Linear Operators in Hilbert Space

The basic definition of the linear operators acting on Hilbert space reads as  

Such feature implies important consequences in bra-ket formalism. For instance, if one has the operatorial-ket equation 

In the same manner further operatorial properties may be unfolded, as follows. 

the hemiiticity property of operators is formalized in direct way through fulfilling the auto-adjunct condition  

The direct consequences regard the hermiticity and anti-hermiticity of the next combinations  

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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... 

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... 

Akhiezer, N. and Glazman, I. (1966) The Theory of Linear Operators in Hilbert Space. Nauka Moscow (in Russian). 

We may only raise these questions. The development of the main mathematical tools allowing their study (theories of Markoff chains and of linear operators in Hilbert space) may, however, allow us to hope for progress in the coming years. ... 

Toward this end, we have attempted to reduce exposition using vector terminology. Occasionally, this was not possible, so an appendix was developed to provide a review of elementary principles for vector-matrices operation. Linear operators in Hilbert space was briefly mentioned, mainly as a means of coping with the case of infinite eigenfunctions. Last, but not least, a careful... 

M. A. Naimark, On some criteria of completeness of the system of eigen and associated vectors of a linear operator in Hilbert space, Dokl. Akad. Nauk SSSR (N.S.) vol. 98 (1954) pp. 727-730 (in Russian). 

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 ... 

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

Stone, M. H. (1930). Linear transformations in Hilbert space. HI. operational methods and group theory. ProcNatl AcadSciUSA. 16 172-175. 

Naimark, M. A., Linear Differential Operators, Part II Linear Differential Operators in Hilbert Space, New York Ungar Publishing, 1968. 

In the discussion that follows, it will be assumed that the reader is familiar with the theory of linear operators on Hilbert spaces, as presented in standard textbooks on quantum theory (for example, Messiah). Whilst this is adequate for most purposes, the somewhat more mathematical presentation to be found in books such as those by Richtrayer, Kato or Reed and Simon S will be followed here. The reader may wish to consult these books for background information. The presentation in this section will be informal and reference to these works avoided. 

Here D and A stand for linear operators in a Hilbert space H, z(t) and Tp i) refer to abstract functions of the argument t (E u) with the values in the... 

Lemma 1 If A > 0 is a linear operator in a Hilbert space H, then... 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

Tlie mechanical quantities tliat describe the particle (energy, the components of vectors of position, momentum, angular momentum, etc.) are represented by linear operators acting in Hilbert space (see Appendix B). There are two important examples of the operators the operator of the particle s position x = x (i.e. multiplication by x, or x = x-. Fig. 1.5), as well as the operator of the (x-component) momentum px = —where i stands foi the imaginary unit. 

One can show that the eigenvectors of a Hermitian operator form the complete basis set in Hilbert space, i.e. any function of class can be landed in a linear eombination of the basis set. 

In quantum mechanics, dynamical variables are represented by linear Hermitian operators 0 that operate on state vectors in Hilbert space. The spectra of these operators determine possible values of the physical quantities that they represent. Unlike classical systems, specifying the state ) of a quantum system does not necessarily imply exact knowledge of the value of a dynarttical variable. Only for cases in which the system is in an eigenstate of a dynamical variable will the knowledge of that state IV ) provide an exact value. Otherwise, we can only determine the quantum average of the dynamical variable. 

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... 

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