On the Boundary of the Spatial Numerical Range

We prove a theorem that generalizes various results ([4, Theorem 1], [9, Theorem 1-8], [7], [8, Lemma 4.1]) concerning eigenvalues and spectral points in the boundary of the spatial numerical range and numerical range of a continuous linear operator T on a complex normed space X. The proof is similar to the proof of [4, Theorem 1] and uses a fixed point theorem as do the proofs of [4, Theorem 1] and [7, Proposition 1]. If T is a continuous linear operator on the normed space X, then the spatial numerical range V (T) of T is defined by