THE NUMERIC RANGE OF THE TWO-PARAMETER EIGENVALUE PROBLEM

In this paper we investigate the following two-parameter eigenvalue problem {lambda(1) K-i,K-1 phi i + lambda(2) K-i,(2 phi i) = phi(i), phi(i) is an element of H-i (1) i = 1; 2, where lambda(1), lambda(2) are spectral parameters and K-i,K-1, K-i,K-2, i = 1; 2 are compact, self-adjoint operators on Hilbert space H-i i = 1; 2. It is proved that under the corresponding conditions of definiteness the numerical range of problem (1) is either a convex polygon (bounded or unbounded), or a union of two convex polygons (bounded or unbounded).