The numerical radius of fractional powers of matrices

2025/09/24 by Eman Aldabbas, Mohammad Sababheh, Aldabbas, Eman +1
Computer Science · Mathematics · Physics and Astronomy · #15A60 #47A12 #47B44 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2509.19882

openalex publication_date 2025/09/24 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix A and any t ∈ (0,1), the matrix \(A^t\) is accretive-dissipative, and that ω(A^t)≥ ω^t(A) , where \(ω(⋅)\) is the numerical radius. This inequality complements the well-known power inequality ω(A^k)≤ ω^k(A), valid for any square matrix and positive integer power k. As an application, we prove that if A is accretive, then the above fractional inequality holds if 0

Citations

• Accretive Partial Transpose Matrices and Their Connections to Matrix Means
• Further Properties of Accretive Matrices
• Operator inequalities via accretive transforms
• On the weighted Geometric mean of accretive matrices
• Numerical radii of accretive matrices
• From positive to accretive matrices

Related

• Numerical Radius of Products of Special Matrices
• Numerical radii of accretive matrices
• q-Numerical radius of sectorial matrices and 2 × 2 operator matrices
• General Numerical Radius for Products of Sectorial Matrices
• New lower bound for numerical radius for off-diagonal 2× 2 matrices