Adjoint operator, definition

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. 

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

In the sequel the form (47) is more convenient for the case of self-adjoint operators A and the form (48) - for the case of non-self-adjoint positive definite operators A = A(f). We will elaborate on this later. 

The operator A is self-adjoint, positive definite and satisfies the estimate (see Chapter 2, Section 4)... 

Stability theory is the central part of the theory of difference schemes. Recent years have seen a great number of papers dedicated to investigating stability of such schemes. Many works are based on applications of spectral methods and include ineffective results given certain restrictions on the structure of difference operators. For schemes with non-self-adjoint operators the spectral theory guides only the choice of necessary stability conditions, but sufficient conditions and a priori estimates are of no less importance. An energy approach connected with the above definitions of the scheme permits one to carry out an exhaustive stability analysis for operators in a prescribed Hilbert space Hh-... 

If J is a positive definite quadratic form with a closed extension, its closure J and hence the self-adjoint operator If belonging to J are also positive definite. 

Definition. Let H% be a self-adjoint operator depending on a real parameter k, and let... 

Definition of pseudo-eigenvalues. Let II be a self-adjoint operator with the spectral decomposition... 

The definition of the operator (3 7) leads to the following relation for the adjoint operator... 

Consider now a self-adjoint positive-definite operator A Its eigenvalues are all positive, and the positive square root can be defined in terms of the spectral resolution ... 

It is clear from the definition that the adjoint operator Ta is independent of the choice of basis, but is also illustrative to show that the expression (2.25) is invariant under the transformation X = X.a. In order to understand the expression (2.25) somewhat better, one should observe that the reciprocal basis Xr has its own metric matrix ... 

Even in this case, it is possible to introduce "adjoint operators" through the special definitions... 

For sufficiently smooth functions

boundary conditions, the energy functional H(formal operator H to find the self-adjoint operator in L2( 2). As a result, the Hamiltonian H may be defined for the set Du, the domain of definition of H, being a dense set in 2.2(D). Different boundary conditions generate different self-adjoint operators H with different domains. 

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