What problems other then spectral theory are related to the development of the idea of a algebra homomorphism $\pi_{A}: f \rightarrow f(A)$
Also what functions are important other then projections(characteristic funtions)?
https://en.wikipedia.org/wiki/Holomorphic_functional_calculus#Spectral_projections
The functional calculus for selfadjoint and normal operators requires projections and integrals.
The holomoprhic functional calculus deals with general linear operators, but less general functions of those operators. For example, if $C$ is a positively-oriented simple closed rectifiable curve enclosing the spectrum of a bounded operator $A$ on a complex Banach space, then $$ F(A) = \frac{1}{2\pi i}\oint_{C}f(z)\frac{1}{zI-A}dz, $$ where $\frac{1}{zI-A}$ suggestively for $(zI-A)^{-1}$. The function $F$ may be holomorphic on a neighborhood of the spectrum, which may require using a system of contours enclosing components of $\sigma(A)$. Even for matrices, this is not simple. For a matrix $A$ on $\mathbb{C}^N$, the operator $(zI-A)^{-1}$ has poles at the eigenvalues of $A$, and the order of the pole is the size of the largest Jordan block associated with $A$. Nilpotent operators play an important role in the functional calculus for $A$, the series expansion of $\frac{1}{zI-A}$ around $\lambda$ has the form $$ \frac{1}{zI-A} = \sum_{n=-N}^{\infty}A_n(z-\lambda)^{n}, $$ where \begin{align} A_{-n} & = \frac{1}{2\pi i}\oint_{|z-\lambda| = r} (z-\lambda I)^{n-1}(zI-A)^{-1}dz \\ & = (A-\lambda I)^{n-1}\frac{1}{2\pi i}\oint_{|z-\lambda|=r}(zI-A)^{-1}dz \\ & = (A-\lambda I)^{n-1}P, \end{align} where $P = \frac{1}{2\pi i}\oint_{|z-\lambda|=r}(zI-A)^{-1}dz$ is a projection. Only the singular terms contribute to the integral for $F(A)$, and these give $$ F(A) = F(\lambda)P + \frac{F'(\lambda)}{1!}(A-\lambda I)P+\cdots+\frac{F^{(N-1)}(\lambda)}{(N-1)!}(A-\lambda I)^{N-1}P + \mbox{ terms at other eigenvalues }. $$ The absence of higher or terms happens because $(A-\lambda I)^{N}P=0$, where $N$ is the size of largest Jordan block with eigenvalue $\lambda$. The operator $(A-\lambda I)P$ is nilpotent of order $N$: because $(A-\lambda I)P=P(A-\lambda I)$ and $P^2=P$, then $$ [(A-\lambda I)P]^{n} = (A-\lambda I)^{n}P,\;\;\; n \ge 1 \\ [(A-\lambda I)P]^{N} = 0. $$
So projections are not enough, even for the functional calculus of a finite square matrix.