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Keywords:
Banach and Hilbert space; homogeneous operator; polynomial operator; symmetric operator; monotone operator; numerical range; spectrum; eigenvalue
Banach and Hilbert space; homogeneous operator; polynomial operator; symmetric operator; monotone operator; numerical range; spectrum; eigenvalue
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.
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