  Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space,
$\mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators acting on
$\mathcal{H}$ and $\mathcal{J}$ a proper two-sided ideal of
$\mathcal{B}(\mathcal{H})$. Denote by $\mathcal{U}_\mathcal{J}(\mathcal{H})$
the group of all unitary operators of the form $I+\mathcal{J}$. Recall that an
operator $A \in \mathcal{B}(\mathcal{H})$ is diagonalizable if there exists a
unitary operator $U$ such that $UAU^*$ is diagonal with respect to some
orthonormal basis. A more restrictive notion of diagonalization can be
formulated with respect to a fixed orthonormal basis $\mathrm{e}=\{
e_n\}_{n\geq 1}$ and a proper operator ideal $\mathcal{J}$ as follows: $A \in
\mathcal{B}(\mathcal{H})$ is called restricted diagonalizable if there exists
$U\in \mathcal{U}_\mathcal{J}(\mathcal{H})$ such that $UAU^*$ is diagonal with
respect to $\mathrm{e}$. In this work we give necessary and sufficient
conditions for a diagonalizable operator to be restricted diagonalizable. Our
conditions become a characterization of those diagonalizable operators which
are restricted diagonalizable when the ideal is arithmetic mean closed. Then we
obtain results on the structure of the set of all restricted diagonalizable
operators. In this way we answer several open problems recently raised by
Belti\c{t}$\breve{\text{a}}$, Patnaik and Weiss.
