I am interested in effective and computations for finding approximate spectral decompositions in some suitable format.

Namely, let $A: H \rightarrow H$ be a Hermitian operator on an $n-$dimensional Hilbert space $H$ with the spectrum $\{\lambda_1, \ldots \lambda_m\}, m \leq n$. Then, $A$ can be decomposed as:

$$ A = \sum_{i=1}^{m}\lambda_i P_i,$$

where $P_i, i=1,\ldots m$ are orthogonal projections with $P_i, P_j = 0, i \neq j$ onto the eigenspaces $H_i = \ker \{ \lambda_i I - A \}$ such that:

$$ H= \displaystyle \underset{i=1}{\overset{m}{\oplus}} H_i.$$

In an approximate format, the theorem can be stated as follows (p. 380):

for any $\varepsilon > 0$, there exist projections $P_i, i=1, \ldots n$ with $P_i, P_j = 0, i \neq j$, and real numbers $\alpha_1, \ldots \alpha_n$ such that $\big|\big| A - \displaystyle \sum_{i=1}^{n} \alpha_i P_i \big|\big| \leq \varepsilon$.

**What about the approximate eigenspaces?**

A particular example is this article, but it addresses **exact** spectral decomposition at the cost of additional input (cardinality of spectrum).