In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ:[Figure not available: see fulltext.], where Cyc A=R:R is a subspace of Χ, dim R<+∞. Spqn (AⁿR:n≥0)=χ. Always disc A ≥μ_A=(the multiplicity of the spectrum of the operator[Figure not available: see fulltext.] (dim R:R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ≤ disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties of disc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H²); disc S=2, (but μ_S=i); disc S_n^*=n (but μ=1), where S_n=S⊕. ⊕S; disc S=2, (but μ_S=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_N=1 {mapping} (the operator N is reductive). There are other results also, and also a list of unsolved problems.