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Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H. Let B A (H) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several A -numerical radius inequalities for operator matrices in B A (H ⨁ H) . In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption 'A is strictly positive'. We then establish several new upper and lower bounds for the A -numerical radius of an operator matrix whose entries are operators in B A (H). Further, we prove some refinements of earlier A-numerical radius inequalities for operators in B A (H) . [ABSTRACT FROM AUTHOR]

In this article, we obtain some upper bounds for A-numerical radius of operators in semi-Hilbertian spaces which improve the existing ones. Furthermore, we discuss some generalizations of earlier A-numerical radius inequalities. We also establish several inequalities for B-numerical radius of n × n operator matrices, where B is an n × n diagonal matrix whose diagonal entries are positive operator A. [ABSTRACT FROM AUTHOR]