关于Hilbert空间的不变子空间问题
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摘要
有界线性算子在无穷维可分复Hilbert空间上的不变子空间问题至今仍是一个未解难题。通过引进该Hilbert空间的一组正交基,在该组正交基满足一定的条件下,有界线性算子在该Hilbert空间上一定有一个非平凡的不变子空间。
It is still an unsolved problem that whether every bounded linear operator on an infinite-dimensional separable complex Hilbert space has a nontrivial invariant subspace. By introducing a set of orthogonal basis in the Hilbert space, this paper proves that if the orthogonal base met certain conditions, the corresponding bounded linear operator must have a nontrivial invariant subspace in the Hilbert space.
It is still an unsolved problem that whether every bounded linear operator on an infinite-dimensional separable complex Hilbert space has a nontrivial invariant subspace. By introducing a set of orthogonal basis in the Hilbert space, this paper proves that if the orthogonal base met certain conditions, the corresponding bounded linear operator must have a nontrivial invariant subspace in the Hilbert space.
引文
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[7]PEARCY C.Some recent developments in operator theory[M].Providence:Amer.Math.Soc.,1977.
[8]PEARCY C,SHIELDS A.A survey of Lomonosov technique in the theory of invariant subspaces[M].Providence:Amer.Math.Soc.,1974.
[9]SABABHEH M,YOUSEF A,KHALIL R.On the invariant subspace problem[J].Bull.Malays.Math.Sci.Soc.,2016,39(2):699-705.
[2]CHALENDAR I,PARTINGTON J.An overview of some recent developments on the invariant subspace problem[J].Concrete Operators,2013,1:1-10.
[3]ARONSZAJN N,SMITH K.Invariant subspaces of completely continuous operators[J].Ann.Math.,1954,60(2):345-390.
[4]BERNSTEIN A,ROBINSON A.Solution of an invariant subspace problem of K.T.Smith and P.R.Halmos[J].Pacific J.Math.,1966,12(3):421-431.
[5]HALMOS P.Invariant subspaces of polynomially compact operators[J].Pacific J.Math.,1966,12(3):433-437.
[6]LOMONSOV V.Invariant subspaces of the family of operators that commute with a completely continuous operators[J].Functional Anal.i Prilozen,1973,7(3):55-56.
[7]PEARCY C.Some recent developments in operator theory[M].Providence:Amer.Math.Soc.,1977.
[8]PEARCY C,SHIELDS A.A survey of Lomonosov technique in the theory of invariant subspaces[M].Providence:Amer.Math.Soc.,1974.
[9]SABABHEH M,YOUSEF A,KHALIL R.On the invariant subspace problem[J].Bull.Malays.Math.Sci.Soc.,2016,39(2):699-705.