[Paper Review] Approximate Birkhoff-James orthogonality in the space of bounded linear operators

Kallol Paul, ‎Debmalya Sain+1 more|arXiv (Cornell University)|Jun 28, 2017

Advanced Banach Space Theory11 references21 citations

TL;DR

This paper provides a complete characterization of two notions of approximate Birkhoff-James orthogonality—denoted ⊥ǫ_D and ⊥ǫ_B—for bounded linear operators on normed and Hilbert spaces. It establishes necessary and sufficient conditions using the norm attainment set MT, extending prior results to infinite-dimensional Hilbert spaces and general normed spaces, with key results showing that T⊥ǫ_B A holds if and only if there exist vectors in MT satisfying specific quadratic inequalities involving A, even when MT is not symmetric or connected.

ABSTRACT

There are two notions of approximate Birkhoff-James orthogonality in a normed space. We characterize both the notions of approximate Birkhoff-James orthogonality in the space of bounded linear operators defined on a normed space. A complete characterization of approximate Birkhoff-James orthogonality in the space of bounded linear operators defined on Hilbert space of any dimension is obtained which improves on the recent result by Chmieli\'nski et al. [ J. Chmieli\'nski, T. Stypula and P. W\'ojcik, extit{Approximate orthogonality in normed spaces and its applications}, Linear Algebra and its Applications, extbf{531} (2017), 305--317.], in which they characterized approximate Birkhoff-James orthogonality of linear operators on finite dimensional Hilbert space and also of compact operators on any Hilbert space.

Motivation & Objective

• To characterize two distinct notions of approximate Birkhoff-James orthogonality—⊥ǫ_D and ⊥ǫ_B—for bounded linear operators on normed spaces.
• To extend previous results on finite-dimensional and compact operators to general Hilbert spaces of any dimension and reflexive Banach spaces.
• To establish necessary and sufficient conditions for approximate orthogonality in terms of the norm attainment set MT, particularly for operators on Hilbert spaces.
• To provide a unified framework that includes cases where the norm attainment set is not symmetric or connected, such as D ∪ (−D).
• To resolve limitations in prior work by characterizing orthogonality for non-compact, non-finite-dimensional operators, including those not satisfying the conditions of earlier corollaries.

Proposed method

• Uses the norm attainment set MT = {x ∈ SX : ∥Tx∥ = ∥T∥} as a central tool to characterize orthogonality relations.
• Applies two definitions of approximate orthogonality: ⊥ǫ_D (based on ∥x + λy∥ ≥ √(1−ǫ²)∥x∥) and ⊥ǫ_B (based on ∥x + λy∥² ≥ ∥x∥² − 2ǫ∥x∥∥λy∥).
• Employs the concept of x+ and x− (for λ ≥ 0 and λ ≤ 0) and their ǫ-variants x+(ǫ) and x−(ǫ) to define directional behavior of operators.
• For compact operators on reflexive Banach spaces, proves that T⊥ǫ_D A holds iff certain norm-attaining vectors satisfy ∥T + λA∥ ≥ √(1−ǫ²)∥T∥ for specific λ intervals.
• For the ⊥ǫ_B case, derives conditions using quadratic inequalities: ∥Tx + λAx∥² ≥ ∥T∥² − 2ǫ∥T∥∥λA∥ for λ ≥ 0 and λ ≤ 0.
• Uses sequential approximation via sequences {xn}, {yn} with ∥Txn∥ → ∥T∥ to handle cases where MT may be empty or non-compact, especially in general normed spaces.

Experimental results

Research questions

• RQ1 Under what conditions does a bounded linear operator T satisfy approximate Birkhoff-James orthogonality with A under the ⊥ǫ_D definition?
• RQ2 How can ⊥ǫ_B orthogonality be characterized when the norm attainment set MT is not symmetric or connected?
• RQ3 Can the characterization of approximate orthogonality be extended beyond finite-dimensional and compact operators to general Hilbert spaces?
• RQ4 What role does the norm attainment set MT play in determining approximate orthogonality for non-compact operators?
• RQ5 How do the two different definitions of approximate orthogonality (⊥ǫ_D and ⊥ǫ_B) relate in general normed spaces, and when do they coincide?

Key findings

• For T, A ∈ K(X, Y) with X reflexive, T⊥ǫ_D A if and only if either (a) there exists x ∈ MT with Ax ∈ (Tx)+ and ∥Txλ + λAxλ∥ ≥ √(1−ǫ²)∥T∥ for λ in a specific interval, or (b) a similar condition holds for y ∈ MT with Ay ∈ (Ty)−.
• The paper provides an alternative proof of Theorem 2.2 in [7], showing that in finite-dimensional spaces, T⊥B A iff there exist x, y ∈ MT with Ax ∈ (Tx)+ and Ay ∈ (Ty)−.
• For T ∈ K(H) on a Hilbert space H, T⊥ǫ_B A holds if and only if there exists x ∈ MT such that Tx⊥ǫ_B Ax, even when MT is not contained in MA.
• In the general case for T, A ∈ B(X, Y) on any normed space, T⊥ǫ_B A holds iff either (a) there is a sequence {xn} with ∥Txn∥ → ∥T∥ and lim ∥Axn∥ ≤ ǫ∥A∥, or (b) two sequences {xn}, {yn} satisfy specific quadratic inequalities involving ǫn, δn → 0.
• The characterization in Theorem 3.4 includes operators not covered by earlier results, such as T with MT not symmetric or not contained in MA, demonstrating broader applicability.
• The paper shows that Theorem 3.1 includes a larger class of operators than Corollary 3.1.1 and 3.1.2, as demonstrated by a counterexample in ℓ² with non-symmetric MT.