Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

TLDR: In this paper, it was shown that ωr(T)≤2r−2 √ f2r(|X|)+g2r (|Y∗|)√ 12 √ 12, where X,Y are bounded linear operators on a Hilbert space H, r≥1, and f, g are nonnegative continuous functions satisfying the relation f(t)g(t)=t (t∈[0,∞)).

Abstract: In this article, we establish some upper bounds for numerical radius inequalities, including those of 2×2 operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if T=[0XY0], then ωr(T)≤2r−2‖f2r(|X|)+g2r(|Y∗|)‖12‖f2r(|Y|)+g2r(|X∗|)‖12 and ωr(T)≤2r−2‖f2r(|X|)+f2r(|Y∗|)‖12‖g2r(|Y|)+g2r(|X∗|)‖12, where X,Y are bounded linear operators on a Hilbert space H, r≥1, and f, g are nonnegative continuous functions on [0,∞) satisfying the relation f(t)g(t)=t (t∈[0,∞)). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators T1,…,Tn.