Prove that for every compact $K \subset \{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert space such that $\sigma(T) = \sigma_p(T) = K$

Question

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Compact sets as point spectrum of a bounded operator

Prove that for every compact set $K \subset \mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert space such that $\sigma(T) = \sigma_p(T) = K$ where $\sigma(T)$ is a spectrum of $T$ and $\sigma_p(T)$ is a point spectrum of $T$.

Hint 1 by the teacher. Consider operators on dual space $(AL_2(\mathbb{D}))^*$ for Bergman space $AL_2(\mathbb{D})$. Bergman space $AL_2(\mathbb{D})$ is a space of all holomorphic (on $\mathbb{D}$) functions $f$ such that $f\in L_2(\mathbb{D})$.

Hint 2. Find operator $T$ such that functionals $(AL_2(\mathbb{D}))^*\ni\phi_a = \phi_a(f) = f(a)$ are the eigenvectors of $T$.