AN INDEX FORMULA FOR ELLIPTIC-SYSTEMS IN THE PLANE

An index formula is proved for elliptic systems of P.D.E.'s with bound ary values in a simply connected region Omega in the plane. Let A deno te the elliptic operator and B the boundary operator. In an earlier pa per by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the bo undary, a square matrix function Delta(B)(+) defined on the unit cotan gent bundle of partial derivative Omega was constructed from the princ ipal symbols of the coefficients of the boundary operator and a spectr al pair for the family of matrix polynomials associated with the princ ipal symbol of the elliptic operator. The Lopatinskii condition is equ ivalent to the condition that the function at Delta(B)(+) have inverti ble values. In the present paper, the index of (A, B) is expressed in terms of the winding number of the determinant of Delta(B)(+).