^\alpha_\eta \partial^\beta_{yy'} a(y,y', \eta)\bigr |_{q \over-\gamma +|\alpha |}\leq C_{\alpha \beta}NEWLINE\]NEWLINE where \(\|\cdot \|\) indicates the operator norm, \(\gamma\leq 0\), \(q>0\), \(|\cdot |_{q\over-\gamma +|\alpha |}\) is the \({q\over-\gamma+|\alpha |}\)-norm (or quasi-norm) of the sequence of singular numbers, and the estimates must hold for \(|\alpha|\) and \(|\beta|\leq N\) for a sufficiently large nonnegative integer \(N\).NEWLINENEWLINENEWLINEThe iterated integrals in (1) should be interpreted as an oscillatory integral. The author proves for this class relations between amplitudes and symbol, a relation between the symbol of a composition and the symbols of each factor, as well as the existence of parametrices for elliptic operators in the class. The author derives index formulas using the \(K\)-theory of operator algebras and cyclic cohomologies.NEWLINENEWLINENEWLINEThe author applies these formulas to obtain index formulas for other operators, such as Toeplitz operators with operator-valued symbols, and pseudodifferential operators related to manifolds with edge-type singularities, among others.