Sturm Type for Differential Equations of Arbitrary Order with Operator-Valued Coefficients

The operators under consideration are generated by differential expressions with locallybounded operator coefficients both on finite and infinite intervals. A relation between spectral andoscillation properties of such differential operators of arbitrary order (even and odd) is investigated. The operators in question may be both self-adjoint and non self-adjoint with block-triangularmatrix coefficients. Several well-known Sturm type Theorems, which were obtained by topologicalmethods, are obtained also by operator-theoretical methods. The results contained in this survey for infinite systems, being applied to either finite systems or scalar problems, are at least as preciseas the results already known for those cases; and sometimes appear to be even more precise.