The norm of an elementary operator has been studied by many mathematicians. Varied results have been established especially on the lower bound of this norm. Here, we attempt the same problem for finite dimensional ope...The norm of an elementary operator has been studied by many mathematicians. Varied results have been established especially on the lower bound of this norm. Here, we attempt the same problem for finite dimensional operators.展开更多
Let E(x) = be an elementary operator on a C-algebra A. We prove that if A is prime with soc(A) = 0, or there is a family of irreducible representation of A such that is faithful and (A) does not contain compact operat...Let E(x) = be an elementary operator on a C-algebra A. We prove that if A is prime with soc(A) = 0, or there is a family of irreducible representation of A such that is faithful and (A) does not contain compact operator,or A has a faithful repre sentation π such that π(A)″with no central portions of type In for n>1 then E is positive if and only if E is completely positive.展开更多
We shall point out the further properties of the opera-tion→.And some uses of the operation→in the Fuzzy Equa-tions are discussed.Moreover a sufficient and necessarycondition is obtained.
Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and ...Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.展开更多
文摘The norm of an elementary operator has been studied by many mathematicians. Varied results have been established especially on the lower bound of this norm. Here, we attempt the same problem for finite dimensional operators.
文摘Let E(x) = be an elementary operator on a C-algebra A. We prove that if A is prime with soc(A) = 0, or there is a family of irreducible representation of A such that is faithful and (A) does not contain compact operator,or A has a faithful repre sentation π such that π(A)″with no central portions of type In for n>1 then E is positive if and only if E is completely positive.
文摘We shall point out the further properties of the opera-tion→.And some uses of the operation→in the Fuzzy Equa-tions are discussed.Moreover a sufficient and necessarycondition is obtained.
文摘Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.
