In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators P _a of order 2a, with type and factorization index a ∈ R₊, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations A_χ,Σ+ in L₂(Ω) of mixed problems for a second-order strongly elliptic symmetric differential operator A on a bounded smooth set Ω ⊂ Rⁿ; here the boundary ∂Ω=Σ is partioned smoothly into Σ=Σ_∪Σ₊, the Dirichlet condition _γ0u=0 is imposed on Σ_, and a Neumann or Robin condition χu=0 is imposed on Σ₊. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 1/2 with factorization index 1/2, relative to Σ₊. The above theory allows a detailed description of D (_{Aχ,Σ_+}) with singular elements outside of Η3/2 (Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference A ⁻¹_{χ,Σ_+} − A⁻¹_ϒ.