Applying the Operator Inequality (Section 9.24, p.281) on the right-hand side of the above equation, [Pg.118]

On account of Theorem 1 with regard to (65) the operator inequalities [Pg.441]

Let JsT be a closed convex subset of V. We consider the operator inequal- [Pg.33]

Ays = —z2a=iy a.ra the space H, where A is a (2p+ l)-point difference Laplace operator, we rely in the further derivation on the Green formula and condition (63), whose combination gives the operator inequalities (32). Having involved the same operator R as was done in problem (53)-(54), we obtain the constant u> and the operator B in terms of known members 8 and A. Just for this reason the same algorithm as in a) is workable for determination of the ( + l)th iteration for either of the components k 4 l [Pg.698]

Proof In complete agreement with the preceding theorem with known coefficients 7i and arising from the operator inequalities 7i5 < < 72-S> we deduce from (27) and (32) that [Pg.687]

The main problem here is connected with selecting the regularizator R. Since regularity conditions became operator inequalities, it seems reasonable to choose as R operators of the most simplest structures which are energetically equivalent to the operator A. Let, for instance, A and Aq be energetically equivalent operators with constants 7, and 7j, so that [Pg.456]

For real Hilbert spaces this statement fails to be true. As far as only real Hilbert spaces are considered, we will use the operator inequalities for non-self-adjoint operators as well. [Pg.45]

Scheme (4) is stable if S) < 1 + c t for all j = 0,1,..., ng — 1. In practical applications of this sufficient stability criterion one needs to reveal some properties of the operators A and B ensuring condition (21). Such conditions are established in Section 2 of the present chapter. They asquire the form of linear operator inequalities for the operators A and B acting in the Hilbert space H = [Pg.394]

A necessary and sufficient condition for a two-layer scheme to be stable can be written as the operator inequality [Pg.780]

Necessity. Suppose that scheme (la) is stable and estimate (15) is satisfied. We are going to show that this leads to the operator inequality (14), that is, [Pg.400]

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. [Pg.151]

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