Cyclic sums of comparative indices and their applications

Abstract

In this paper we generalize the notion of the comparative index for the pair of Lagrangian subspaces which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We introduce cyclic sums $\mu_c^{\pm}(Y_1,Y_2,\dots,Y_m),\,m\ge 2$ of the comparative indices for the set of $n-$ dimensional Lagrangian subspaces. We formulate and prove main properties of the cyclic sums, in particular, we state connections of the cyclic sums with the Kashiwara index. The main results of the paper connect the cyclic sums of the comparative indices with the number of positive and negative eigenvalues of $mn\times mn$ symmetric matrices defined in terms of the Wronskians $Y_i^T \,J\, Y_j,$ $i,j=1,\dots,m.$ We also present first applications of the cyclic sums of the comparative indices in the oscillation theory of the discrete symplectic systems connecting the number of focal points of their principal solutions with the negative and positive inertia of symmetric matrices.