New aspects of the $L$-condition for elliptic systems

In this paper, the L-condition for an elliptic system (A,B) in a bounded domain Ω ⊂ R is reformulated in a new algebraic form. A square matrix function, ∆+B , defined on the unit cotangent bundle, ST ∗(∂Ω), is constructed from the principal symbols of the coefficients of the boundary operator, B, and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator, A. The L-condition is equivalent to the condition that the function, ∆+B , have invertible values. This paper is divided into three sections. In Section 1 we give the definition of elliptic systems and the L-condition. In Section 3 the L-condition is reformulated in various equivalent forms, which include (in addition to the new form indicated above) the Lopatinskĭı condition, the complementing condition of Agmon–Douglis–Nirenberg and a condition of Fedosov. The purpose of Section 2 is to briefly state the definition of a spectral triple for a matrix polynomial, which is needed for the proof of the equivalence of all these conditions.