Consider following operator from this paper;
Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ixh(x) + K\int_{-\infty}^\infty h(x)g(x)dx.$$
Let $\sigma_d(B)$ be the discrete spectrum and $\sigma_c(B)$ the continuous spectrum. It was shown in the paper that $$\sigma_c(B) =\{\lambda:\text{Re}[\lambda]=0\}$$ and that $$\sigma_d(B) = \{\lambda: K^{-1} =\int_{-\infty}^\infty(\lambda - i\omega)^{-1}g(\omega)d\omega\}.$$
We can assume that, for $g(w)$ nice enough, the discrete spectrum only has one element.
I am learning spectral theory, and my functional analysis is a bit rusty, but i understand this as an eigenvalue-eigenvector problem. So, how would i construct a projection operator onto the subspace associated with the discrete spectrum?
The sort of answer I am looking for will show how to construct such a projection operator, call it $P_d$, and show how I project the linear operator on to this subspace by evaluating $(P_dBh)(x)$ and $(B P_d h)(x)$.