Operators self-adjoint

So far we have not made any assumptions about the properties of the operator A, but we will now assume that A is either self-adjoint or commutes with its hermitean adjoint operator Ht, so that... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

Observe that the equality u, Lv)q — (n, Lu) means the self-adjointness of the operator L. 

Any nonnegative operator A in a complex Hilbert space H is self-adjoint ... 

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. 

Theorem 2 The product AB of two commuting nonnegative self-adjoint operators A and B is also a nonnegative self-adjoint operator. 

Theorem 3 There exists a unique nonnegative self-adjoint square root B of any nonnegative self-adjoint operator A commuting with any operator which commutes with A. 

Let A be a positive self-adjoint linear operator. By introducing on the space H the inner product x,y) = Ax,y) and the associated norm x) we obtain a Hilbert space Ha, which is usually called the energetic space Ha- It is easy to show that the inner product... 

Axioms (2) and (3) are met by virtue of the linearity property. The validity of (4) is stipulated by the fact that the operator A is positive. The meaning of the self-adjointness of the operator A is that we should have... 

Lemma 2 For any positive self-adjoint operator A in a real Hilbert space the generalized Cauchy-Bunyakovskii inequality holds ... 

If A is a self-adjoint operator for which A exists, its negative norm can be defined by... 

With this, for any self-adjoint operator A the following relations occur ... 

Lemma 4 If A is a self-adjoint positive hounded operator, then the estimate is valid ... 

The matrix of a self-adjoint operator in any orthonormal basis is a symmetric matrix. 

Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

A self-adjoint operator A in the space Rn possesses n mutually orthogonal eigenvectors, ... We assume that all the j. s are normalized, that is, Mi II = 1 for k = I,..., n. Then ( j, i,) = The corresponding eigenvalues are ordered with respect to absolute values ... 

If a linear operator A given on Rn possesses n mutually orthogonal eigenvalues, then A is a self-adjoint operator A = A. ... 

If self-adjoint operators A and B are commuting AB = BA), then they possess a common system of eigenvectors. 

These formulae confirm that the operator A is self-adjoint. 

While solving the operator equations (2) we establish the basic properties of the operator A such as self-adjointness, positive definiteness, the lower bound of the operator and its norm and more. The operator A constructed in Example 1 will be frequently encountered in the sequel. Before stating the main results, will be sensible to list its basic properties. 

Indeed, the norm of a self-adjoint positive operator in a finite-dimensional space 17, is equal to its greatest eigenvalue j A = A,v-i- case, in... 

Non-self-adjoint difference operators appear, for example, in the approximation of second-order elliptic operators with the first derivatives. The operator Lu = u x) + bu (x), a [0, 1], 6 = const, is approximated... 

We will assume that problem (37) is solvable for any right-hand sides (p H there exists an operator A with the domain V A ) = H. All the constants below are supposed to be independent of h. In what follows the space H is equipped with an inner product (, ) and associated norm II. T II = /i x, x ). The writing A = A > 0 means that A is a self-adjoint... 

Consider first the simplest case when A is a non-self-adjoint positive definite operator ... 

Assuming A to be a non-self-adjoint operator subject to the inequality 0 0 0... 

The requirement that scheme (4) of Section 1 should be conservative ( divergent ) is equivalent to being self-adjoint of the appropriate difference operator. To make sure of it, we refer to a second-order operator... 

Let T be a self-adjoint positive definite linear operator in Hilbert space H equipped with an inner product (,) and let / be a given element of the space H. The problem of minimizing the functional... 

Suppose that the inverse operators and A exist. Moreover, we assume that A and A are self-adjoint positive operators. Substitutions of u = A f and u = A f into (15) yield... 

Theorem Let u be a solution to equation (11) and u be a solution to equation (14), where A, A and Aq are self-adjoint positive operators for which the inverse operators exist. If condition (18) and the inequality A > CjTo, Cj > 0 hold, then the estimates are valid ... 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

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