On ( n , m ) - A -normal and ( n , m ) - A -quasinormal semi-Hilbertian space operators

Samir Al Mohammady; Sid Ahmed Ould Beinane; Sid Ahmed Ould Ahmed Mahmoud

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 2, page 169-186
  • ISSN: 0862-7959

Abstract

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The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let ℋ be a Hilbert space and let A be a positive bounded operator on ℋ . The semi-inner product 〈 h ∣ k 〉 A : = 〈 A h ∣ k 〉 , h , k ∈ ℋ , induces a semi-norm ∥ · ∥ A . This makes ℋ into a semi-Hilbertian space. An operator T ∈ ℬ A ( ℋ ) is said to be ( n , m ) - A -normal if [ T n , ( T ♯ A ) m ] : = T n ( T ♯ A ) m - ( T ♯ A ) m T n = 0 for some positive integers n and m .

How to cite

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Al Mohammady, Samir, Ould Beinane, Sid Ahmed, and Ould Ahmed Mahmoud, Sid Ahmed. "On $(n,m)$-$A$-normal and $(n,m)$-$A$-quasinormal semi-Hilbertian space operators." Mathematica Bohemica 147.2 (2022): 169-186. <http://eudml.org/doc/297917>.

@article{AlMohammady2022,
abstract = {The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $\{\mathcal \{H\}\}$ be a Hilbert space and let $A$ be a positive bounded operator on $\{\mathcal \{H\}\}$. The semi-inner product $\langle h\mid k\rangle _A:=\langle Ah\mid k\rangle $, $h,k \in \{\mathcal \{H\}\}$, induces a semi-norm $\Vert \{\cdot \}\Vert _A$. This makes $\{\mathcal \{H\}\}$ into a semi-Hilbertian space. An operator $T\in \{\mathcal \{B\}\}_A(\{\mathcal \{H\}\})$ is said to be $(n,m)$-$A$-normal if $[T^n,(T^\{\sharp _A\})^m]:=T^n(T^\{\sharp _A\})^m-(T^\{\sharp _A\})^mT^n=0$ for some positive integers $n$ and $m$.},
author = {Al Mohammady, Samir, Ould Beinane, Sid Ahmed, Ould Ahmed Mahmoud, Sid Ahmed},
journal = {Mathematica Bohemica},
keywords = {semi-Hilbertian space; $A$-normal operator; $(n,m)$-normal operator; $(n,m)$-quasinormal operator},
language = {eng},
number = {2},
pages = {169-186},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $(n,m)$-$A$-normal and $(n,m)$-$A$-quasinormal semi-Hilbertian space operators},
url = {http://eudml.org/doc/297917},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Al Mohammady, Samir
AU - Ould Beinane, Sid Ahmed
AU - Ould Ahmed Mahmoud, Sid Ahmed
TI - On $(n,m)$-$A$-normal and $(n,m)$-$A$-quasinormal semi-Hilbertian space operators
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 2
SP - 169
EP - 186
AB - The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let ${\mathcal {H}}$ be a Hilbert space and let $A$ be a positive bounded operator on ${\mathcal {H}}$. The semi-inner product $\langle h\mid k\rangle _A:=\langle Ah\mid k\rangle $, $h,k \in {\mathcal {H}}$, induces a semi-norm $\Vert {\cdot }\Vert _A$. This makes ${\mathcal {H}}$ into a semi-Hilbertian space. An operator $T\in {\mathcal {B}}_A({\mathcal {H}})$ is said to be $(n,m)$-$A$-normal if $[T^n,(T^{\sharp _A})^m]:=T^n(T^{\sharp _A})^m-(T^{\sharp _A})^mT^n=0$ for some positive integers $n$ and $m$.
LA - eng
KW - semi-Hilbertian space; $A$-normal operator; $(n,m)$-normal operator; $(n,m)$-quasinormal operator
UR - http://eudml.org/doc/297917
ER -

References

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