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.

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 $H$ ($k < r$, $f\in H$).

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

scholarly journals On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 903
Author(s):  
Marat V. Markin ◽  
Edward S. Sichel
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Symmetric Operators ◽  
Straightforward Proof

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

scholarly journals Inequality of Taikov type for powers of normal operators in Hilbert space

2021 ◽  
Vol 19 ◽  
pp. 3
Author(s):  
V.F. Babenko ◽  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Intermediate Derivative ◽  
Higher Derivative ◽  
Normal Operators

The Taikov inequality, which estimates $L_{\infty}$-norm of intermediate derivative by $L_2$-norms of a function and its higher derivative, is extended on arbitrary powers of normal operator acting in Hilbert space.

scholarly journals On (P*-N) quasi normal operators Of order "n" In Hilbert space

2021 ◽  
Vol 32 (1) ◽  
pp. 10
Author(s):  
Salim Dawood M. ◽  
Jaafer Hmood Eidi
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Normal Operator ◽  
Sufficient Conditions ◽  
Basic Operation ◽  
Properties And Characterization ◽  
Normal Operators

Through this paper, we submitted  some types of quasi normal operator is called be (k*-N)- quasi normal operator of order n defined on a Hilbert space H, this concept is generalized of some kinds of  quasi normal operator appear recently form most researchers in the  field of functional analysis, with some properties  and characterization of this operator   as well as, some basic operation such as addition and multiplication of these operators had been given, finally the relationships of this operator proved with some examples to illustrate conversely and introduce the sufficient conditions to satisfied this case with other types had been studied.

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.

On the spectral theorem for normal operators

1970 ◽  
Vol 68 (2) ◽  
pp. 393-400 ◽  
Author(s):  
R. G. Douglas ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Measure Theory ◽  
Normal Operator ◽  
The Other ◽  
Spectral Theorem ◽  
Ideal Space ◽  
Von Neumann ◽  
Normal Operators ◽  
Abelian Von Neumann Algebra

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

scholarly journals Spectral properties of n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5063-5069 ◽  
Author(s):  
Muneo Chō ◽  
Biljana Nacevska
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Properties ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Isolated Point ◽  
Zero Number

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

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