In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Fundamental theories of physicsJan 1, 20171 citation

Verify conditions of explicit formula solutions correlated with the operating system acting on a Hilbert space

Salim Dawood Mohsen, Aqeel Naeem Mhr

The requirements for obtaining a common solution of the system of Bounded operators’ equations, Ψ∗𝑋Ψ=Ω, Ψ𝑋∗Ψ∗=Ω, are presented in this article, where any operator on the Hilbert spaces 𝐻 is represented by 𝑋, and 𝐵 is a closed range operator. The inquiry results in the demonstration of a number of theorems that give specific descriptions for the solutions and requires have been necessary in the same time sufficient for their existence. As principal findings are the express solution for the operator equations in general additionally bounded answers for situations in which 𝐵 possesses a pseudo-inverse operator. For linear bounded operators, this response is given by the operator's pseudo-inverse on Hilbert spaces. Significant achievements include the generic formula solution of the equations has operators. Using pseudo inverse operators, we may achieve bounded solutions in situations where Ψ possesses a pseudo inverse operator, Further provide the subscriber solution for the next equations of operator Ψ∗𝑋Ψ=Ω1, Ψ𝑋Ψ∗=Ω2, Ψ𝑋∗Ψ∗=Ω3, Ψ∗𝑋∗Ψ=Ω4 where in (Ψ, Ω1, Ω2, Ω3, Ω4) are bounded noninvertible operators formed on Hilbert space. Likewise for varying the values of 𝑋, 𝑋0, we have given specific and comprehensive answers. See what makes the two solutions different. For each of the theorems, offer proofs.The requirements for obtaining a common solution of the system of Bounded operators’ equations, Ψ∗𝑋Ψ=Ω, Ψ𝑋∗Ψ∗=Ω, are presented in this article, where any operator on the Hilbert spaces 𝐻 is represented by 𝑋, and 𝐵 is a closed range operator. The inquiry results in the demonstration of a number of theorems that give specific descriptions for the solutions and requires have been necessary in the same time sufficient for their existence. As principal findings are the express solution for the operator equations in general additionally bounded answers for situations in which 𝐵 possesses a pseudo-inverse operator. For linear bounded operators, this response is given by the operator's pseudo-inverse on Hilbert spaces. Significant achievements include the generic formula solution of the equations has operators. Using pseudo inverse operators, we may achieve bounded solutions in situations where Ψ possesses a pseudo inverse operator, Further provide the subscriber solution for the next equations of operator Ψ∗𝑋Ψ=Ω1, Ψ𝑋Ψ∗=Ω2, Ψ𝑋∗Ψ∗=Ω3, Ψ∗𝑋∗Ψ=Ω4 where in (Ψ, Ω1, Ω2, Ω3, Ω4) are bounded noninvertible operators formed on Hilbert space. Likewise for varying the values of 𝑋, 𝑋0, we have given specific and comprehensive answers. See what makes the two solutions different. For each of the theorems, offer proofs.

AIP conference proceedingsJan 1, 20260 citations

Hilbert Spaces and Classes of Operators

Valter Moretti

Springer eBooksJan 1, 20190 citations

The Necessary and Sufficient Conditions of Separability for Bipartite Pure States in Infinite Dimensional Hilbert Spaces

Su Hu, Zong‐Wen Yu

In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ketψ$ , we prove $M$ is a compact operator. We also prove $\ketψ$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ketψ$, we prove $M$ is a compact operator. We also prove $\ketψ$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.

arXiv:0704.0969Apr 7, 20071 citation

Operators on Hilbert space

Charles R. Giardina

Elsevier eBooksJan 1, 20240 citations

In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let

M

be the matrix of the amplitudes of

\ketψ

, we prove

M

is a compact operator. We also prove

\ketψ

is separable if and only if

M

is a bounded linear operator with rank 1, that is the image of

M

is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.