[Paper Review] On A-numerical radius inequalities for $2 imes 2$ operator matrices

Nirmal Chandra Rout, Satyajit Sahoo+1 more|arXiv (Cornell University)|Apr 16, 2020

Mathematical Inequalities and Applications18 references4 citations

TL;DR

This paper establishes new upper and lower bounds for the B-numerical radius of 2×2 operator matrices, where B is a block-diagonal operator matrix with A on the diagonal. It refines existing A-numerical radius inequalities using A-operator seminorms and A-adjoints, providing sharp estimates through spectral and norm-based analysis in semi-Hilbertian spaces.

ABSTRACT

Let ($\mathcal{H}, \langle . , . angle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,. angle_A),$ where $\langle x, y angle_A:=\langle Ax, y angle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2 imes 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlier $A$-numerical radius inequalities for operators.

Motivation & Objective

• To derive tighter upper and lower bounds for the B-numerical radius of 2×2 operator matrices in the context of A-semi-inner product spaces.
• To refine existing A-numerical radius inequalities by incorporating A-operator seminorms and A-adjoint operators.
• To investigate the behavior of numerical radius under block matrix structures induced by a positive operator A.
• To establish sharp inequalities that generalize classical numerical radius bounds in Hilbert space operator theory.

Proposed method

• Define the A-operator seminorm ‖T‖_A = sup{‖Tx‖_A / ‖x‖_A : x ∈ R(A), x ≠ 0} for T ∈ B_A(H).
• Introduce the B-numerical radius w_B(M) for 2×2 operator matrices M = [T₁ T₂; T₃ T₄] with B = diag(A,A).
• Use A-adjoint operators T#_A = A†T*A to define real and imaginary parts: Re_A(T) = (T + T#_A)/2, Im_A(T) = (T - T#_A)/(2i).
• Apply unitary invariance and spectral norm identities to derive bounds involving w_A(T₁ ± T₂) and w_A(T₁ ± iT₂).
• Employ matrix norm identities and submultiplicativity of ‖·‖_A to bound the B-numerical radius in terms of individual operator norms.
• Use the identity w_B(M) = sup{|⟨Mξ, ξ⟩_B| : ‖ξ‖_B = 1} and unitary transformations to reduce the problem to symmetric forms.

Experimental results

Research questions

• RQ1 What are the optimal upper and lower bounds for the B-numerical radius of a 2×2 operator matrix in terms of the A-numerical radii of its entries?
• RQ2 How can classical A-numerical radius inequalities be refined using A-operator seminorms and A-adjoints?
• RQ3 Under what conditions does equality hold in the derived B-numerical radius bounds?
• RQ4 Can the B-numerical radius of a block matrix be bounded using only the A-norms and A-numerical radii of its entries?
• RQ5 What is the role of A-unitary operators in preserving the B-numerical radius under similarity transformations?

Key findings

• For M = [T₁ T₂; 0 0], the inequality w_B(M) ≥ (1/2) max{w_A(T₁ ± T₂), w_A(T₁ ± iT₂)} holds, providing a sharp lower bound.
• For general 2×2 matrices, w_B(M) ≤ min{α, β}, where α and β are expressions involving ‖T_i‖_A and spectral norms of products like T₁#_A T₂.
• In the case T₁#_A T₂ = 0 and T₄ T₃#_A = 0, the bound simplifies to w_B(M) ≤ max(‖T₁‖_A, ‖T₂‖_A) + max(‖T₃‖_A, ‖T₄‖_A).
• The inequality w_B([0 T₁; T₂ 0]) ≤ w_A(T₁) + w_A(T₂) − (1/2)|w_A(T₁+T₂) − w_A(T₁−T₂)| provides a refined upper bound.
• The refined inequality includes a correction term involving the real and imaginary parts of T₁#_A, yielding w_A(T₁) ≥ (1/2)‖T₁‖_A + (1/2)(‖Re(T₁#_A)‖_A − ‖Im(T₁#_A)‖_A).
• Equality in the bounds is achieved when T₁ = I and T₂ = T₃ = T₄ = 0, confirming the sharpness of the derived inequalities.