Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Refuting a Recent Proof of the Invariant Subspace Problem

Ahmed Ghatasheh

TL;DR

For a bounded linear operator $T$ on an infinite-dimensional separable Hilbert space $H$, the invariant subspace problem asks whether $H$ always contains a nontrivial closed invariant subspace. This paper critically examines a recent purported proof by Khalil et al., breaking it into claims to isolate a key error. Using standard Hilbert space machinery—including the construction of $\{T^n(h)\}$, the Gram–Schmidt process to obtain a complete orthonormal sequence $\{\theta_n\}$, and weak convergence arguments—it demonstrates that the alleged Fourth Claim hinges on an ill-defined functional, leading to a false contradiction. Consequently, the paper concludes that Khalil et al.'s argument does not resolve the invariant subspace problem and highlights subtle pitfalls in nonconstructive invariant-subspace proofs.

Abstract

This article demonstrates that the recent proof of the invariant subspace problem, as presented by Khalil et al., is incorrect.

Refuting a Recent Proof of the Invariant Subspace Problem

TL;DR

For a bounded linear operator on an infinite-dimensional separable Hilbert space , the invariant subspace problem asks whether always contains a nontrivial closed invariant subspace. This paper critically examines a recent purported proof by Khalil et al., breaking it into claims to isolate a key error. Using standard Hilbert space machinery—including the construction of , the Gram–Schmidt process to obtain a complete orthonormal sequence , and weak convergence arguments—it demonstrates that the alleged Fourth Claim hinges on an ill-defined functional, leading to a false contradiction. Consequently, the paper concludes that Khalil et al.'s argument does not resolve the invariant subspace problem and highlights subtle pitfalls in nonconstructive invariant-subspace proofs.

Abstract

This article demonstrates that the recent proof of the invariant subspace problem, as presented by Khalil et al., is incorrect.

Paper Structure

This paper contains 2 sections, 12 theorems, 16 equations.

Table of Contents

  1. Basics of Hilbert Spaces
  2. The Proof by Khalil Et Al.

Key Result

Theorem 1.2

Let $(u_j)_{j=1}^{\infty}$ be a linearly independent sequence in an infinite-dimensional inner product space $V$. Set $w_1=u_1$ and for each $n\in\mathbb{N}$. Then the following statements hold:

Theorems & Definitions (27)

  • Definition 1.1: Independence, Orthogonality, and Orthonormality of Sequences
  • Theorem 1.2: Gram-Schmidt Process
  • Theorem 1.3: Finite-Dimensional Subspaces
  • proof
  • Definition 1.4: Convergence in Inner Product Spaces
  • Theorem 1.5
  • proof
  • Theorem 1.6: Bessel's Inequality and Parseval's formula
  • proof
  • Definition 1.7: Complete Orthonormal Sequences
  • ...and 17 more