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On Invariant subspace for hyponormal operators

Junfeng Liu

Abstract

There is a resent paper claiming that every hyponormal operator which is not a multiple of the identity (operator) has a nontrivial hyperinvariant subspace. If this claim is true, then every hyponormal operator has a nontrivial invariant subspace, and every subnormal operator which is not multiple of the identity has a nontrivial hyperinvariant subspace. But we find out that the proof of the above claim go wrong. Therefore the invariant subspace problem of the hyponormal operator and the hyperinvariant problem of subnormal operator remains unresolved. Moreover, it is well known that these are two important research topics in operator theory that have not been solved for a long time, and that a lot of people have been trying to solved the two problems. So it is very meaningful to clarify whether these two problems have been solved.

On Invariant subspace for hyponormal operators

Abstract

There is a resent paper claiming that every hyponormal operator which is not a multiple of the identity (operator) has a nontrivial hyperinvariant subspace. If this claim is true, then every hyponormal operator has a nontrivial invariant subspace, and every subnormal operator which is not multiple of the identity has a nontrivial hyperinvariant subspace. But we find out that the proof of the above claim go wrong. Therefore the invariant subspace problem of the hyponormal operator and the hyperinvariant problem of subnormal operator remains unresolved. Moreover, it is well known that these are two important research topics in operator theory that have not been solved for a long time, and that a lot of people have been trying to solved the two problems. So it is very meaningful to clarify whether these two problems have been solved.
Paper Structure (2 sections, 2 theorems, 21 equations)

This paper contains 2 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction and Preliminary
  2. Main Results

Key Result

Theorem 1

Let $H$ be a Hilbert space and let $T$ be a bounded linear operator on $H$. Assume that $T$ is not a multiple of the identity operator $I$ and unitary equivalent to an upper triangular operator matrix on $H$, say If $T_{11}$ and $T_{nn}$ are M-hyponormal operators, then $T$ has a nontrivial hyperinvariant subspace.

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1