Abstract
Abstract In this paper we consider the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.
Highlights
Let (E, · ) be a normed linear space over the real or complex number field K
In this paper we introduce the hypo-q-operator norms and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces
A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy–Buniakowski– Schwarz inequality are given
Summary
Let (E, · ) be a normed linear space over the real or complex number field K. If E = H, where H is an inner product space over K, the hypo-Euclidean norm on Hn will be denoted by n x e := sup λjxj. X 2, i.e., · 2 and · e are equivalent norms on Hn. The following representation result for the hypo-Euclidean norm plays a key role in obtaining various bounds for this norm: Theorem 1.2 (Dragomir, 2007, [6]). The following representation result for the hypo-q-norms on En plays a key role in obtaining different bounds for these norms (see [7]): Theorem 1.3 (Dragomir, 2017, [7]). A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy–Buniakowski– Schwarz inequality are given
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