A-numerical radius and product of semi-Hilbertian operators

DOI10.1007/S41980-020-00388-4arXiv1912.11168MaRDI QIDQ6331719

Ali Zamani

Publication date: 23 December 2019

Abstract: Let $A$ be a positive bounded operator on a Hilbert space $$. The semi-inner product ${langlex,yangle}_A:=langleAx,yangle$, $x,yinmathcalH,$ induces a seminorm ${|cdot|}_A$ on $mathcalH$. Let $w_A(T)$ denote the $A$-numerical radius of an operator $T$ in the semi-Hilbertian space $$. In this paper, for any semi-Hilbertian operators $T$ and $S$, we show that $w_A(TR)=w_A(SR)$ for all ($A$-rank one) semi-Hilbertian operator $R$ if and only if $A^{1/2}T=lambda{A}^{1/2}S$ for some complex unit $lambda$. From this result we derive a number of consequences.

Mathematics Subject Classification ID

Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) (46C05) Numerical range, numerical radius (47A12) General (adjoints, conjugates, products, inverses, domains, ranges, etc.) (47A05) Positive linear operators and order-bounded operators (47B65)

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