Putnam-Fuglede theorem and the range-kernel orthogonality of derivations
Let ℬ(H) denote the algebra of operators on a Hilbert space H into itself. Let d=δ or Δ, where δAB:ℬ(H)→ℬ(H) is the generalized derivation δAB(S)=AS−SB and ΔAB:ℬ(H)→ℬ(H) is the elementary operator ΔAB(S)=ASB−S. Given A,B,S∈ℬ(H), we say that the pair (A,B) has the property PF(d(S)) if dAB(S)=0 implie...
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201006159 |
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| Summary: | Let ℬ(H) denote the algebra of operators on a Hilbert
space H into itself. Let d=δ or Δ, where δAB:ℬ(H)→ℬ(H) is the generalized derivation δAB(S)=AS−SB and ΔAB:ℬ(H)→ℬ(H) is the elementary operator ΔAB(S)=ASB−S. Given A,B,S∈ℬ(H), we say that the pair (A,B) has the property PF(d(S)) if dAB(S)=0 implies dA∗B∗(S)=0. This paper characterizes operators A,B, and S for which the pair (A,B) has property PF(d(S)), and establishes a relationship between the PF(d(S))-property of the pair (A,B) and the range-kernel orthogonality of the operator dAB. |
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| ISSN: | 0161-1712 1687-0425 |