[Paper Review] Orthogonality and Numerical radius inequalities of operator matrices

‎Arpita Mal, Kallol Paul+1 more|arXiv (Cornell University)|Mar 16, 2019

Matrix Theory and Algorithms21 references8 citations

TL;DR

This paper establishes a complete characterization of Birkhoff-James orthogonality with respect to the numerical radius norm in the space of bounded linear operators on a complex Hilbert space. Using this characterization, it derives improved and generalized lower bounds for the numerical radius of $n\times n$ operator matrices, particularly enhancing existing bounds for upper triangular matrices and block structures, with numerical examples demonstrating tighter estimates than prior results.

ABSTRACT

We completely characterize Birkhoff-James orthogonality with respect to numerical radius norm in the space of bounded linear operators on a complex Hilbert space. As applications of the results obtained, we estimate lower bounds of numerical radius for $n imes n$ operator matrices, which improve on and generalize existing lower bounds. We also obtain a better lower bound of numerical radius for an upper triangular operator matrix.

Motivation & Objective

• To fully characterize Birkhoff-James orthogonality with respect to the numerical radius norm in $\mathbb{B}(\mathbb{H})$ for complex Hilbert spaces.
• To develop sharper lower bounds for the numerical radius of $n\times n$ operator matrices, improving upon known existing bounds.
• To generalize and refine lower bounds for upper triangular operator matrices using orthogonality properties.
• To demonstrate through examples that the proposed bounds are tighter than those from prior works, such as [10, Th. 3.7], [7, Cor. 3.3], and [8].

Proposed method

• Introduces and defines numerical radius orthogonality $T \perp_w A$ via the condition $w(T + \lambda A) \geq w(T)$ for all $\lambda \in \mathbb{C}$ (or $\mathbb{R}$ in the real case).
• Establishes equivalence conditions for $T \perp_w A$, including $T^* \perp_w A^*$ and scalar invariance under $\alpha T \perp_w \beta A$.
• Proves that for self-adjoint $T$, $T \perp_w A$ implies $T \perp_B A$ (Birkhoff-James orthogonality), and for nilpotent $T$ with $T^2 = 0$, $T \perp_B A$ implies $T \perp_w A$.
• Applies the orthogonality characterization to derive lower bounds for numerical radius of block matrices, especially upper triangular ones, using the structure of off-diagonal blocks.
• Uses the inequality $w(T) \geq \max\{w(A), w(D), \|B\|/2\}$ for $T = \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$ to generalize and improve upon [10, Th. 3.7].
• Applies Theorem 3.5 to scalar matrices, showing $w(A) \geq \max_i w(T_i)$, where $T_i$ is $A$ with the $i$-th row and column zeroed out.

Experimental results

Research questions

• RQ1 How can Birkhoff-James orthogonality with respect to the numerical radius norm be fully characterized in $\mathbb{B}(\mathbb{H})$ for complex Hilbert spaces?
• RQ2 What are the implications of this characterization for deriving lower bounds on the numerical radius of $n \times n$ operator matrices?
• RQ3 Can the new bounds improve upon existing ones, particularly for upper triangular or block-structured matrices?
• RQ4 In what cases do the new bounds strictly exceed previously known estimates, such as those in [10, Th. 3.7] or [7, Cor. 3.3]?

Key findings

• The paper establishes that for self-adjoint $T$, $T \perp_w A$ implies $T \perp_B A$, and for nilpotent $T$ with $T^2 = 0$, $T \perp_B A$ implies $T \perp_w A$, showing a connection between the two orthogonality concepts.
• For an upper triangular operator matrix $T = \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$, the bound $w(T) \geq \max\{w(A), w(D), \|B\|/2\}$ is proven, which is tighter than the bound $w(T) \geq \max\{w(A), w(D), \frac{1}{2}w(B)\}$ from [10, Th. 3.7].
• Theorem 3.5 provides a new lower bound for scalar matrices: $w(A) \geq \max_i w(T_i)$, where $T_i$ is $A$ with the $i$-th row and column removed, which outperforms the bound $w(T) \geq w(B)$ from [8] when $B$ is zero.
• In Example 3.5.1, the proposed bound yields $w(T) \geq 4.55$ for a specific $3 \times 3$ matrix, which exceeds all prior estimates (ranging from 2.236 to 3.654), demonstrating a significant improvement.
• The bound from Theorem 3.4 is shown to be strictly better than [7, Cor. 3.3] in cases where $m(A_j) = m(A_j^*) = 0$, and remains competitive even when these minimum moduli are non-zero.