Basic Hilbert Space Theory

1998 ◽  
pp. 1-18 ◽  
Author(s):  
Avraham Feintuch
Keyword(s):  
Hilbert Space ◽  
Space Theory

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

scholarly journals Non-unitary scattering and capture. I. Hilbert space theory

1980 ◽  
Vol 71 (3) ◽  
pp. 277-288 ◽  
Author(s):  
E. B. Davies
Keyword(s):  
Hilbert Space ◽  
Space Theory

Quantum electrodynamics and Hilbert space theory

1956 ◽  
Vol 52 (4) ◽  
pp. 719-733
Author(s):  
J. G. Taylor
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Quantum Electrodynamics ◽  
Quantum Field ◽  
Space Theory ◽  
Electrons And Positrons ◽  
Simple Quantum ◽  
Interacting Electrons ◽  
Space Formalism

ABSTRACTThis paper describes an attempt to formulate quantum field theory, in particular quantum electrodynamics, in terms of Hilbert space theory. The work of Cook (1) is extended to give a precise description of non-interacting electrons and positrons. The hole interpretation is not required in this extension, and no subtraction formalism is required. It is shown that the formalism can never reduce to that of intuitive quantum field theory except by an abuse of language associated with the δ-function. Interaction cannot be introduced in a simple manner into the rigorous formalism, so it seems extremely difficult to develop the Hilbert space formalism for quantum field theory in any useful manner.These difficulties indicate that an investigation of the Hilbert space basis of simple quantum theory is necessary before a rigorous mathematical formalism for intuitive quantum field theory can be developed.

scholarly journals Boolean valued interpretation of Hilbert space theory

1983 ◽  
Vol 35 (4) ◽  
pp. 609-627 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Hilbert Space ◽  
Space Theory
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