scholarly journals An optimum cubically convergent iterative method of inverting a linear bounded operator in Hilbert space

1960 ◽  
Vol 10 (4) ◽  
pp. 1107-1113 ◽  
Author(s):  
M. Altman
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Bounded Operator ◽  
Linear Bounded Operator

scholarly journals Continuous frames for unbounded operators

2021 ◽  
Vol 6 (2) ◽  
Author(s):  
Giorgia Bellomonte
Keyword(s):  
Hilbert Space ◽  
Unbounded Operator ◽  
Atomic System ◽  
Bounded Operator ◽  
Unbounded Operators ◽  
Linear Bounded Operator ◽  
Continuous Frames ◽  
Densely Defined ◽  
Continuous Setting

AbstractFew years ago Găvruţa gave the notions of K-frame and atomic system for a linear bounded operator K in a Hilbert space $$\mathcal {H}$$ H in order to decompose $$\mathcal {R}(K)$$ R ( K ) , the range of K, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator A on a Hilbert space in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.

scholarly journals General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nopparat Wairojjana ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Linear Bounded Operator ◽  
New Methods

This paper deals with new methods for approximating a solution to the fixed point problem; findx̃∈F(T), whereHis a Hilbert space,Cis a closed convex subset ofH,fis aρ-contraction fromCintoH,0<ρ<1,Ais a strongly positive linear-bounded operator with coefficientγ̅>0,0<γ<γ̅/ρ,Tis a nonexpansive mapping onC,andPF(T)denotes the metric projection on the set of fixed point ofT. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality〈(A-γf)x̃+τ(I-S)x̃,x-x̃〉≥0forx∈F(T), whereτ∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

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 Modified Iterative Algorithms for Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Hilbert Space ◽  
Approximate Solution ◽  
Real Number ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Contractive Mapping ◽  
Linear Bounded Operator ◽  
Special Cases

Let H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)∩F(T)≠∅, let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1−βn)Syn, yn=αnγf(xn)+(I−αnA)Txn, n≥0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following control conditions: (C1) lim⁡n→∞ αn=0, (C3) 0<lim⁡inf⁡n→∞ βn≤lim⁡sup⁡n→∞ βn<1, then ‖xn+1−xn‖→0. We also discuss several special cases of this iterative algorithm.

A note on nth roots of positive operators

1969 ◽  
Vol 66 (1) ◽  
pp. 27-30
Author(s):  
John H. Jowett
Keyword(s):  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Spectral Representation ◽  
Bounded Operator ◽  
Positive Operators

The existence and uniqueness of a positive self-adjoint nth. root of a positive, self-adjoint, not necessarily bounded operator on a Hilbert Space H can be readily demonstrated using the spectral representation of the transformation.

scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set&#x1D4AE;r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the&#x1D4AE;r(ℬ)setting.

scholarly journals Explicit Scheme for Fixed Point Problem for Nonexpansive Semigroup and Split Equilibrium Problem in Hilbert Space

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Pei Zhou ◽  
Gou-Jie Zhao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Explicit Scheme ◽  
Common Element ◽  
Nonexpansive Semigroup ◽  
Point Problem ◽  
Split Equilibrium Problem

We establish an iterative method for finding a common element of the set of fixed points of nonexpansive semigroup and the set of split equilibrium problems. Under suitable conditions, some strong convergence theorems are proved. Our works improve previous results for nonexpansive semigroup.

scholarly journals A New Iterative Method for Equilibrium Problems and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Abdul Latif ◽  
Mohammad Eslamian
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Common Element ◽  
Continuous Mappings ◽  
New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

scholarly journals Hyponormality on general Bergman spaces

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5737-5741 ◽  
Author(s):  
Houcine Sadraoui
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Toeplitz Operators ◽  
Bounded Operator ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

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