Generalized Cartesian decomposition and numerical radius inequalities

Bhunia, P and Sen, A and Paul, K (2023) Generalized Cartesian decomposition and numerical radius inequalities. In: Rendiconti del Circolo Matematico di Palermo .

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Abstract

Let T= { λ∈ C: ∣ λ∣ = 1 }. Every linear operator T on a complex Hilbert space H can be decomposed as T=T+λT∗2+iT-λT∗2i(λ∈T), designated as the generalized Cartesian decomposition of T. Using the generalized Cartesian decomposition we obtain several lower and upper bounds for the numerical radius of bounded linear operators which refine the existing bounds. We prove that if T is a bounded linear operator on H, then w(T)≥12∥T+λ+μ2T∗∥,for allλ,μ∈T. This improves the existing bounds w(T)≥12‖T‖ , w(T) ≥ ‖ Re(T) ‖ , w(T) ≥ ‖ Im(T) ‖ and so w2(T)≥14‖T∗T+TT∗‖, where Re(T) and Im(T) denote the the real part and the imaginary part of T, respectively. Further, using a lower bound for the numerical radius of a bounded linear operator, we develop upper bounds for the numerical radius of the commutator of operators which generalize and improve on the existing ones. © 2023, The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature.

Item Type:	Journal Article
Publication:	Rendiconti del Circolo Matematico di Palermo
Publisher:	Springer-Verlag Italia s.r.l.
Additional Information:	The copyright for this article belongs to the Springer-Verlag Italia s.r.l.
Keywords:	Bounded linear operator; Inequality; Numerical radius; Usual operator norm
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	17 Dec 2023 10:13
Last Modified:	17 Dec 2023 10:13
URI:	https://eprints.iisc.ac.in/id/eprint/83472

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