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.

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 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.

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).

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

scholarly journals A note on d-symmetric operators

1981 ◽  
Vol 23 (3) ◽  
pp. 471-475
Author(s):  
B. C. Gupta ◽  
P. B. Ramanujan
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Commutator Ideal ◽  
Derivation Operator ◽  
Double Commutant ◽  
Symmetric Operators ◽  
The Ideal

An operator T on a complex Hilbert space is d-symmetric if , where is the uniform closure of the range of the derivation operator δT(X)=TX−XT. It is shown that if the commutator ideal of the inclusion algebra for a d-symmetric operator is the ideal of all compact operators then T has countable spectrum and T is a quasidiagonal operator. It is also shown that if for a d-symmetric operator I(T) is the double commutant of T then T is diagonal.

scholarly journals On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

2020 ◽  
Vol 53 (1) ◽  
pp. 352-359
Author(s):  
Marat V. Markin
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Continuous Group ◽  
Left Translation ◽  
Bounded Linear ◽  
Scalar Type ◽  
Operator Group

Abstract Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection { e t A } t ≥ 0 {\{{e}^{tA}\}}_{t\ge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C 0 {C}_{0} -semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group { e t A } t ∈ ℝ {\{{e}^{tA}\}}_{t\in {\mathbb{R}}} of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L 2 ( ℝ ) {L}_{2}({\mathbb{R}}) , the anti-self-adjoint differentiation operator A ≔ d d x A:= \tfrac{\text{d}}{\text{d}x} with the domain D ( A ) ≔ W 2 1 ( ℝ ) D(A):= {W}_{2}^{1}({\mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

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 Remarks on n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5441-5451 ◽  
Author(s):  
Muneo Chō ◽  
Ji Lee ◽  
Kotaro Tanahashi ◽  
Atsushi Uchiyama
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Normal Operators

Let T be a bounded linear operator on a complex Hilbert space and n,m ? N. Then T is said to be n-normal if T+Tn = TnT+ and (n,m)-normal if T+mTn = TnT+m. In this paper, we study several properties of n-normal, (n,m)-normal operators. In particular, we prove that if T is 2-normal with ?(T) ? (-?(T)) ? {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n,m)-normal, then T is decomposable and Weyl?s theorem holds for f (T), where f is an analytic function on ?(T) which is not constant on each of the components of its domain.

scholarly journals Extensions of dissipative operators with closable imaginary part

2021 ◽  
Vol 41 (3) ◽  
pp. 381-393
Author(s):  
Christoph Fischbacher
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Operator ◽  
Dissipative Operators ◽  
First Order ◽  
Necessary And Sufficient ◽  
Symmetric Operators

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

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