Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 35', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 206 Algebra The rank of a linear operator A is the dimension of its range rank A dim im A . Properties of the rank of a linear operator rank AB min rank A rank B rank A rank B - n rank AB where A and B are linear operators in L V V and n dim V. Remark. If rank A n then rank AB rank BA rank B . Theorem. Let A V V be a linear operator Then the following statements are equivalent 1. A is invertible i. e. there exists A-1 . 2. ker A 0. 3. im A V. 4. rank A dim V. . Notion of a adjoint operator. Hermitian operators. Let A 6 L V V be a bounded linear operator in a Hilbert space V. The operator A in L V V is called its adjoint operator if Ax y x A y for all x and y in V. Theorem. Any bounded linear operator A in a Hilbert space has a unique adjoint operator Properties of adjoint operators A B A B AA AA AB B A O O A-1 A -1 A A Ax By x A By B Ax y A A I I l A A A 2 for all x and y in V where A and B are bounded linear operators in a Hilbert space V A is the complex conjugate of a number . A linear operator A e L V V in a Hilbert space V is said to be Hermitian self-adjoint if A A or Ax y x Ay . A linear operator A g V V in a Hilbert space V is said to be skew-Hermitian if A -A or Ax y -x Ay . . Unitary and normal operators. A linear operator U e L V V in a Hilbert space V is called a unitary operator if for all x and y in V the following relation holds Ux Uy x y. This relation is called the unitarity condition. . Linear Operators 207 Properties of a unitary operator U U U-1 or U U UU I Ux x for all x in V. A linear operator A in L V V is said to be normal if A A AA . Theorem. A bounded linear operator A is normal if and only if Ax A x . Remark. Any unitary or Hermitian operator is normal. . Transpose symmetric and orthogonal operators. The transpose operator of a bounded linear operator As L V V in a real Hilbert space V is the operator AT e L V V such that for all x y in V the following relation holds Ax y x Aty . Theorem. Any bounded linear .
