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We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.Keywords: quasielliptic operator, weighted Sobolev space, isomorphism, Sobolev type equations § 1. IntroductionIn this article we consider a class of matrix quasielliptic operators,on the whole space R n . For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces W l p,σ and apply these properties to proving the solvability of the initial value problem for the class of Sobolev type equations(1.2)The theorems on the isomorphism properties of differential operators have numerous applications in the theory of partial differential equations. However, in many cases their statements are far from obvious. For instance, the differential operator Δ − εI : W 2 p (R n ) → L p (R n ), 1 < p < ∞, for ε > 0 establishes an isomorphism, but this is false for the Laplace operator Δ : W 2 p (R n ) → L p (R n ), 1 < p < ∞. The situation is similar for the polyharmonic operator Δ m : W 2m p (R n ) → L p (R n ), 1 < p < ∞. This operator is not an isomorphism for any m. In particular, using the results of [1, Chapter 12], we can show that the solvability of the equationsin the Sobolev space W 2m p (R n ) with p ≤ n n−2m requires that the right-hand side f (x) satisfy the orthogonality conditions of the form R n f (x)x α dx = 0, |α| ≤ s.