Several financial intermediate operators directly control big logistics groups; as a consequence, many of their investments should be considered vertical or horizontal.
To achieve this purpose, we use the estimates for the norms of intermediate derivative operators by the norm of an operator generated by the principal part of the considered equation and the given boundary conditions.
We also show the relationship between these conditions and the exact estimates for the norms of intermediate derivatives operators in the subspaces and with respect to the norm of the operator generated by the principal part of equation (1.1).
Mirzoev [16] was the first who paid detailed attention to such relation (for more details about the calculation of the norms of intermediate derivatives operators, see [17]).
We find relations between the estimates of the norms of intermediate derivatives operators in the subspace W 2 3 ( R + ; H ) and the solvability conditions.
Besides, the estimates for the norms of intermediate derivative operators in a Sobolev-type space are obtained, and their close relationship with the solvability conditions is established.
Note that the method offered in [23, 36] and later developed in [31] to calculate the exact values of the norms of intermediate derivative operators is not directly applicable in our case.
And, as the intermediate derivative operators A^{j} frac{d^{4-j} }{dt^{4-j} } :W_{2,K}^{4} ( mathbb{R}_ ;H to L_{2} (mathbb{R}_ ;H), quad j=1,2,3,4, are continuous (see [35]), the norms of these operators can be estimated through the norm (Vert P_{0} uVert _{L_{2} (mathbb{R}_ ;H )} ).
The approach relies on an analysis of intermediate non-self-adjoint operator algebras and the classifications are given in terms of K0 invariants, partial isometry homology, and scales in the composite invariant K0⊕H1.
Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspaces in relation to the norms of the operator generated by the main part of this equation.
