We present a numerical radius bound for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> operator matrices that improves the bound of Abu-Omar and Kittaneh (Linear Algebra Appl 468:18–26, 2015). As a significant application, we derive an estimate for the numerical radius of the Kronecker products <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A\otimes B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊗</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, where A is an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix and B is a bounded linear operator. This result refines Holbrook’s classical bound <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$w(A\otimes B) \le w(A) \Vert B\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊗</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mi>B</mi> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> in the special case when all entries of A are non-negative. In addition, we establish spectral radius inequalities for the sums, products, and commutators of operators, improving upon the bounds of Kittaneh (Proc Am Math Soc 134:385–390, 2006) and Abu-Omar and Kittaneh (Stud Math 216(1):69–75, 2013). We further obtain an estimate for the zeros of an algebraic equation via Frobenius companion matrix, strengthening the bound of Abdurakhmanov (Mat Sb (N.S.) 131(173)(1):40–51, 126, 1986; translation in Math. USSR-Sb. 59(1):39–51, 1988). Furthermore, the Berezin radius inequalities are established, supported by several illustrative examples.

Strengthening of spectral radius, numerical radius, and Berezin radius inequalities

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2025年10月9日

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Pintu Bhunia

摘要

We present a numerical radius bound for $n\times n$ $n \times n$ operator matrices that improves the bound of Abu-Omar and Kittaneh (Linear Algebra Appl 468:18–26, 2015). As a significant application, we derive an estimate for the numerical radius of the Kronecker products $A\otimes B$ $A \otimes B$ , where A is an $n\times n$ $n \times n$ matrix and B is a bounded linear operator. This result refines Holbrook’s classical bound $w(A\otimes B) \le w(A) \Vert B\Vert $ $w (A \otimes B) \leq w (A) ‖ B ‖$ in the special case when all entries of A are non-negative. In addition, we establish spectral radius inequalities for the sums, products, and commutators of operators, improving upon the bounds of Kittaneh (Proc Am Math Soc 134:385–390, 2006) and Abu-Omar and Kittaneh (Stud Math 216(1):69–75, 2013). We further obtain an estimate for the zeros of an algebraic equation via Frobenius companion matrix, strengthening the bound of Abdurakhmanov (Mat Sb (N.S.) 131(173)(1):40–51, 126, 1986; translation in Math. USSR-Sb. 59(1):39–51, 1988). Furthermore, the Berezin radius inequalities are established, supported by several illustrative examples.