For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_455_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$T \in \mathcal {B}_d({\mathcal {H}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <msub> <mi mathvariant="script">B</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (the set of Drazin invertible operators), we say that T is polynomially Drazin normal if there exits a non trivial complex polynomial P such that <Equation ID="Equ3"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_455_Article_Equ3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="191" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} P(T^D)T^-T^P(T^D)=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mi>D</mi> </msup> <mo stretchy="false">)</mo> </mrow> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>-</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mi>D</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>or equivalently, <Equation ID="Equ4"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_455_Article_Equ4.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="266" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{0\le k\le n}a_k\bigg (\big (T^D)^kT^-T^\big (T^D\big )^k\bigg )= 0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mi>k</mi> </msub> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">(</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msup> <mi>T</mi> <mi>D</mi> </msup> <msup> <mrow> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>-</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msup> <mi>T</mi> <mi>D</mi> </msup> <msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>k</mi> </msup> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper we study some structural properties of polynomially Drazin normal operators. Our motivation for this study comes from the problem of finding operators that their Drazin inverses are polynomially normal operators.

Some results on polynomially Drazin normal operators

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发表日期：

2025年1月11日

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Sid Ahmed Ould Ahmed Mahmoud,
Messaoud Guesba,
Naeem Ahmad

摘要

For $T \in \mathcal {B}_d({\mathcal {H}})$ $T \in B_{d} (H)$ (the set of Drazin invertible operators), we say that T is polynomially Drazin normal if there exits a non trivial complex polynomial P such that $\begin{aligned} P(T^D)T^*-T^*P(T^D)=0, \end{aligned}$ $\begin{matrix} P (T^{D}) T^{*} - T^{*} P (T^{D}) = 0, \end{matrix}$ or equivalently, $\begin{aligned} \sum _{0\le k\le n}a_k\bigg (\big (T^D)^kT^*-T^*\big (T^D\big )^k\bigg )= 0. \end{aligned}$ $\begin{matrix} \sum_{0 \leq k \leq n} a_{k} ((T^{D})^{k} T^{*} - T^{*} (T^{D})^{k}) = 0 . \end{matrix}$ In this paper we study some structural properties of polynomially Drazin normal operators. Our motivation for this study comes from the problem of finding operators that their Drazin inverses are polynomially normal operators.