## Kernels of generalized derivations.(English)Zbl 0731.47038

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.