Hilbert space operators

That the use of symbolic dynamics to study the behavior of complex or chaotic systems in fact heralds a new epoch in physics wris boldly suggested by Joseph Ford in the foreword to this Physics Reports review. Ford writes, Just as in that earlier period [referring to 1922, when The Physical Review had published a review of Hilbert Space Operator Algebra] physicists will shortly be faced with the arduous task of learning some new mathematics... For make no mistake about it, the following review heralds a new epoch. Despite its modest avoidance of sweeping claims, its theorems point like arrows toward the physics of the second half of the twentieth century. ... 

The superscript H on the square bracket in (8.3.4) indicates that the Liouville expression inside the bracket needs to be transformed back into Hilbert space before multiplication with the Hilbert-space operator 1 = 1 - ily = I- and formation of the trace. Inverse Fourier transformation over k produces an image of the spin density. Given the form of the density matrix after the initial pulse, the spin density corresponds to 1+ (r). It is weighted by the phase evolution under the internal Hamiltonians Hx(r) during the space-encoding time ti,... 

We next briefly survey some properties of Liouville space superoperators that will be useful in the following derivations [49]. The elements of the Hilbert space N XN density matrix, p(t), are arranged as a Liouville space vector (bra or ket) of length N. Operators of N X N dimension in this space are denoted as superoperators. With any Hilbert space operator A, we associate two superoperators Al (left) and Ar (right) defined through their... 

Vol. 1820 F. Hiai, H. Kosaki, Means of Hilbert Space Operators (2003)... 

The question to be asked is Under what conditions (if at all) do the components of X fulfill Eq. (B.8) In [34] it is proved that this relation holds for any full Hilbert space. Here, we shall show that this relation holds also for the P sub-Hilbert space of dimension M, as defined by Eq. (10). To show that we employ, again, the Feshbach projection operator foraialism [79] [see Eqs. (11)]. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

As we can see, penalty operators can be built easier in Hilbert spaces. In applications H is often a Hilbert space such that... 

On the other hand, the duality mapping I is defined by the scalar product in the Hilbert space V. Assume that the operator A is self-conjugate. Then we can define the scalar product in V as follows ... 

Theorem 1.23. If A V V is a linear, self-conjugate, strongly monotonous and Lipschitz continuous operator in a Hilbert space V, then there exists a unique solution u G K of the variational inequality (1.126) given by the formula... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

The von Neumann Projection Operators.—Consider the eigenstates n > in the Hilbert space of N particles with the properties ... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

These in operators now allow us to build up a complete set of states that span the Hilbert space of physical states in the manner discussed in Section 9.6 of the previous chapter for the free-field case. These... 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

If A induces a one-to-one mapping of the Hilbert space on itself then the inverse operator A 1 exists. It is an antilinear operator with the property that... 

Hermitian operators for electric and magnetic field intensities, 561 Herzfeld, C. M., 768 Hessenberg form, 73 Hessenberg method, 75 Heteroperiodic oscillation, 372 Hilbert space abstract, 426... 

Linear algebraic problem, 53 Linear displacement operator, 392 Linear manifolds in Hilbert space, 429 Linear momentum operator, 392 Linear operators in Hilbert space, 431 Linear programming, 252,261 diet problem, 294 dual problem, 304 evaluation of methods, 302 in matrix notation, simplex method, 292... 

Appendix C). The space of Hilbert-Schmidt operators, 2 > is a Hilbert... 

The space of trace class operators acting in the Hilbert space... 

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... 

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