On an operator equation in Hilbert space

1974 ◽  
Vol 25 (6) ◽  
pp. 660-661
Author(s):  
A. T. Zaplitnaya
Keyword(s):  
Hilbert Space ◽  
Operator Equation

scholarly journals An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems

1988 ◽  
Vol 31 (1) ◽  
pp. 99-105 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Space Applications ◽  
Bounded Linear ◽  
Image Position

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the typeare given in terms of solutions of an algebraic operator equation

10.—An Abstract Relation for Multiparameter Eigenvalue Problems

1976 ◽  
Vol 74 ◽  
pp. 135-143 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Image Position ◽  
Symmetric Operators ◽  
Densely Defined

SynopsisConsider the multiparameter systemwhere ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equationwhere B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

Application of g-frames in conjugate gradient

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Hassan Jamali ◽  
Neda Momeni
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Operator Equation ◽  
Conjugate Gradient ◽  
Separable Hilbert Space

AbstractThis paper proposes an iterative method for solving an operator equation on a separable Hilbert space

scholarly journals On the operator equationα+α−1=β+β−1

1986 ◽  
Vol 9 (4) ◽  
pp. 767-770 ◽  
Author(s):  
A. B. Thaheem
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Von Neumann Algebra ◽  
Central Projection ◽  
Bounded Operators ◽  
New Techniques ◽  
Von Neumann ◽  
Alpha Beta

Letα,βbe∗-automorphisms of a von Neumann algebraMsatisfying the operator equationα+α−1=β+β−1. In this paper we use new techniques (which are useful in noncommutative situations as well) to provide alternate proofs of the results:- Ifα,βcommute then there is a central projectionpinMsuch thatα=βonMPandα=β−1onM(1−P); IfM=B(H), the algebra of all bounded operators on a Hilbert spaceH, thenα=βorα=β−1.

scholarly journals Generalized Fuglede-Putnam Theorem and $m$-quasi-class $A(k)$ operators

2019 ◽  
pp. 73
Author(s):  
Mohammad H.M. Rashid
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Operator Equation ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Dense Range ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Class A ◽  
Dimensional Hilbert Space

For a bounded linear operator $T$ acting on acomplex infinite dimensional Hilbert space $\h,$ we say that $T$is $m$-quasi-class $A(k)$ operator for $k>0$ and $m$ is apositive integer (abbreviation $T\in\QAkm$) if$T^{*m}\left((T^*|T|^{2k}T)^{\frac{1}{k+1}}-|T|^2\right)T^m\geq0.$ The famous {\it Fuglede-Putnam theorem} asserts that: the operator equation$AX=XB$ implies $A^*X=XB^*$ when $A$ and $B$ are normal operators.In this paper, we prove that if $T\in \QAkm$ and $S^*$ isan operator of class $A(k)$ for $k>0$. Then $TX=XS$, where $X\in\bh$ is an injective with dense range implies $XT^*=S^*X$.

scholarly journals SOLUTION OF A NONLINEAR OPERATOR EQUATION OF THE FIRST KIND WITH A CONTINUOUS POSITIVE LINEAR OPERATOR

2021 ◽  
pp. 118-123
Author(s):  
I.A. Usenov ◽  
R.K. Usenova ◽  
A. Nurkalieva
Keyword(s):  
Hilbert Space ◽  
Exact Solution ◽  
Approximate Solution ◽  
Linear Operator ◽  
Operator Equation ◽  
Nonlinear Operator ◽  
Regularization Parameter ◽  
Positive Linear Operator ◽  
Nonlinear Operator Equation ◽  
The Right

In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately. Based on the method of Lavrent'ev M.M. an approximate solution of the equation in Hilbert space is constructed. The dependence of the regularization parameter on errors was selected. The rate of convergence of the approximate solution to the exact solution of the original equation is obtained.

scholarly journals An iterative method for solution of nonlinear operator equation

2016 ◽  
Vol 13 (1) ◽  
pp. 6-10
Author(s):  
Nguyễn Bường
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Integral Equations ◽  
Operator Equation ◽  
Iterative Process ◽  
Nonlinear Operator ◽  
Nonlinear Operator Equation ◽  
Nonlinear Integral Equations

In the note, for finding a solution of nonlinear operator equation of Hammerstein’s type an iterative process in infinite-dimentional Hilbert space is shown, where a new iteration is constructed basing on two last steps. An example in the theory of nonlinear integral equations is given for illustration.

The operator equation TS = 1 and representations of l1 of the bicyclic semigroup

1977 ◽  
Vol 79 (1-2) ◽  
pp. 131-136
Author(s):  
John B. Conway ◽  
Bernard B. Morrel ◽  
Joseph G. Stampfli
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Bounded Operators ◽  
Bicyclic Semigroup ◽  
Power Bounded Operators ◽  
Power Bounded

SynopsisThe operator equation TS = 1 is studied for power bounded operators T, S on Hilbert space, and its relation to *—representations of the bicyclic semigroup is explored.

scholarly journals On the Positive Operator Solutions to an Operator Equation X − A ∗ X − t A = Q

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Kaifan Yang
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Equation ◽  
Positive Operator ◽  
Iterative Sequence ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

In this paper, the positive operator solutions to operator equation X − A ∗ X − t A = Q (t > 1) are studied in infinite dimensional Hilbert space. Firstly, the range of norm and the spectral radius of the solution to the equation are given. Secondly, by constructing effective iterative sequence, it gives some conditions for the existence of positive operator solutions to operator equation X − A ∗ X − t A = Q (t > 1). The relations of these operators in the operator equation are given.

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