Representation of Symmetries and Observables in Functional Hilbert Space. II.

1972 ◽  
Vol 27 (1) ◽  
pp. 7-22 ◽  
Author(s):  
A. Rieckers
Keyword(s):  
Hilbert Space ◽  
Relativistic Quantum ◽  
Preceding Paper ◽  
Complex Hilbert Space ◽  
Unbounded Operators ◽  
Test Function ◽  
Test Function Space ◽  
Quantum Observables ◽  
Set Up ◽  
Real Test

Abstract The representation of infinitesimal generators corresponding to the group representation dis-cussed in the preceding paper is analyzed in the Hilbert space of functionals over real test functions. Explicit expressions for these unbounded operators are constructed by means of the functio-nal derivative and by canonical operator pairs on dense domains. The behaviour under certain basis transformations is investigated, also for non-Hermitian generators. For the Hermitian ones a common, dense domain is set up where they are essentially selfadjoint. After having established a one-to-one correspondence between the real test function space and a complex Hilbert space the theory of quantum observables is applied to the functional version of a relativistic quantum field theory.

scholarly journals Traces of localisation operators with two admissible wavelets

2003 ◽  
Vol 45 (1) ◽  
pp. 17-25 ◽  
Author(s):  
M. W. Wong ◽  
Zhaohui Zhang
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Resolution Of The Identity

AbstractThe resolution of the identity formula for a localisation operator with two admissible wavelets on a separable and complex Hilbert space is given and the traces of these operators are computed.

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

scholarly journals Necessity of Complex Hilbert Space for Quantum Mechanics

1973 ◽  
Vol 49 (3) ◽  
pp. 707-713 ◽  
Author(s):  
Toshiyuki Toyoda
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Complex Hilbert Space

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

On Operator Algebras and Invariant Subspaces

1969 ◽  
Vol 21 ◽  
pp. 1178-1181 ◽  
Author(s):  
Chandler Davis ◽  
Heydar Radjavi ◽  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Elementary Proof ◽  
Operator Algebras ◽  
Equivalent Form ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
Closed Algebra ◽  
Weakly Closed ◽  
Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

scholarly journals Quantum mechanics: why complex Hilbert space?

2017 ◽  
Vol 375 (2106) ◽  
pp. 20160393 ◽  
Author(s):  
G. Cassinelli ◽  
P. Lahti
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Complex Hilbert Space ◽  
Complex Field ◽  
Quantum Revolution

We outline a programme for an axiomatic reconstruction of quantum mechanics based on the statistical duality of states and effects that combines the use of a theorem of Solér with the idea of symmetry. We also discuss arguments favouring the choice of the complex field. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

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