The closure of the numerical range contains the spectrum

Summary:Notions as the numerical range $W(S,T)$ and the spectrum $\s(S,T)$ of couple $(S,T)$ of homogeneous operators on a Banach space are used to derive theorems on solvability of the equation $Sx-lTx=y.$ Conditions for the existence of eigenvalues of the couple $(S,T)$ are given