Operator inequality

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

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. 

With the aid of the above operator inequalities we are able to produce the necessary a priori estimates and justify the convergence with the rate 0(1/r ) for the scheme in hand. Observe that for p = 2 operator (16) coincides with operator (15). 

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 =... 

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,... 

On account of Theorem 1 with regard to (65) the operator inequalities... 

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... 

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... 

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

Chapter 6 includes a priori estimates expressing stability of two-layer and three-layer schemes in terms of the initial data and the right-hand side of the corresponding equations. It is worth noting here that relevant elements of functional analysis and linear algebra, such as the operator norm, self-adjoint operator, operator inequality, and others are much involved in the theory of difference schemes. For the reader s convenience the necessary prerequisities for reading the book are available in Chapters 1-2. 

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. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

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... 

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

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