Properties and Applications of Generalized Numerical Radius in Block Matrix Structures

In this paper, we prove several results that generalize fundamental properties of numerical radius and generalized numerical radius. Among our proven inequalities, we establish theorems for matrices A ∈ Mm(Mn), where Mm(Mn) represents the set of all m × m block complex matrices with each block belonging to Mn(C). Furthermore, we demonstrate that if A ∈ M2(Mn) is a positive semidefinite matrix, then w(2)(A) is positive semidefinite, and if A ∈ Mm(M2) is a positive semidefinite matrix, thenw(1)(A) is positive semidefinite. Here, w(1)(A) denotes the first partial matrix of numerical radius and w(2)(A) denotes the second partial matrix of numerical radius, respectively. Additionally, we develop relationships with classical matrix parameters and establish structural theorems for block matrices.

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