Norm of the operator

Definition 28 The norm of an operator A is the minimum value of all possible M that satisfy the inequality (A.23)  

Based on the last formula, we can write an equivalent expression for the norm of operator A as 

Taking into account this definition and inequality (A.22), we can write 

Note that the expression on the right hand side of the inequality (A.27) is called the Frobenius norm of a matrix, HA1  

Definition 29 A linear operator A is called a bounded operator if it has a bounded norm  

---

In this regard, rounding errors can be treated as possible perturbations of the right-hand side of equation (1) at every step. The iteration scheme (14) with parameters (29), (41) or (42) becomes unstable with respect to the right-hand side by exactly the same reasoning as before the norm of the operator Sk = E — Tf.A for the transition from the k — l)th iteration to the fcth iteration may exceed 1 for negative values of tf., since... 

As known, the norm of the operator Tn is ecjual to the greatest eigenvalue max Aj,(Tn). Further replacements of Q/. and in (11) by continuous variables a and / lead to increased maximum of the right-hand side of (11) meaning... 

Consequently, we can carry out the BCH expansion to arbitrarily high order without any increase in the complexity of the terms in the effective Hamiltonian. In practice, the expansion is carried out until convergence in a suitable norm of the operator coefficients is achieved, as illustrated in Table I. Rapid convergence is usually observed. Note that through the decomposition (23), the effective Hamiltonian depends on the one- and two-particle density matrices and therefore becomes state specific, much like the Fock operator in Hartree-Fock theory. 

The minimal constant M satisfying condition (1) is called the norm of the operator A and is denoted by A x y or simply A. ... 

Definition 62 The smallest constant M for which condition (B.l) holds for any X E X is called the norm of the operator ... 

This is very seldom the case, of course, and—in this situation—one may instead try to minimize the Hilbert-Schmidt norm of the operator... 

Let Hhea Hilbert space. It will be noted by L(H) the set of all bordered operators on H, equipped with the norm of the operators, such as ... 

The smallest such k is said to be the norm of the operator A, /41, and we have the well known inequality... 

A typical regularization method for the Fredholm integral equation of the first kind is to add a term Q(f> y) (with q a fixed number) to the left-hand side of Eq. (66), thus obtaining a Fredholm integral equation of the second kind, whose solution is stable. The difiieulty of this method is related to the fact that the formula expressing such a solution (the Neumann series ) does converge only for q > A, where A is the norm of the operator A [45]. 

---