scholarly journals The best approximation of classes, defined by powers of self-adjoint operators acting in Hilbert space, by other classes

2021 ◽  
Vol 17 ◽  
pp. 23
Author(s):  
V.F. Babenko ◽  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Adjoint Operator ◽  
The Best Approximation ◽  
Adjoint Operators

The best approximation of class of elements such that $\| A^k x \| \leqslant 1$ by classes of elements such that $\| A^r x \| \leqslant N$, $N > 0$ for powers $k < r$ of self-adjoint operator $A$ in Hilbert space $H$ is found.

scholarly journals Approximation of unbounded functionals by the bounded ones in Hilbert space

2012 ◽  
Vol 20 ◽  
pp. 3
Author(s):  
V.F. Babenko ◽  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Adjoint Operator ◽  
The Best Approximation

We obtained the value of the best approximation of unbounded functional $F_f(x) = (A^kx, f)$ on the class $\{ x\in D(A^r) \colon \| A^r x \| \leqslant 1 \}$ by linear bounded functionals ($A$ is a self-adjoint operator in the Hilbert space $H$, $f\in H$, $k < r$).

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.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals Approximation of unbounded functional, defined by grades of normal operator, on the class of elements of Hilbert space

2016 ◽  
Vol 24 ◽  
pp. 3
Author(s):  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Normal Operator ◽  
The Best Approximation

We obtain the best approximation of unbounded functional $(A^k x; f)$ on the class $\{ x\in D(A^r) \colon \| A^r x \| \leqslant 1 \}$ by linear bounded functionals for a normal operator $A$ in the Hilbert space $H$ ($k < r$, $f\in H$).

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.

scholarly journals Self adjoint operators and matrix measures

1971 ◽  
Vol 4 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Spectral Representation ◽  
Adjoint Operator ◽  
Multiplication Operator ◽  
Linear Operators ◽  
Positive Matrix ◽  
Adjoint Operators ◽  
Matrix Measures

Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

A Spectral Theorem for Hermitian Operators of Meromorphic Type on Banach Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 703-708
Author(s):  
T. Owusu-Ansah
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Adjoint Operator ◽  
Spectral Projection ◽  
Spectral Theorem ◽  
Adjoint Operators

It is well known that if T is a compact self-adjoint operator on a Hilbert space whose distinct non-zero eigenvalues {λn} are arranged so that |λn|≥|λn+1| for n = 1, 2…. and if En in the spectral projection corresponding to λn, then with convergence in the uniform operator topology. With the generalisation of self-adjoint operators on Hilbert spaces to Hermitian operators on Banach spaces by Vidav and Lumer, Bonsall gave a partial analogue of this result for Banach spaces when he proved the following theorem.

scholarly journals ON THE BEST APPROXIMATION OF THE INFINITESIMAL GENERATOR OF A CONTRACTION SEMIGROUP IN A HILBERT SPACE

2017 ◽  
Vol 3 (2) ◽  
pp. 40-45 ◽  
Author(s):  
Elena E. Berdysheva ◽  
Maria A. Filatova
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Infinitesimal Generator ◽  
Contraction Semigroup ◽  
The Best Approximation

scholarly journals Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1918
Author(s):  
Oleh Lopushansky ◽  
Renata Tłuczek-Piȩciak
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Adjoint Operator ◽  
Invariant Subspaces ◽  
Accurate Estimate ◽  
Complex Hilbert Space ◽  
Best Approximations ◽  
Adjoint Operators ◽  
Increasing Sequences

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.

scholarly journals Some problems of approximation theory for powers of normal operators in Hilbert space

2021 ◽  
Vol 18 ◽  
pp. 59
Author(s):  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Approximation Theory ◽  
Best Approximation ◽  
Unbounded Operator ◽  
Normal Operator ◽  
The Best Approximation ◽  
Normal Operators

The best approximation of unbounded operator $A^k$ in class with $\| A^r x \| \leqslant 1$ and the best approximation of class with $\|A^k x \| \leqslant 1$ by class with $\| A^r x \| \leqslant N$, $N > 0$ for powers $k < r$ of normal operator $A$ in the Hilbert space $H$ are found.

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