scholarly journals More on Inequalities for Weaving Frames in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 141 ◽  
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
Triangle Inequality ◽  
Bounded Operator ◽  
Point Of View ◽  
Real Numbers ◽  
Linear Bounded Operator ◽  
Bessel Sequences ◽  
New Inequalities

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

C∗-algebra consisting of all g-Bessel sequences in a Hilbert space

2017 ◽  
Vol 15 (01) ◽  
pp. 1750004
Author(s):  
Z. L. Chen ◽  
H. X. Cao ◽  
Z. H. Guo
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Small Perturbation ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Operator Space ◽  
Riesz Bases ◽  
C Algebra ◽  
Linear Bijection ◽  
Bessel Sequences

For Hilbert spaces [Formula: see text] and [Formula: see text], we use the notations [Formula: see text], [Formula: see text] and [Formula: see text] to denote the sets of all [Formula: see text]-Bessel sequences, [Formula: see text]-frames and Riesz bases in [Formula: see text] with respect to [Formula: see text], respectively. By defining a linear operation and a norm, we prove that [Formula: see text] becomes a Banach space and is isometrically isomorphic to the operator space [Formula: see text], where [Formula: see text]. In light of operator theory, it is proved that [Formula: see text] and [Formula: see text] are open sets in [Formula: see text]. This implies that both [Formula: see text]-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection [Formula: see text] from [Formula: see text] onto the [Formula: see text]-algebra [Formula: see text], a multiplication and an involution on the Banach space [Formula: see text] are defined so that [Formula: see text] becomes a unital [Formula: see text]-algebra that is isometrically isomorphic to the [Formula: see text]-algebra [Formula: see text], provided that [Formula: see text] and [Formula: see text] are isomorphic.

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.

scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Numerical Analysis ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Bounded Operator ◽  
Fractional Part ◽  
Equivalent Formulation ◽  
Linear Bounded Operator ◽  
Pure Mathematics ◽  
The One ◽  
Bode Integral

Abstract Using the equivalent formulation of RH given by Beurling ([4],1955), Alcantara-Bode showed ([2], 1993) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel defined by fractional part function of the expressionbetween brackets {y/x}, is injective.Since then, the injectivity of the integral operator used in equivalentformulation of RH has not been addressed nor has been dissociatedfrom RH and, a pure mathematics solution for RH is not ready yet.Here is a numerical analysis approach of the injectivity of the linearbounded operators on separable Hilbert spaces addressing the problemslike the one presented in [2]. Apart of proving the injectivity of theBeurling - Alcantara-Bode integral operator, we obtained the followingresult: every linear bounded operator (or its associated Hermitian),strict positive definite on a dense family of including approximationsubspaces in L2(0,1) built on simple functions, is injective if the rateof convergence to zero of its unbounded sequence of inverse conditionnumbers on approximation subspaces is o(n-s) for some s ≥ 0. Whens = 0, the sequence is inferior bounded by a not null constant, that isthe case in the Beurling - Alcantara-Bode integral operator.In the Theorem 4.1 we addressed with numerical analysis toolsthe injectivity of the integral operator in [2] claiming that - even if asolution in pure mathematics is desired, together with the Theorem 1,pg. 153 in [2], the RH holds.

scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Fractional Part ◽  
Positive Definite ◽  
Its Sequence ◽  
Bounded Operators ◽  
Linear Bounded Operator

Abstract In 1993, Alcantara-Bode showed ([2]) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel function defined by the fractional part of (y/x), isinjective. Since then, the injectivity of the integral operator used inequivalent formulation of RH has not been addressed nor has beendissociated from RH.We provided in this paper methods for investigating the injectivityof linear bounded operators on separable Hilbert spaces using theirapproximations on dense families of subspaces.On the separable Hilbert space L2(0,1), an linear bounded operator(or its associated Hermitian), strict positive definite on a dense familyof including approximation subspaces in built on simple functions, isinjective if the rate of convergence of its sequence of injectivity pa-rameters on approximation subspaces is inferior bounded by a not nullconstant, that is the case with the Beurling - Alcantara-Bode integraloperator.We applied these methods to the integral operator used in RHequivalence proving its injectivity.

scholarly journals Some Inequalities for g-Frames in Hilbert C*-Modules

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 25
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Operator Theory ◽  
Bessel Sequences ◽  
New Inequalities

In this paper, we obtain new inequalities for g-frames in Hilbert C * -modules by using operator theory methods, which are related to a scalar λ ∈ R and an adjointable operator with respect to two g-Bessel sequences. It is demonstrated that our results can lead to several known results on this topic when suitable scalars and g-Bessel sequences are chosen.

scholarly journals A New Inequality for Frames in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Bessel Sequences

We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.

scholarly journals On New Inequalities of Simpson’s Type for Functions Whose Second Derivatives Absolute Values are Convex

2013 ◽  
Vol 9 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Mehmet Zeki Sarikaya ◽  
Erhan. Set ◽  
M. Emin Ozdemir
Keyword(s):  
Real Numbers ◽  
Special Means ◽  
Second Derivatives ◽  
New Inequalities

Abstract In this note, we obtain new some inequalities of Simpson’s type based on convexity. Some applications for special means of real numbers are also given.

scholarly journals Fractional Ostrowski type inequalities for differentiable harmonically convex functions

2021 ◽  
Vol 7 (3) ◽  
pp. 3939-3958
Author(s):  
Thanin Sitthiwirattham ◽  
◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Sotiris K. Ntouyas ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Real Numbers ◽  
Special Means ◽  
Special Cases ◽  
Harmonically Convex Functions ◽  
New Inequalities

<abstract><p>In this paper, we prove some new Ostrowski type inequalities for differentiable harmonically convex functions using generalized fractional integrals. Since we are using generalized fractional integrals to establish these inequalities, therefore we obtain some new inequalities of Ostrowski type for Riemann-Liouville fractional integrals and $ k $-Riemann-Liouville fractional integrals in special cases. Finally, we give some applications to special means of real numbers for newly established inequalities.</p></abstract>

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