scholarly journals On linear operators for which TTD is normal

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2695-2704
Author(s):  
Ramesh Yousefi ◽  
Mansour Dana
Keyword(s):  
Matrix Representation ◽  
Normal Operator ◽  
Invertible Operator ◽  
Linear Operators ◽  
Commutativity Theorem ◽  
Normal Matrices ◽  
Basic Properties ◽  
The Matrix ◽  
Normal Operators ◽  
Drazin Invertible Operator

A Drazin invertible operator T ? B(H) is called skew D-quasi-normal operator if T* and TTD commute or equivalently TTD is normal. In this paper, firstly we give a list of conditions on an operator T; each of which is equivalent to T being skew D-quasi-normal. Furthermore, we obtain the matrix representation of these operators. We also develop some basic properties of such operators. Secondly we extend the Kaplansky theorem and the Fuglede-Putnam commutativity theorem for normal operators to skew D-quasi-normal matrices.

scholarly journals On numerical ranges of generalized derivations and related properties

1984 ◽  
Vol 36 (1) ◽  
pp. 134-142 ◽  
Author(s):  
Sen-Yen Shaw
Keyword(s):  
Numerical Range ◽  
Normal Operator ◽  
Linear Operators ◽  
Minimal Norm ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Normal Operators ◽  
Schmidt Operator ◽  
The Difference ◽  
Norm Ideal

AbstractThis paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.

scholarly journals Property $$(UW{\scriptstyle \Pi })$$ under perturbations

2019 ◽  
Vol 11 (1) ◽  
pp. 29-46
Author(s):  
P. Aiena ◽  
M. Kachad
Keyword(s):  
Analytic Function ◽  
Banach Spaces ◽  
Invertible Operator ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Weyl’S Theorem ◽  
Bounded Linear ◽  
Browder’S Theorem ◽  
The Stability ◽  
Drazin Invertible Operator

AbstractProperty $$(UW {\scriptstyle \Pi })$$(UWΠ), introduced in Berkani and Kachad (Bull Korean Math Soc 49:1027–1040, 2015) and studied more recently in Aiena and Kachad (Acta Sci Math (Szeged) 84:555–571, 2018) may be thought as a variant of Browder’s theorem, or Weyl’s theorem, for bounded linear operators acting on Banach spaces. In this article we study the stability of this property under some commuting perturbations, as quasi-nilpotent perturbation and, more in general, under Riesz commuting perturbations. We also study the transmission of property $$(UW {\scriptstyle \Pi })$$(UWΠ) from T to f(T), where f is an analytic function defined on a neighborhood of the spectrum of T. Furthermore, it is shown that this property is transferred from a Drazin invertible operator T to its Drazin inverse S.

Quasi-normal matrices and products

1970 ◽  
Vol 11 (3) ◽  
pp. 329-339 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Canonical Form ◽  
Normal Matrix ◽  
Complex Conjugate ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Basic Properties ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Class Of Matrices

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

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).

MRHS solver based on linear algebra and exhaustive search

2018 ◽  
Vol 12 (3) ◽  
pp. 143-157 ◽  
Author(s):  
Håvard Raddum ◽  
Pavol Zajac
Keyword(s):  
Linear Algebra ◽  
Matrix Representation ◽  
Equation System ◽  
Exhaustive Search ◽  
Binary Matrix ◽  
Algebraic Attacks ◽  
The Matrix ◽  
Speed Up ◽  
Symmetric Key

Abstract We show how to build a binary matrix from the MRHS representation of a symmetric-key cipher. The matrix contains the cipher represented as an equation system and can be used to assess a cipher’s resistance against algebraic attacks. We give an algorithm for solving the system and compute its complexity. The complexity is normally close to exhaustive search on the variables representing the user-selected key. Finally, we show that for some variants of LowMC, the joined MRHS matrix representation can be used to speed up regular encryption in addition to exhaustive key search.

scholarly journals Quasi-posinormal operators

2010 ◽  
Vol 7 (3) ◽  
pp. 1282-1287
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Basic Properties ◽  
Hyponormal Operators ◽  
The Relationship

In this paper, we introduce a class of operators on a Hilbert space namely quasi-posinormal operators that contain properly the classes of normal operator, hyponormal operators, M–hyponormal operators, dominant operators and posinormal operators . We study some basic properties of these operators .Also we are looking at the relationship between invertibility operator and quasi-posinormal operator .

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