The paper deals with the mappings of Banach space E given in a form of quasilinear difference equation. Side by side with the above equation we consider the linear difference equation of the first approximation defined by bouded operator A. We will discuss the assertions which guarantee local stability or instability for the trivial solution of quasilinear difference equation if the above linear difference equation to be of this specificity. The proposal paper not only generalizes well known finite dimensional stability analysis results for quasilinear difference equations. Using spectral properties of operator A as a basis, our paper shows that the infinite dimension of the space E not only strongly complicates computations and proofs of relevant theorems on stability analysis by the first approximation but also can have significant influence to statement of these results.