Observables must be hermitian operators b/c we associate experimental measurable values with eigenvalues, therefore eigenvalues must be real - which is ensured by hermitian operators.
For every hermitian operator O you can construct a family of unitary operators U(s) = exp(iOs) with a real parameter s. These U(s) may be symmetries which act on Hilbert space states as unitary operators. One example is a hermitian angular momentum operator L_{i} which generates rotations w.r.t. to the i-axis. A special case are time translations which are generated by the hermitian Hamiltonian H, i.e. U(t) = exp(iHt).
Important operators which are neither hermitian nor unitary are a) the creation and annihilation operators of the harmonic oscillator and b) the ladder operators for angular momentum.
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