scholarly journals On properties of inner product type integral transformers

2020 ◽  
Vol 4 (2) ◽  
pp. 160-169
Author(s):  
Benard Okelo ◽  
Keyword(s):  
Quantum Theory ◽  
Product Type ◽  
Inner Product ◽  
Grüss Inequality ◽  
Norm Inequalities ◽  
Unitarily Invariant Norms ◽  
Type Integral

In this paper, we give characterizations of certain properties of inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Grüss inequality. Lastly, we explore some of the applications in quantum theory.

scholarly journals Some Hermite-Hadamard type integral inequalities for operator AG-preinvex functions

2016 ◽  
Vol 8 (2) ◽  
pp. 312-323
Author(s):  
Ali Taghavi ◽  
Haji Mohammad Nazari ◽  
Vahid Darvish
Keyword(s):  
Integral Inequalities ◽  
Unitarily Invariant Norm ◽  
Norm Inequalities ◽  
Invariant Norm ◽  
Type Integral ◽  
Preinvex Functions ◽  
Inequalities For Operators

Abstract In this paper, we introduce the concept of operator AG-preinvex functions and prove some Hermite-Hadamard type inequalities for these functions. As application, we obtain some unitarily invariant norm inequalities for operators.

scholarly journals NORM AND ANTI-NORM INEQUALITIES FOR POSITIVE SEMI-DEFINITE MATRICES

2011 ◽  
Vol 22 (08) ◽  
pp. 1121-1138 ◽  
Author(s):  
JEAN-CHRISTOPHE BOURIN ◽  
FUMIO HIAI
Keyword(s):  
Positive Operators ◽  
Norm Inequalities ◽  
Block Matrices ◽  
Determinantal Inequality ◽  
Unitarily Invariant Norms ◽  
Analytic Criterion

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if [Formula: see text] is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, [Formula: see text] To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then [Formula: see text] for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.

scholarly journals Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 174
Author(s):  
Koen Thas
Keyword(s):  
Quantum Theory ◽  
Finite Fields ◽  
Hilbert Spaces ◽  
Inner Product ◽  
Unitary Operators ◽  
Evolution Operators ◽  
Quantum Theories ◽  
Classical Quantum ◽  
Absolute Quantum ◽  
Made In

In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).

scholarly journals Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 707
Author(s):  
Nicuşor Minculete
Keyword(s):  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Grüss Inequality ◽  
Chebyshev Function ◽  
Ostrowski’S Inequality

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshev function and the Grüss inequality are presented.

scholarly journals Anti-PT Transformations and Complex PT-Symmetric Superpartners

2021 ◽  
Author(s):  
Sehban Kartal ◽  
Taha Koohrokhi ◽  
Ali Mohammadi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Inner Product ◽  
Quantum Mechanical ◽  
Probabilistic Interpretation ◽  
Quantum Mechanical System ◽  
Complex Conjugation ◽  
New Formalism ◽  
Shape Invariant ◽  
Symmetric Quantum

Abstract A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.

scholarly journals Some Improvements of the Cauchy-Schwarz Inequality Using the Tapia Semi-Inner-Product

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2112
Author(s):  
Nicuşor Minculete ◽  
Hamid Reza Moradi
Keyword(s):  
Hilbert Space ◽  
Normed Space ◽  
Triangle Inequality ◽  
Schwarz Inequality ◽  
Inner Product ◽  
Numerical Radius ◽  
Real Numbers ◽  
Norm Inequalities ◽  
Hilbert Space Operators ◽  
New Inequalities

The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain some new inequalities for the numerical radius and norm inequalities for Hilbert space operators.

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