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Green’s formula for right invertible operators. (English) Zbl 0534.47019

Let X be a D-algebra, i.e. a commutative algebra with a right invertible operator D satisfying the condition: If \(x,y\in dom D\) then \(xy\in D.\) Let \(F_{\alpha},F_{\beta}\) (\(\beta\neq \alpha)\) be initial operators for D corresponding to its right inverses \(R_{\alpha},R_{\beta}\), respectively. Let \(Q(D)=\sum^{N}_{k=0}Q_ kD^ k,\quad Q^+(D)=\sum^{N}_{k=0}(-1)^ kD^ kQ_ k,\) where \(Q_ 0,...,Q_ N\) are linear operators defined on dom \(D^ N\), \(N\geq 1\) and with ranges in X. Then for all \(x,y\in dom D^ N\) the following identity holds: \[ F_{\beta}R_{\alpha}[xQ(D)y-yQ^+(D)x]= \]
\[ \sum^{N}_{j=0}(- 1)^ jc_ D^{-(j+1)}(F_{\beta}-F_{\alpha})D^ j(yQ_{j+1}x)+F_{\beta}R_{\alpha}[g_{Q(D)}(x,y)+h_{Q(D)}(x,y)] \] where \[ g_{Q(D)}(x,y)=\sum^{N}_{k=1}((-1)^ kc_ D^{-k}f_ D^{(k)}(Q_ kx,y)+[1+(-1)^ k](Q_ kx)D^ ky)+yQ_ 0x, \]
\[ h_{Q(D)}(x,y)=\sum^{N}_{k=0}[xQ_ k-Q_ kx]D^ ky,\quad f_ D^{(1)}(x,y)=D(x,y)-c_ D(xDy+yDx) \] and for \(k\geq 2\) \[ f_ D^{(k)}(x,y)=c_ D^ k[(Dx)(D^{k-1}y)+(D^{k-1}x)(Dy)]+ \]
\[ +c_ D[f_ D^{(k-1)}(x,Dy)+f_ D^{(k-1)}(Dx,y)]+D^{k-1}f_ D(x,y). \] \(f_ D^{(1)}\) is said to be a non-Leibniz component for D and \(c_ D\neq 0\) is a scalar dependent on D only. The formula for \(f_ D^{(k)}\) here is corrected (with respect to the paper, where there are two misprintings, without any influence for further proofs). If \(Q_ kx=xQ_ k\) \((k=0,1,...,N)\) then \(h_{Q(D)}=0\). There are given examples of applications to the Picard and Cauchy problems for hyperbolic equations.

MSC:

47C05 Linear operators in algebras
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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