A note on hyponormal operators

In this paper, We introduce a new class of multivariable operators known as joint m-quasihyponormal tuple of operators. It is a naturel extension of joint normal and joint hyponormal tuples of operators. An m-tuple of operators $$\mathbf{S}=(S_1, \ldots ,S_m)\in {{\mathcal {B}}}({{\mathcal {H}}})^m$$
 is said to be joint m-quasihyponormal tuple if $$\mathbf{S}$$
 satisfying $$\begin{aligned} \displaystyle \sum _{1\le l,\;k\;\le m}\big \langle S_k^*\big [S_k^*,\;\; S_l\big ]S_lu_k\;|\;u_l\big \rangle \ge 0, \end{aligned}$$
 for each finite collections $$(u_l)_{1\le l\le m}\in {{\mathcal {H}}}.$$
 Some properties of this class of multivariable operators are studied.

Joint m-quasihyponormal operators on a Hilbert space

Abstract This paper is concerned with studying a new class of multivariable commuting operators know as left $(m,C)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> -invertible p -tuples and related to a given conjugation transformation C on a Hilbert space $\mathcal{Y}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> . Some structural properties of some members of this class are given. 

/pdf/class-of-operators-related-to-a-m-c-m-c-isometric-tuple-of-syp7olib.pdf

Class of operators related to a 
              
                
              
              
                (
                m
                ,
                C
                )
              
              $(m,C)$
            -isometric tuple of commuting operators

/pdf/class-of-operators-related-to-a-m-c-documentclass-12pt-18cezh19.pdf

Class of operators related to a (m,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(m,C)$\end{document}-isometric tup

/pdf/joint-n-normality-of-linear-transformations-4fd9w1jmto.pdf

Joint n-normality of linear transformations

The purpose of the paper is to introduce and study a class of multivariable operators on semi-Hilbertian spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Let T=(T1,…,Tm) a...

Joint A-hyponormality of operators in semi-Hilbert spaces

Our aim in this paper is to consider a generalization of the concept of n-quasi-m-isometric operators of a single operator done in Mahmoud Sid Ahmed et al. (Results Math 73:511–531, 2018) and Mechri and Mahmoud Sid Ahmed (Oper Matrices 14(1): 145–157, 2020) to the multi-dimensional operators. We discuss the most interesting results concerning these classes of tuples of operators by extending some results obtained in several recent papers for a single operator.

$$(n_1,\ldots ,n_p)$$ ( n 1 , … , n p ) -quasi-m-isometric commuting tuple of operators on a Hilbert space

On joint posinormality of operators

<jats:p>The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operators. We give a characterization for any operator to be <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operator. Using this characterization, we prove that any power of an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operator is also an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operator. Furthermore, we study the perturbation of an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetric operator with a nilpotent operator. The product and tensor products of two <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-quasi-<jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>m</mi>
                              <mo>,</mo>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-isosymmetries are investigated.</jats:p>

Structure of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                     <mi>k</mi>
                  </math>-Quasi-<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                     <mfenced open="(" close=")" separators="|">
                        <mrow>
           

e investigation of new operators belonging to some specic classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called k-quasi(m, n)-isosymmetric operators on Hilbert spaces. is new class of operators emerges as a generalization of the (m, n)-isosymmetric operators. We give a characterization for any operator to be k-quasi-(m, n)-isosymmetric operator. Using this characterization, we prove that any power of an k-quasi-(m, n)-isosymmetric operator is also an k-quasi-(m, n)-isosymmetric operator. Furthermore, we study the perturbation of an k-quasi-(m, n)-isosymmetric operator with a nilpotent operator. e product and tensor products of two k-quasi-(m, n)-isosymmetries are investigated.

El Moctar Ould Beiba

Papers

Joint A-hyponormality of operators in semi-Hilbert spaces

$$(n_1,\ldots ,n_p)$$ ( n 1 , … , n p ) -quasi-m-isometric commuting tuple of operators on a Hilbert space

On joint posinormality of operators

Structure of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>k</mi> </math>-Quasi-<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")" separators="|"> <mrow>

Structure of k-Quasi-( m , n )-Isosymmetric Operators