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Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators

  • Pintu Bhunia , Raj Kumar Nayak and Kallol Paul EMAIL logo
Published/Copyright: August 9, 2022
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 4

Abstract

Let 𝓗 be a complex Hilbert space and A be a non-zero positive bounded linear operator on 𝓗. The main aim of this paper is to discuss a general method to develop A-operator seminorm and A-numerical radius inequalities of semi-Hilbertian space operators using the existing corresponding inequalities of bounded linear operators on 𝓗. Among many other inequalities we prove that if S, T, X ∈ 𝓑A(𝓗), i.e., if A-adjoint of S, T, X exist, then

2∥S♯AXT∥A≤∥SS♯AX+XTT♯A∥A.

Further, we prove that if T ∈ 𝓑A(𝓗), then

14∥T♯AT+TT♯A∥A≤18(∥T+T♯A∥A2+∥T−T♯A∥A2)≤18(∥T+T♯A∥A2+∥T−T♯A∥A2)+18cA2(T+T♯A)+18cA2(T−T♯A)≤wA2(T).

Here wA(⋅), cA(⋅) and ∥⋅∥A denote A-numerical radius, A-Crawford number and A-operator seminorm, respectively.

MSC 2010: Primary 47A12; Secondary 47A30; 47A63
Keywords: A-numerical radius; A-adjoint operator; A-selfadjoint operator; positive operator

  1. ( Communicated by Michal Zajac )

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Received: 2021-05-20
Accepted: 2021-08-16
Published Online: 2022-08-09
Published in Print: 2022-08-26

© 2022 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2022-0067
Keywords for this article
A-numerical radius; A-adjoint operator; A-selfadjoint operator; positive operator

Articles in the same Issue

  1. Regular Papers
  2. Variants of Booleanness: Congruences of a partial frame versus those of its free frame
  3. Models, coproducts and exchangeability: Notes on states on Baire functions
  4. Inverse tangent series involving pell and pell-lucas polynomials
  5. A new family of two-variable polynomials based on hermite polynomials
  6. Decreasing property and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function
  7. On the solvability of a fourth-order differential evolution equation on singular cylindrical domain in R4
  8. Investigation of an implicit Hadamard fractional differential equation with Riemann-Stieltjes integral boundary condition
  9. Stability criteria for systems of two first-order linear ordinary differential equations
  10. A perturbed eigenvalue problem in exterior domain
  11. On extensions of bilinear maps
  12. Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
  13. Weighted composition operators from the Besov space into nth weighted type spaces
  14. Fuzzy ideal topological vector spaces
  15. Partial actions on convergence spaces
  16. On quasi-small loop groups
  17. Topologies generated by symmetric porosity on normed spaces
  18. The Alpha Power Rayleigh-G family of distributions
  19. An alternative for Laplace Birnbaum-Saunders distribution
  20. Existence of positive solutions for boundary value problems with p-Laplacian operator
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