Adjoint space

A dual space or adjoint space of a vector space X, denoted X, is the space of all functions on X. 

An alternative expression for the perturbation in the fission kernel can be obtained in the adjoint space. In this space... 

Alternatively, the reactivity effect of the flux perturbation can be taken into account in the adjoint space. We define a generalized adjoint function 46)... 

As regards the importance vector, the theory follows quite directly by a passage from the space of expected neutron distributions N to the dual ( adjoint ) space A of linear functionals. The effect of P and Q on this dual space is expressed by their transposes P and Q, respectively, as always. 

Abstract Hilbert space, 426 Accuracy of computed root, 78 Acharga, R., 498,539,560 Additive Gaussian noise channel, 242 Adjoint spinor transformation under Lorentz transformation, 533 Admissible wave function, 552 Aitkin s method, 79 Akhiezer, A., 723 Algebra, Clifford, 520 Algebraic problem, 52 linear, 53... 

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. 

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... 

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

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 ... 

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... 

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... 

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... 

We thus have shown that under conditions (26) inequality (14) is sufficient for the stability of scheme (la) in the space Ha, that is, relation (29) occurs. Let us stress that the requirement of self-adjointness of the operator B is necessary here, while the energy method demands only the positivity of B and no more. 

Let now // be a complex space and S be a non-self-adjoint operator. Then a necessary and sufficient condition for the stability in the space Ha with respect to the initial data of the scheme... 

The general theory of iterative methods is presented in the next sections with regard to an operator equation of the first kind Au — f, where T is a self-adjoint operator in a finite-dimensional Euclidean space. The applications of such theory to elliptic grid equations began to spread to more and more branches as they took on an important place in real-life situations. 

Three-layer iteration schemes. So far we have considered two-layer iteration schemes available for solving operator equations of the form Au = / with a self-adjoint operator A under the assumption that the spectral bounds and for the operator A are known in advance either in a space H or in a space Hb, where B = B > 0 is some stabilizator. Other iterative methods find a wide range of applications in some or other aspects. 

Dunford, N. and Schwartz, J. (1971b) Linear Operators (second edition). Part II Spectral Theory. Self-Adjoint Operators in Hilbert Space. Wiley New York. 

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-... 

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j... 

For every linear operator that transforms ipi) in ket space into 0/) = Atpi), there is a corresponding linear operator in bra space which transforms tpi into (0, = Axpi. This operator is called the adjoint of A. In bra space the transformation is expressed as... 

The domain of the Schrodinger operator on the graph is the L2 space of differentiable functions which are continuous at the vertices. The operator is constructed in the following way. On the bonds, it is identified as the one dimensional Laplacian — It is supplemented by boundary conditions on the vertices which ensure that the resulting operator is self adjoint. We shall consider in this paper the Neumann boundary conditions ... 

The equation above is written using the units h c 1. The quantity y is a vector of Dirac matrices, m is the electron mass multiplied by a Dirac matrix. Fermi level. With this definition the energy functional is... 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

Definition 3.9 Suppose V and W are finite-dimensional complex scalar product spaces, and let ( , > v and , denote their complex scalar products. Suppose T.V W is a linear transformation, that is, suppose T Hom (V, VP). Then the adjoint of T is the unique linear transformation T W V such that for all v e V and all w e W we have... 

Although our definition of adjoint applies only to finite-dimensional vector spaces, we cannot resist giving an inhnite-dimensional example. The proof of uniqueness works for infinite-dimensional spaces as well, but our proof of existence fails. Fix an element a e L (W) and consider the linear transformation T c defined by... 

Dehnition 3.10 Suppose V is a complex scalar product space. A Hermihan linear operator (also known as a Hermihan symmetric operator or self-adjoint operator) on V is a linear operator T V V such that for all Vi, V2 E V we have... 

With this complex scalar product on the dual space V in hand, we can make the relationship between the dual and the adjoint clear. The definition... 

So when there is one fixed complex scalar product on a vector space V, it is consistent to use the notation v for both dual and adjoint. In a unitary basis, the asterisk means coordinate transpose. 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

---