FINITE OPERATORS AND ORTHOGONALITY

Abstract. Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Let A,B ∈ L(H) we define the generalized derivation δA,B: L(H) 7 → L(H) by δA,B(X) = AX −XB, we note δA,A = δA. If for all X ∈ L(H) and for all T ∈ ker δA the inequality ||T −(AX−XA)| | ≥ ||T ||(*) holds, then we say that the range of δA is orthogonal to kernel δA in the sense of Birkhoff. The operator A ∈ L(H) is said to be finite [17] if ||I − (AX −XA)| | ≥ 1(**) for all X ∈ L(H), where I is the identity operator. The well-known Inequality (**) due to J.P.Williams [17] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [18]). This topic deals with minimizing the distance, measured by some norms or other, between a varying commutator XX ∗ − X∗X and some fixed operator [12]. In this paper we prove that a paranormal operator is finite and we present some generalized finite operators. An extension of Inequality (*) is also given. 1