Bound operation

The set of bounded operators acting in a Hilbert space form a normed linear space. The norm is given by the bound on the operator... 

The set of trace class operators form a subset of the set of bounded operators defined by... 

It is worth noting here that in a finite-dimensional space any linear operator is bounded. All of the linear bounded operators from X into Y constitute what is called a normed vector space, since the norm j4 of an operator A satisfies all of the axioms of the norm ... 

We will denote by A X the set of linear bounded operators with the domain coinciding with A and the range belonging to A. On the set A —> A... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

Theorem 4 Let A he a linear bounded operator in Hilbert space H, V[A) = H. In order that the operator A possess an inverse operator A with the domain V(A ) = H, it is necessary and sufficient the existence of a constant 5 > 0 such that for all x H the following inequalities hold ... 

Lemma 3 If S = S is a linear bounded operator and n is a positive integer, then... 

The competent authorities of the Netherlands, the task force member responsible for coordinating Project Prism activities in Europe, have launched a specific time-bound operation focusing on backtracking investigations. The operation is aimed at identifying companies and individuals responsible for the manufacture and diversion of precursors of amphetamine-type stimulants, specifically P-2-P, in the region. If successful, the operation may be expanded to include areas beyond Europe. 

Perturbation method as asymptotic expansion. In the preceding section we have shown that the solution power series of k, which is absolutely convergent for every k and for every initial state bounded operator. But this is the only case in which we have succeeded in proving the convergence of the series (16. 9). 

Corollary Let A be a positive definite linear bounded operator with the domain V(A) = H. Then there exists a bounded inverse operator A J with the domain VIA-1) = H. 

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

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

self-adjoint problems D(W) = D(H) and for bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

For such a bounded operator, the spectral and general stability properties of the linear operator T stay essentially invariant under the transformation Eq. (1.2), and, for self-adjoint and normal operators, one has a series of well-known theorems. Before proceeding to the unbounded transformations, we will briefly review some additional properties of the bounded similarity transformations. 

If, further, T is a linear operator defined on the L2 space , one says that an element F = F(X) belongs to the domain D(T) of the operator T, if both T and its image TT belong to L2 the set T P is then referred to as the range of the operator T. It should be observed that a bounded operator T as a rule has the entire Hilbert space as its domain—or may be extended to achieve this property—whereas an unbounded operator T has a more restricted domain. In the latter case, we will let the symbol C(T) denote the complement to the domain D(T) with respect to the Hilbert space. A function F = (X) is hence an element of C(T), if it belongs to L2, whereas this is not true for its image 7 F. 

Since, in physics, the Hamiltonian H is usually not bounded from above, it is in principle an unbounded operator with a specific domain D(H), and it is thus in certain ways more complicated to handle than a bounded operator. Instead of the Hamiltonian, one often considers its resolvent... 

If u is a bounded operator, it may be defined over the entire one-particle Hilbert space, which we will assume to be of type L2, so that... 

As a result, we conclude that there exists an inverse operator, L, and it is a bounded operator because of the condition (4.25) (see Appendix B) ... 

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

It is easy to show that a linear bounded operator is a continuous operator, i.e. that the small variations of the argument of the operator will result in a small variation of its values. Clearly, from equation (A.25) we have... 

Definition 77 The operator A is called differentiable at some point x E X if there exists a linear bounded operator F., acting from X to Y, such that... 

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