Find the spectrum of the operator $T: l^p\to l^p$, $1\leq p\leq \infty$ $$ T(x_1, x_2, x_3, x_4, \ldots )=(-x_2, x_1, -x_4, x_3, \ldots). $$
My attempt
I have that $Tx = \lambda x$ $$ \implies \lambda (x_1, x_2, x_3, x_4, \ldots )=(-x_2, x_1, -x_4, x_3, \ldots)$$ $$ \implies \lambda (x_1, x_2, x_3, x_4, \ldots ) - (-x_2, x_1, -x_4, x_3, \ldots) = 0$$ $$ \implies (\lambda x_1 + x_2, \lambda x_2 - x_1, \lambda x_3 + x_4, \lambda x_4 - x_3 , \ldots ) = 0$$ $$ \implies x_1 = \lambda x_2, x_2 = - \lambda x_1$$ and similarly for $x_3, x_4 \cdots \cdots$. we then have $$ \implies \lambda = +i, -i.$$ Then what is the spectrum? Does it only consist of the eigen spectrum?