Adjoint operators, stability problem

Brief Review of some of the Results in the Previous Studies of the Stability Problem for a Pair of Adjoint Operators. 

The general stability problem for a pair of adjoint many-particle operators T and T has been discussed in a previous paper2, which will be referred to as reference B. The Hartree-Fock scheme for a pair of such operators has also been discussed, and this paper will be referred to as reference C. Some of the most important results in the references A, B, and C will be briefly reviewed here to make the presentation more self-contained. 

The stability problem for the adjoint operator T may now be expressed in a similar form ... 

Clearly, stability is an intrinsic property of schemes regardless of approximations and interrelations between the resulting schemes and relevant differential equations. Because of this, any stability condition should be imposed as the relationship between the operators A and B. More specifically, let a family of schemes specified by the restrictions on the operators A and B be given A = A > 0 or Ay, v) = y, Av) and Ay, y) > 0 for any y, v H, where (, ) is an inner product in H, B > 0 and B B B is non-self-adjoint). The problem statement consists of extracting from that family a set of schemes that are stable with respect to the initial data, having the form... 

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