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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
$$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$
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.
References
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