An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

2023/07/20 by M. H. M. Rashid, Rashid, M. H. M, Feras Bani-Ahmad +1
Decision Sciences · Mathematics · #47A12 #47A30 #47B15 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #I.1 #Mathematical Inequalities and Applications #Multi-Criteria Decision Making #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2307.11135

openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if \A_i,\B_i,\X_i∈\bh (i=1,2,⋯,n), m∈\N, p,q>1 with (1)/(p)+(1)/(q)=1 and ϕ and ψ are non-negative functions on [0,∞) which are continuous such that ϕ(t)ψ(t)=t for all t ∈ [0,∞), then w^2r\bra∑_i=1ⁿ\X_i\A_i^m\B_i≤ \fracn^2r-1m∑_j=1^m\norm∑_i=1ⁿ(1)/(p)S_i,j^pr+(1)/(q)T_i,j^qr-r₀inf_\normx=1ρ(ξ), where r₀=min\(1)/(p),(1)/(q)\, S_i,j=\X_iϕ²\bra\abs\A_i^j*\X_i^*, T_i,j=\bra\A_i^m-j\B_i^*ψ²\bra\abs\A_i^j\A_i^m-j\B_i and ρ(x)=\fracn^2r-1m∑_j=1^m∑_i=1ⁿ\bra\seqS_i,j^rξ,ξ^(p)/(2)-\seqT_i,j^rξ,ξ^(q)/(2)2.

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