Abstract. The inner derivation $\delta_{A}$ implemented by an element $A$ of the algebra $L(H)$ of all bounded linear operators on the separable complex Hilbert space $H$ into itself is the map $X\mapstochar\longrightarrow AX-XA$ for $X\in L(H)$. In this paper, we are interested in the class of operators $A\in L(H)$ which satisfy the following property $AT=TA$ implies $A^{\ast }T=TA^{\ast }$ for all $T\in C_{1}(H)$(trace class operators). Such operators are termed P-symmetric. We establish some properties of this class. We also turn our attention to commutants and derivation ranges. Hence, we obtain new results concerning the intersection of the kernel and the closure of the range of an inner derivation.

AMS Subject Classification (1991): 47B47, 47B10; 47A30

Keyword(s): Derivations, P-symmetric operators, subnormal operators, cyclic vector

Received February 2, 2006, and in revised form May 12, 2006. (Registered under 5945/2009.)