Let ${mathcal N}$ be a nest on a complex Banach space $X$ and let $mbox{Alg}{mathcal N}$ be the associated nest algebra. We say that an operator $Zinmbox{ Alg}{mathcal N}$ is an all-derivable point of $mbox{ Alg}{mathcal N}$if every linear map $delta$ from $mbox{ Alg}{mathcal N}$ into itselfderivable at $Z$ (i.e. $delta$ satisfies $delta(A)B+Adelta(B)=delta(Z)$ forany $A,B in mbox{ Alg}{mathcal N}$ with $AB=Z$) is a derivation. In thispaper, it is shown that every injective operator and every operator with denserange in $mbox{Alg}{mathcal N}$ are all-derivable points of$mbox{Alg}{mathcal N}$ without any additional assumption on the nest.
展开▼
机译:令$ { mathcal N} $为复Banach空间$ X $上的嵌套,令$ mbox {Alg} { mathcal N} $为相关的嵌套代数。我们说算子$ Z in mbox {Alg} { mathcal N} $是$ mbox {Alg} { mathcal N} $的全导数,如果每个线性映射$ delta $来自$ mbox {Alg} { mathcal N} $本身可以以$ Z $导出(即$ delta $满足$ delta(A)B + A delta(B)= delta(Z)$的任何$ A,B mbox {Alg} { mathcal N} $中的$ AB = Z $)是推导。在本文中,我们证明了每个注入型运算符和$ mbox {Alg} { mathcal N} $中具有密集范围的每个运算符都是$ mbox {Alg} { mathcal N} $的所有可导出点,没有任何其他假设在巢上。
展开▼