On the range-kernel orthogonality of elementary operators

Said Bouali; Youssef Bouhafsi

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 3, page 261-269
  • ISSN: 0862-7959

Abstract

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Let L ( H ) denote the algebra of operators on a complex infinite dimensional Hilbert space H . For A , B ∈ L ( H ) , the generalized derivation δ A , B and the elementary operator Δ A , B are defined by δ A , B ( X ) = A X - X B and Δ A , B ( X ) = A X B - X for all X ∈ L ( H ) . In this paper, we exhibit pairs ( A , B ) of operators such that the range-kernel orthogonality of δ A , B holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A , B with respect to the wider class of unitarily invariant norms on L ( H ) .

How to cite

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Bouali, Said, and Bouhafsi, Youssef. "On the range-kernel orthogonality of elementary operators." Mathematica Bohemica 140.3 (2015): 261-269. <http://eudml.org/doc/271582>.

@article{Bouali2015,
abstract = {Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A, B\in L(H)$, the generalized derivation $\delta _\{A,B\}$ and the elementary operator $\Delta _\{A,B\}$ are defined by $\delta _\{A,B\}(X)=AX-XB$ and $\Delta _\{A,B\}(X)=AXB-X$ for all $X\in L(H)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of $\delta _\{A,B\}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $\Delta _\{A,B\}$ with respect to the wider class of unitarily invariant norms on $L(H)$.},
author = {Bouali, Said, Bouhafsi, Youssef},
journal = {Mathematica Bohemica},
keywords = {derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property},
language = {eng},
number = {3},
pages = {261-269},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the range-kernel orthogonality of elementary operators},
url = {http://eudml.org/doc/271582},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Bouali, Said
AU - Bouhafsi, Youssef
TI - On the range-kernel orthogonality of elementary operators
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 3
SP - 261
EP - 269
AB - Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A, B\in L(H)$, the generalized derivation $\delta _{A,B}$ and the elementary operator $\Delta _{A,B}$ are defined by $\delta _{A,B}(X)=AX-XB$ and $\Delta _{A,B}(X)=AXB-X$ for all $X\in L(H)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of $\delta _{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $\Delta _{A,B}$ with respect to the wider class of unitarily invariant norms on $L(H)$.
LA - eng
KW - derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property
UR - http://eudml.org/doc/271582
ER -

References

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  8. Kittaneh, F., 10.1090/S0002-9939-1995-1242091-2, Proc. Am. Math. Soc. 123 (1995), 1779-1785. (1995) Zbl0831.47036MR1242091DOI10.1090/S0002-9939-1995-1242091-2
  9. Tong, Y., Kernels of generalized derivations, Acta Sci. Math. 54 (1990), 159-169. (1990) Zbl0731.47038MR1073431
  10. Turnšek, A., 10.1007/s006050170039, Monatsh. Math. 132 (2001), 349-354. (2001) MR1844072DOI10.1007/s006050170039
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