算子的乘积
|
摘要
本文是一篇综述类论文,通过阅读和整理相关文献,对于算子乘积及相关问题进行了综合分析.主要总结了两类特殊算子(自伴算子、正算子)的乘积.给出了Hilbert空间上有界线性算子可表示成两个自伴算子乘积的充分必要条件,并给出了哪些算子可表示成n个正算子的乘积及最少的因子数.
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.
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.
引文
[1]D.H.Carlson.On real eigenvalues of complex matrices[J]. Pacific J. Math.,1965,15: 1119-1129.
[2]F.F.Bonsall,J.Duncan.Complete normed algebras[M].New York:Springer-Verlag, 1973.
[3]R.Bouldin.The essential minimum modulus [J].Indiana Univ,Math.J.,1981,30:513-517.
[4]J.B.Conway.A course in functional analysis[M].New York:Springer-Verlag,1985.
[5]C.R.Putnam.Commutation properties of Hilbert space Operators and related topics [M].Berlin:Springer,1967.
[6]H.Radjavi,P.Rosenthal.Invariant subspaces for Products of Hermitian Opera-tors[M].Proc. Amer. Math.Soc.,1974,43:483-484.
[7]S.Izumino,Y.Kato.The closure of invertible operators on a Hilbert space[J].Acta Sci.Math.,1985,49:321-327.
[8]H.Radjavi,J.P.Williams.Products of self-adjoint operators [J]. Michigan Math.J.P., 1969,16;177-185.
[9]L.J.Gray.Products of Hermitian operators[J].Proc.Amer.Math.Soc.,1976,59;123-126.
[10]Chung-Chou Jiang.On products of two Hermitian operators[J]. Linear Algebra and its Applications,2009,430:553-557.
[11]H.Radjavi.On self-adjoint factorization of operators[J].Can.J.Math.,1969,21:1421-1426.
[12]Pei Yuan Wu.Unitary Dilations and Numerical range[J].NSC,1997,38:25-42.
[13]C.S.Ballantine.Products of positive definite matrices, Ⅳ [J].Linear Alg.Appl., 1970,3:79-114.
[14]Chandler Davis.Separation of two liner subspaces[J].Acta.Sic.Math.,1958,19:172-184.
[15]M.Khalkali,C.Laurie,B.Mathes,H.Radjavi.Approximation byproducts of positive operators[J].J.P.Operator Theory,1993,29:237-247.
[16]H.Radjavi.Products of Hermitian matrices and symmetries[J].Proc.Amer.Math. Soc.,1969,21:369-372;1970,26:701.
[17]P.Y.Wu.Products of normal operators[J].Canadian J.P.Math.,1988,40:1322-1330.
[18]P.Y.Wu.Products of positive semi-definite matrices[J].Linear Alg.Appl.to appear.
[19]孙善利,纪友清.泛函分析[M].长春：吉林大学出版社, [时间不详].
[20]N.C.Phillips.Every invertible Hilbert-space operator is a product of seven positive operators[J].Canad.Math.Bull.,1995,38:230-236.
[21]Heydar Radjavi,Peter Rosenthal.Invariant Subspaces[M].New York:Dover Publi-cations,2003.
[22]D.A.Herrero.Approximation of Hilbert space operators[M].New York:John Whey and Sons,1982.
[23]L.Gillman,M.Jerison.Rings of continous function[M].New York:Springer-Verlag, 1976.
[24]P.R.Halmos.A Hilbert Space Problem Book[M].New York:Springed,1990.
[25]江泽坚,吴智泉.实变函数论[M].北京：高等教育出版社,1994.
[26]G.J.MURPHY.Product of positive operators[J].Amer.Math.Soc.,1997,125:3675-3677.
[27]P.A.Fillmore.on products of symmetries[J].Canad.J.Math.,1966,18:897-900.
[28]A.R.Sourour.A factorization theorem for matrices[J].Linear and Multilinear Al-gebra,1986,19:141-147.
[29]J.P.Williams.Operators similar to their adjoints[J].Proc.Amer.Math.Soc.,1969, 20:121-123.
[30]G.J.Murphy and N.C.Phillips.C*-algebras with the approximate positive factor-ization property[J]. Trans. Amer. Math. Soc.,1996,348:2291-2306.
[31]C.Davis.Separation of two linear subspace[J].Acta.Sci.Math.,1958,19:172-187.
[32]C.R.Putman.On normal operators in Hilbert space[J].Amer.J.Math.,1951, 73:357-362.
[33]H.Radjavi.Structure of A*A-AA*[J].J.Math.Mech.,1966,16:19-26.
[34]F.Riesz and B.Sz.-Nagy.Functional analysis[M].New York:Ungar,1955.
[2]F.F.Bonsall,J.Duncan.Complete normed algebras[M].New York:Springer-Verlag, 1973.
[3]R.Bouldin.The essential minimum modulus [J].Indiana Univ,Math.J.,1981,30:513-517.
[4]J.B.Conway.A course in functional analysis[M].New York:Springer-Verlag,1985.
[5]C.R.Putnam.Commutation properties of Hilbert space Operators and related topics [M].Berlin:Springer,1967.
[6]H.Radjavi,P.Rosenthal.Invariant subspaces for Products of Hermitian Opera-tors[M].Proc. Amer. Math.Soc.,1974,43:483-484.
[7]S.Izumino,Y.Kato.The closure of invertible operators on a Hilbert space[J].Acta Sci.Math.,1985,49:321-327.
[8]H.Radjavi,J.P.Williams.Products of self-adjoint operators [J]. Michigan Math.J.P., 1969,16;177-185.
[9]L.J.Gray.Products of Hermitian operators[J].Proc.Amer.Math.Soc.,1976,59;123-126.
[10]Chung-Chou Jiang.On products of two Hermitian operators[J]. Linear Algebra and its Applications,2009,430:553-557.
[11]H.Radjavi.On self-adjoint factorization of operators[J].Can.J.Math.,1969,21:1421-1426.
[12]Pei Yuan Wu.Unitary Dilations and Numerical range[J].NSC,1997,38:25-42.
[13]C.S.Ballantine.Products of positive definite matrices, Ⅳ [J].Linear Alg.Appl., 1970,3:79-114.
[14]Chandler Davis.Separation of two liner subspaces[J].Acta.Sic.Math.,1958,19:172-184.
[15]M.Khalkali,C.Laurie,B.Mathes,H.Radjavi.Approximation byproducts of positive operators[J].J.P.Operator Theory,1993,29:237-247.
[16]H.Radjavi.Products of Hermitian matrices and symmetries[J].Proc.Amer.Math. Soc.,1969,21:369-372;1970,26:701.
[17]P.Y.Wu.Products of normal operators[J].Canadian J.P.Math.,1988,40:1322-1330.
[18]P.Y.Wu.Products of positive semi-definite matrices[J].Linear Alg.Appl.to appear.
[19]孙善利,纪友清.泛函分析[M].长春：吉林大学出版社, [时间不详].
[20]N.C.Phillips.Every invertible Hilbert-space operator is a product of seven positive operators[J].Canad.Math.Bull.,1995,38:230-236.
[21]Heydar Radjavi,Peter Rosenthal.Invariant Subspaces[M].New York:Dover Publi-cations,2003.
[22]D.A.Herrero.Approximation of Hilbert space operators[M].New York:John Whey and Sons,1982.
[23]L.Gillman,M.Jerison.Rings of continous function[M].New York:Springer-Verlag, 1976.
[24]P.R.Halmos.A Hilbert Space Problem Book[M].New York:Springed,1990.
[25]江泽坚,吴智泉.实变函数论[M].北京：高等教育出版社,1994.
[26]G.J.MURPHY.Product of positive operators[J].Amer.Math.Soc.,1997,125:3675-3677.
[27]P.A.Fillmore.on products of symmetries[J].Canad.J.Math.,1966,18:897-900.
[28]A.R.Sourour.A factorization theorem for matrices[J].Linear and Multilinear Al-gebra,1986,19:141-147.
[29]J.P.Williams.Operators similar to their adjoints[J].Proc.Amer.Math.Soc.,1969, 20:121-123.
[30]G.J.Murphy and N.C.Phillips.C*-algebras with the approximate positive factor-ization property[J]. Trans. Amer. Math. Soc.,1996,348:2291-2306.
[31]C.Davis.Separation of two linear subspace[J].Acta.Sci.Math.,1958,19:172-187.
[32]C.R.Putman.On normal operators in Hilbert space[J].Amer.J.Math.,1951, 73:357-362.
[33]H.Radjavi.Structure of A*A-AA*[J].J.Math.Mech.,1966,16:19-26.
[34]F.Riesz and B.Sz.-Nagy.Functional analysis[M].New York:Ungar,1955.