The Products Of Operators

The products of operators problem has been well known and not been solved completely. The problem is whether T can be factorized into the product of some good operatoers has been discussed by many authors? Such as normal operators, self-adjoint operators, positive operators and so on.Many mathematicians are studying this problem. Products of operators in Hilbert space play a role in several different areas of mathematics. We shall give two examples:(1). Every bounded operator T may be written T = A + iB, where A and B are self-adjolnts. If T is known to be semi-normal,Putnam[5]proved that normality and self-adjointedness of AB are the same.(2). Radjavi and Rosenthal[6]proved that the product of a self-adjoint and a positive operator always has a non-trivial invariant subspace. However,it has not been decided whether the product of two self-adjoint operators or of a unitary and a positive operator has an invariant subspace(this is the famous" invariant subspace problem").Hence, the products of operators problem play a central role in the operator theory. So the products of operators problem is naturally an attracting topic.In section 2 we prove that T is similar to its adjoint T* if and only if T can be decomposed as a product of two self-adjoint operators. We obtain some new results which extend and improve the related known works.Theorem 1. If (?) is a finite-dimensional Hilbert space,then the following are equiualent conditions for an operator T on (?).(1) T is a product of two self-adjoint operators. (2) T is a product of two self-adjoint operators, one of which is invertible.(3) There exists an invertible self-adjoint operator A such that TA is self-adjoint.(4) There exists an invertible self-adjoint operator A such that A-1TA= T*.(5) There exists a basis of (?) with respect to which the matrix of T is real.(6) T is similar to T*.Theorem 2. If T is a normal operator on a Hilbert space (?), then T is similar to its adjoint T* if and only if T can be decomposed as a product of two self-adjoint operators.Theorem 3. Let A be a normal operator on (?) with pure point spectrum. Then A is the product of two commuting normal operators in(?).Theorem 4. Every normal operator on (?) belongs to (?)2 (?)2.Theorem 5. Let A be an operator on (?) and Letτ(A) be the closure of its numerical range. Then A∈(?) implies thatτ(A) contains either a real or a pure imaginary number.In section 3 we consider the problem of which operators may be factored into products of k positive operators and state that 17 is not smallest factors. We obtain some new results which extend and improve the related known works.Theorem 6. Let A be a normal operator, We have A∈P2 if and only if every component ofσ(A) intersects the non-negative axis.Theorem 7. If A is algebraic, then A∈P4.Theorem 8. The closures of each of the following sets are equal to P5;(1) the set of Fredholm operators with index 0, denoted F0;(2) the set of operators A such that V(A)= V(A*),denoted V;(3) the set of invertible operators,denoted B((?))-1(4) Pn,n≥5;(5) Qn, n≥5;(6) P∞ Theorem 9. Every unitary operator is the product of 16 positive invertible operators.Theorem 10. let M be a properly infinite von Neumann algebra. Then every unitary element of M is a product of 6 positive invertible elements.