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LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS
Published online by Cambridge University Press: 29 September 2021
Abstract
Let
$\mathcal {X}$
be a Banach space over the complex field
$\mathbb {C}$
and
$\mathcal {B(X)}$
be the algebra of all bounded linear operators on
$\mathcal {X}$
. Let
$\mathcal {N}$
be a nontrivial nest on
$\mathcal {X}$
,
$\text {Alg}\mathcal {N}$
be the nest algebra associated with
$\mathcal {N}$
, and
$L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
be a linear mapping. Suppose that
$p_n(x_1,x_2,\ldots ,x_n)$
is an
$(n-1)\,$
th commutator defined by n indeterminates
$x_1, x_2, \ldots , x_n$
. It is shown that L satisfies the rule
for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Aidan Sims
This work is partially supported by the Foreign High-level Cultural and Educational Experts Project of the Beijing Institute of Technology. The work of the first author is supported by National Natural Science Foundation of China under Grant 12071029.