scholarly journals New Čebyšev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Spaces ◽  
Error Bounds ◽  
Čebyšev Functional ◽  
Adjoint Operators

Several new error bounds for the Čebyšev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

2021 ◽  
pp. 2150017
Author(s):  
Yinfeng Zhang ◽  
Guolin Yu
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Feasible Solution ◽  
Variational Inequality Problem ◽  
Gap Functions ◽  
Strong Monotonicity ◽  
Mixed Variational Inequality ◽  
Related Literature ◽  
Quasi Variational Inequality

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

Entropy Numbers for Non-Compact Self-Adjoint Operators in HILBERT Spaces

1981 ◽  
Vol 100 (1) ◽  
pp. 213-219 ◽  
Author(s):  
David E. Edmunds ◽  
Hans Triebel
Keyword(s):  
Hilbert Spaces ◽  
Entropy Numbers ◽  
Adjoint Operators

scholarly journals New bounds for the Cebysev functional of two functions of selfadjoint operators in Hilbert spaces

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 27-39 ◽  
Author(s):  
S.S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Selfadjoint Operators ◽  
Čebyšev Functional ◽  
Subject Classifications ◽  
Mathematics Subject Classifications ◽  
New Inequalities

Some new inequalities for the Cebysev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are given. 2010 Mathematics Subject Classifications. 47A63; 47A99. .

scholarly journals Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

2019 ◽  
Vol 31 (04) ◽  
pp. 1950013 ◽  
Author(s):  
Valter Moretti ◽  
Marco Oppio
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Complex Structure ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
The Real ◽  
Quaternionic Hilbert Space ◽  
Adjoint Operators

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

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