Finite rank operators in Jacobson radical ℛ 𝒩 ⊗ ℳ

Zhe Dong

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 2, page 287-298
  • ISSN: 0011-4642

Abstract

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In this paper we investigate finite rank operators in the Jacobson radical ℛ 𝒩 ⊗ ℳ of A l g ( 𝒩 ⊗ ℳ ) , where 𝒩 , ℳ are nests. Based on the concrete characterizations of rank one operators in A l g ( 𝒩 ⊗ ℳ ) and ℛ 𝒩 ⊗ ℳ , we obtain that each finite rank operator in ℛ 𝒩 ⊗ ℳ can be written as a finite sum of rank one operators in ℛ 𝒩 ⊗ ℳ and the weak closure of ℛ 𝒩 ⊗ ℳ equals A l g ( 𝒩 ⊗ ℳ ) if and only if at least one of 𝒩 , ℳ is continuous.

How to cite

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Dong, Zhe. "Finite rank operators in Jacobson radical ${\mathcal {R}}_{{\mathcal {N}}\otimes {\mathcal {M}}}$." Czechoslovak Mathematical Journal 56.2 (2006): 287-298. <http://eudml.org/doc/31029>.

@article{Dong2006,
abstract = {In this paper we investigate finite rank operators in the Jacobson radical $\mathcal \{R\}_\{\mathcal \{N\}\otimes \mathcal \{M\}\}$ of $\mathop \{\mathrm \{A\}lg\}(\mathcal \{N\}\otimes \mathcal \{M\})$, where $\mathcal \{N\}$, $\mathcal \{M\}$ are nests. Based on the concrete characterizations of rank one operators in $\mathop \{\mathrm \{A\}lg\}(\mathcal \{N\}\otimes \mathcal \{M\})$ and $\mathcal \{R\}_\{\mathcal \{N\}\otimes \mathcal \{M\}\}$, we obtain that each finite rank operator in $\mathcal \{R\}_\{\mathcal \{N\}\otimes \mathcal \{M\}\}$ can be written as a finite sum of rank one operators in $\mathcal \{R\}_\{\mathcal \{N\}\otimes \mathcal \{M\}\}$ and the weak closure of $\mathcal \{R\}_\{\mathcal \{N\}\otimes \mathcal \{M\}\}$ equals $\mathop \{\mathrm \{A\}lg\}(\{\mathcal \{N\}\otimes \mathcal \{M\}\})$ if and only if at least one of $\mathcal \{N\}$, $\mathcal \{M\}$ is continuous.},
author = {Dong, Zhe},
journal = {Czechoslovak Mathematical Journal},
keywords = {Jacobson radical; finite rank operator; Jacobson radical; finite rank operator},
language = {eng},
number = {2},
pages = {287-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite rank operators in Jacobson radical $\{\mathcal \{R\}\}_\{\{\mathcal \{N\}\}\otimes \{\mathcal \{M\}\}\}$},
url = {http://eudml.org/doc/31029},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Dong, Zhe
TI - Finite rank operators in Jacobson radical ${\mathcal {R}}_{{\mathcal {N}}\otimes {\mathcal {M}}}$
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 287
EP - 298
AB - In this paper we investigate finite rank operators in the Jacobson radical $\mathcal {R}_{\mathcal {N}\otimes \mathcal {M}}$ of $\mathop {\mathrm {A}lg}(\mathcal {N}\otimes \mathcal {M})$, where $\mathcal {N}$, $\mathcal {M}$ are nests. Based on the concrete characterizations of rank one operators in $\mathop {\mathrm {A}lg}(\mathcal {N}\otimes \mathcal {M})$ and $\mathcal {R}_{\mathcal {N}\otimes \mathcal {M}}$, we obtain that each finite rank operator in $\mathcal {R}_{\mathcal {N}\otimes \mathcal {M}}$ can be written as a finite sum of rank one operators in $\mathcal {R}_{\mathcal {N}\otimes \mathcal {M}}$ and the weak closure of $\mathcal {R}_{\mathcal {N}\otimes \mathcal {M}}$ equals $\mathop {\mathrm {A}lg}({\mathcal {N}\otimes \mathcal {M}})$ if and only if at least one of $\mathcal {N}$, $\mathcal {M}$ is continuous.
LA - eng
KW - Jacobson radical; finite rank operator; Jacobson radical; finite rank operator
UR - http://eudml.org/doc/31029
ER -

References

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