Kernels of generalized derivations. (English) Zbl 0731.47038

Tong, Yusun

Acta Sci. Math. 54, No. 1-2, 159-169 (1990).

Let H be a Hilbert space and B(H) the Banach algebra of all bounded linear operators on H. For A,B\(\in B(H)\), the generalized derivation \(\delta_{AB}\) and the higher derivations \(\delta_{AB}^{(n)}\) (for any natural number n) are defined by \[ \delta_{AB}(X)=AX-XB,\quad X\in B(H)\text{ and } \delta_{AB}^{(n)}(X)=\sum^{n}_{i=0}(-1)^ iA^ iXB^{n-i},\quad X\in B(H),\quad respectively. \] In this theses, the author studied the kernels of \(\delta_{AB}\) and \(\delta_{AB}^{(n)}\). In general, Ker \(\delta\) \({}_{AB}\subsetneqq Ker \delta_{AB}^{(n)}\), \(n\geq 1\). The author obtained the following results:
(1) For A,B\(\in B(H)\) such that if \(AX=XB\), \(X\in B(H)\), then \(A^*X=XB^*\). In this case Ker \(\delta\) \({}_{AB}^{(n)}=Ker \delta_{AB}\), \(n=1,2,3,....\)
(2) If \(\| Bx\| \leq \| x\| \leq \| Ax\|\) for any \(x\in H\) and A,B\(\in B(H)\) then Ker \(\delta\) \({}_{AB}^{(n)}\cap K(H)=Ker \delta_{AB}\cap K(H)\) holds, where K(H) are the compact operators in B(H).
(3) If \(M_ z\) (z\(\in H)\) is a multiplication operator on some suitably defined Hilbert space,then Ker \(\delta\) \({}_{M_ z}^{(n)}=Ker \delta_{M_ z}\), \(n=1,2,3,....\)
Beside of these, an asymptotic form of this kind of results is obtained and a related theorem concerning compact operators is given.

Reviewer: J.C.Rho (Seoul)

MSC:

47B47

Commutators, derivations, elementary operators, etc.

Keywords:

generalized derivation; kernels; multiplication operator; compact operators