Quantum Mechanics Spectral decomposition misunderstanding

My notes state that the spectral decomposition formula is of the form:

$$ hat{A} = hat{A}hat{1} = sum{hat{A} } |A_iranglelangle A_i | = sum{A_i } |A_iranglelangle A_i | $$

Now consider the Hamiltonian as
$$ hat{H} = begin{bmatrix}E&KK*&Eend{bmatrix}$$
It would be easy to show that the eigenvalues are $$E_+ = E+|K|$$ and$$E_- = E-|K| $$and the eigenstates are as follows:

$$ |+rangle = frac{1}{sqrt{2}}
begin{pmatrix}
frac{k}{|k|}
1
end{pmatrix} $$

$$ |-rangle = frac{1}{sqrt{2}}
begin{pmatrix}
frac{-k}{|k|}
1
end{pmatrix} $$

I was expected then to show that the unitary operator $$hat{U}(t) = e ^{ithat{H}/hbar}$$ can be defined as follows:

$$hat{U}(t)=e^{-ihat{H}t/hbar}=
e^{-iEt/hbar}begin{pmatrix}
cos(|K|t/hbar) & -ifrac{K}{|K|}sin(|K|t/hbar)
-ifrac{K^*}{|K|}sin(|K|t/hbar) & cos(|K|t/hbar)
end{pmatrix}$$

According to my notes the appropriate application of the spectral decomposition formula to do so is

$$ hat{U}(t) = e ^{it{hat{H}}/hbar} = e ^{it{E_+}/hbar} |+ranglelangle +| + e ^{it{E_-}/hbar} |-ranglelangle -| $$

However if we recall the spectral decomposition formula as:
$$ hat{U} = hat{U}hat{1} = sum{hat{U} } |U_iranglelangle U_i | = sum{U_i } |U_iranglelangle U_i | $$

My question therefore is why is it appropriate to use the $|+rangle$ and $|-rangle$ states when they are the eigenstates of the $hat{H}$ operator and not the $hat{U}(t)$ one?