A fixed point method to solve linear operator equations involving self-adjoint operators in Hilbert space

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3249-3251
Author(s):  
Mohammad Khan ◽  
Dinu Teodorescu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Fixed Point Method ◽  
Operator Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Adjoint Operators ◽  
Invertible Matrices ◽  
Linear Operator Equations

In this paper we provide existence and uniqueness results for linear operator equations of the form (I+Am) x = f , where A is a self-adjoint operator in Hilbert space. Some applications to the study of invertible matrices are also presented.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals On the iterative solution of linear operator equations with self-adjoint operators

1972 ◽  
Vol 13 (2) ◽  
pp. 241-255 ◽  
Author(s):  
J. J. Koliha
Keyword(s):  
Hilbert Space ◽  
Approximate Solution ◽  
Linear Operator ◽  
Iterative Method ◽  
Iterative Solution ◽  
Matrix Equations ◽  
Operator Equations ◽  
Adjoint Operators ◽  
Linear Operator Equations ◽  
General Iterative Method

In this paper we deal with a linear equation Au = f in a Hilbert space using a general iterative method with a constant iterative operator for the approximate solution. The method has been studied in many papers [1, 2, 4, 9, 13, 14] and thoroughly treated by Householder [3] for matrix equations and by Petryshyn [7] for operator equations in considerably general and unified manner.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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