Commutants and derivation ranges

Salah Mecheri

Czechoslovak Mathematical Journal (1999)

  • Volume: 49, Issue: 4, page 843-847
  • ISSN: 0011-4642

Abstract

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In this paper we obtain some results concerning the set ℳ = ∪ R ( δ A ) ¯ ∩ { A } ' A ∈ ℒ ( H ) , where R ( δ A ) ¯ is the closure in the norm topology of the range of the inner derivation δ A defined by δ A ( X ) = A X - X A . Here ℋ stands for a Hilbert space and we prove that every compact operator in R ( δ A ) ¯ w ∩ { A * } ' is quasinilpotent if A is dominant, where R ( δ A ) ¯ w is the closure of the range of δ A in the weak topology.

How to cite

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Mecheri, Salah. "Commutants and derivation ranges." Czechoslovak Mathematical Journal 49.4 (1999): 843-847. <http://eudml.org/doc/30529>.

@article{Mecheri1999,
abstract = {In this paper we obtain some results concerning the set $\{\mathcal \{M\}\} = \cup \bigl \lbrace \overline\{R(\delta _A)\}\cap \lbrace A\rbrace ^\{\prime \}\: A\in \{\mathcal \{L\}(H)\}\bigr \rbrace $, where $\overline\{R(\delta _A)\}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal \{H\}$ stands for a Hilbert space and we prove that every compact operator in $\overline\{R(\delta _A)\}^w\cap \lbrace A^*\rbrace ^\{\prime \}$ is quasinilpotent if $A$ is dominant, where $\overline\{R(\delta _A)\}^w$ is the closure of the range of $\delta _A$ in the weak topology.},
author = {Mecheri, Salah},
journal = {Czechoslovak Mathematical Journal},
keywords = {derivation; commutant; nilpotent operator; quasinilpotent operator},
language = {eng},
number = {4},
pages = {843-847},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutants and derivation ranges},
url = {http://eudml.org/doc/30529},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Mecheri, Salah
TI - Commutants and derivation ranges
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 4
SP - 843
EP - 847
AB - In this paper we obtain some results concerning the set ${\mathcal {M}} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal {L}(H)}\bigr \rbrace $, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal {H}$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta _A)}^w\cap \lbrace A^*\rbrace ^{\prime }$ is quasinilpotent if $A$ is dominant, where $\overline{R(\delta _A)}^w$ is the closure of the range of $\delta _A$ in the weak topology.
LA - eng
KW - derivation; commutant; nilpotent operator; quasinilpotent operator
UR - http://eudml.org/doc/30529
ER -

References

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  1. 10.1090/S0002-9939-1973-0312313-6, Proc. Amer. Math. Soc. 38 (1973), 135–140.. (1973) Zbl0255.47036MR0312313DOI10.1090/S0002-9939-1973-0312313-6
  2. 10.1090/S0002-9904-1973-13271-9, Bull. Amer. Math. Soc. 79 (1973), 705–708.. (1973) Zbl0269.47021MR0322518DOI10.1090/S0002-9904-1973-13271-9
  3. 10.1090/S0002-9939-1957-0087914-4, Proc. Amer. Math. Soc. 8 (1957), 535–536.. (1957) Zbl0079.12904MR0087914DOI10.1090/S0002-9939-1957-0087914-4
  4. 10.1007/BF01579599, Monatsh. Math. 84 (1977), 143–153. (1977) Zbl0374.47010MR0458225DOI10.1007/BF01579599
  5. 10.1090/S0002-9939-1979-0512062-0, Proc. Amer. Math. Soc. 73 (1979), 79–82. (1979) Zbl0372.47019MR0512062DOI10.1090/S0002-9939-1979-0512062-0

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