when: Nov. 7th. 2019, 16:00--17:15
where: Bldg.110, Rm. N103

Speaker: Jaewoong Kim (Korea Military Academy)

Title: The invariant subspace problem and the Aluthge transforms of operators

Abstract: Let H be a separable infinite-dimensional complex Hilbert space and B(H) denote the algebra of all bounded linear operators on H. A closed subspace L ⊂H is called an invariant subspace for the operator T ∈ B(H) if TL ⊂ L. The two trivial subspaces, the entire space and the space containing only the zero vector, are invariant for every operator. The invariant subspace problem (due to J. von Neumann) is stated as: Does every operator in B(H) have a nontrivial invariant subspace? This problem remains still open for separable infinite-dimensional complex Hilbert spaces. But there were significant accomplishments on the invariant subspace problem. For example, in 1950, P. Halmos defined a subnormal operator as an operator having a normal extension to some Hilbert space K containing H, and asked whether subnormal operators have nontrivial invariant subspaces. For a long time many mathematicians have made many attempts towards this problem. Eventually, S. Brown found an ingenious proof that subnormal operators do have nontrivial invariant subspaces.

In this talk, we introduce the Aluthge transform of operaotrs on H and its application to the invariant subspace problem. Also, the Aluthge transforms for commuting n-tuples of operators and their applications to common invariant subspaces of commuting n-tuples of operators are introduced. For this, we will start talk with introducing basic concepts and results on operators on H.