PRESERVER OF SOME SPECTRAL DOMAINS
OF PRODUCT OPERATORS
R. Parvinianzadeh
Department of Mathematics
University of Yasouj
Yasouj 75918, IRAN

Abstract

Let $B(X)$ be the algebra of all bounded linear operators on infinite-dimensional complex Banach space $X$. For $T \in B(X)$ and $i \in \{1,2,3\}$, let $\sigma_{i}(T)$ denote any one of the semi-Fredholm domain, the Fredholm domain and the Weyl domain in the spectrum. We prove that if two maps $\varphi_{1}$ and $\varphi_{2}$ from $B(X)$ onto $B(X)$ satisfy

\begin{displaymath}\sigma_{i}(\varphi_{1}(T)\varphi_{2}(S))= \sigma_{i}(TS)\end{displaymath}

for all $T, S \in B(X)$, then either:
(1) there is a bounded linear operator $A:X\rightarrow X$ such that $\varphi_{1}(T) = AT(\varphi_{2}(I)A)^{-1}$ and $\varphi_{2}(T) =\varphi_{2}(I)ATA^{-1}$ for all $T \in B(X)$, or
(2) there is a bounded linear operator $A:X^{\ast}\rightarrow X$ such that $\varphi_{1}(T) = AT^{\ast}(\varphi_{2}(I)A)^{-1}$ and $\varphi_{2}(T) =\varphi_{2}(I)AT^{\ast}A^{-1}$ for all $T \in B(X)$. Where $I$ is identity operator on $X$.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 4
Year: 2023

DOI: 10.12732/ijam.v36i4.3

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References

  1. [1] M. Gonzalez, M. Mbekhta, Linear maps that preserve semi-Fredholm operators acting on Banach spaces, Acta Sci Math., 84, No 12 (2018), 137-149.
  2. [2] S. Hajighasemi, S. Hejazian, Maps preserving some spectral domains of operator products, Linear Multilinear Alg., (2020) 2367-2381.
  3. [3] M. Mbekhta, Linear maps preserving the set of Fredholm operators, Proc Amer Math Soc., 135, No 11 (2007), 3613-3619.482 R. Parvinianzadeh
  4. [4] M. Mbekhta, P. Semrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Alg., 57, No 1 (2009), 55-64.
  5. [5] M. Omladic, P. Semrl, Additive mappings preserving operators of rank one, Linear Algebra Appl., 182 (1993), 239-256.
  6. [6] W. Shi, G. Ji, Additive maps preserving the semi-Fredholm domain in spectrum, Ann Funct Anal., 7 (2016), 254-260.