On the Green operator in relative elliptic theory.

*(English. Russian original)*Zbl 1125.58302
Dokl. Math. 68, No. 1, 57-60 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 391, No. 3, 306-309 (2003).

From the introduction: Let \((M,X)\) be a pair formed by a smooth compact closed manifold \(M\) and a smooth closed submanifold \(X \overset {i}{\hookrightarrow}M\).

The study of the Fredholm property leads to the study of equations on \(M\) of the form \((1+T)u=f\), where \(T\) is a Green operator associated with the pair \((M,X)\). This sort of equation is called Green equation of the second kind.

How can one study the Fredholm property of this equation? Green operators form a nonunital algebra \({\mathfrak G}\), and it is natural to treat \(1+T\) as an element of the algebra \({\mathfrak G}^+\) obtained from \({\mathfrak G}\) by attaching the unit. Then one can seek a two-sided almost inverse of \(1+T\) in the same algebra, that is, in the form \(1+T\), where \(T\) is also a Green operator. It turns out that \(T\) can be viewed as a pseudodifferential operator on the submanifold \(X\) with an operator-valued symbol. This operator-valued symbol, including the Fourier transform with respect to the normal variables, is a family of Fredholm integral operators on the fibers on the conormal bundle \(T^*X\setminus \{0\}\) minus the zero section.

The main theorem of this paper states that the Fredholm property of the Green equation is equivalent to ellipticity, i.e., to the unique solvability of the corresponding family of Fredholm integral equations of the second kind for the principal symbol of the almost inverse operator at all points \((x,p)\in T^*X\setminus\{0\}\). By way of application, we obtain a theorem on the elliptic conditions for relative morphisms in the general case.

The study of the Fredholm property leads to the study of equations on \(M\) of the form \((1+T)u=f\), where \(T\) is a Green operator associated with the pair \((M,X)\). This sort of equation is called Green equation of the second kind.

How can one study the Fredholm property of this equation? Green operators form a nonunital algebra \({\mathfrak G}\), and it is natural to treat \(1+T\) as an element of the algebra \({\mathfrak G}^+\) obtained from \({\mathfrak G}\) by attaching the unit. Then one can seek a two-sided almost inverse of \(1+T\) in the same algebra, that is, in the form \(1+T\), where \(T\) is also a Green operator. It turns out that \(T\) can be viewed as a pseudodifferential operator on the submanifold \(X\) with an operator-valued symbol. This operator-valued symbol, including the Fourier transform with respect to the normal variables, is a family of Fredholm integral operators on the fibers on the conormal bundle \(T^*X\setminus \{0\}\) minus the zero section.

The main theorem of this paper states that the Fredholm property of the Green equation is equivalent to ellipticity, i.e., to the unique solvability of the corresponding family of Fredholm integral equations of the second kind for the principal symbol of the almost inverse operator at all points \((x,p)\in T^*X\setminus\{0\}\). By way of application, we obtain a theorem on the elliptic conditions for relative morphisms in the general case.