Range Kernel Orthogonality and Finite Operators
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Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Let A,B 2 L(H) we define the generalized derivation δA,B : L(H) → L(H) by δA,B(X) = AX - XB, we note δA,A = δA. If the inequality ‖T - (AX - XA)‖≥‖T‖ holds for all X ∈ L(H) and for all T 2 kerδA, then we say that the range of δA is orthogonal to the kernel of δA in the sense of Birkhoff. The operator A 2 L(H) is said to be finite [22] 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 [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
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 into itself. Let A,B 2 L(H) we define the generalized derivation δA,B : L(H) → L(H) by δA,B(X) = AX - XB, we note δA,A = δA. If the inequality ‖T - (AX - XA)‖≥‖T‖ holds for all X ∈ L(H) and for all T 2 kerδA, then we say that the range of δA is orthogonal to the kernel of δA in the sense of Birkhoff. The operator A 2 L(H) is said to be finite [22] 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 [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
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