A note on the A -numerical radius of operators in semi-Hilbert spaces
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Archiv der Mathematik
A note on the A-numerical radius of operators in semi-Hilbert spaces Kais Feki
Abstract. Let A be a positive bounded linear operator acting on a complex Hilbert space H. Our aim in this paper is to prove some A-numerical radius inequalities of bounded linear operators acting on H when an additional semi-inner product structure induced by A is considered. In particular, an alternative proof of a recent result proved in Moslehian et al. (Linear Algebra Appl 591:299–321 2020) is given. Mathematics Subject Classification. 46C05, 47A05, 47B65. Keywords. Positive operator, Numerical radius, Semi-inner product.
1. Introduction and preliminaries. Let H, ·|· be a nontrivial complex Hilbert space equipped with the norm · . Let B (H) stand for the C ∗ -algebra of all bounded linear operators acting on H. Throughout this paper, by an operator we mean a bounded linear operator. The null space of every operator T is denoted by N (T ), its range by R(T ), and the adjoint of T by T ∗ . If S is a linear subspace of H, then S stands for its closure in the norm topology of H. An operator A ∈ B (H) is called positive, denoted by A ≥ 0, if Ax | x ≥ 0 for all x ∈ H. Notice that any positive operator A induces a semi-inner product on H defined by x | yA := Ax | y, ∀ x, y ∈ H. The seminorm induced by ·|·A is given by xA = x | xA = A1/2 x for all x ∈ H. Here A1/2 is denoted to be the square root of A. It can be seen that xA = 0 if and only if x ∈ N (A). This yields that (H, · A ) is a normed space if and only if A is injective. Moreover, one can verify that (H, · A ) is complete if and only if A has closed range.
K. Feki
Arch. Math.
For T ∈ B (H), the A-numerical range of T is defined, as in [3], by WA (T ) = {T x | xA ; x ∈ H, xA = 1} . It should be mentioned that WA (T ) is a nonempty convex subset of C and it is in general not closed even if H is finite dimensional. For more details related to this concept, the reader is referred to [3]. Moreover, the supremum modulus of WA (T ), denoted by ωA (T ), is called the A-numerical radius of T (see [7,12]). Thus ωA (T ) = sup {|T x | xA | ; x ∈ H, xA = 1} . If A = I, we get the classical numerical radius (see [5,8,9]). So, this new concept generalizes the numerical radius of Hilbert space operators and it has recently attracted the attention of many authors. One can consult [3,7,11,12] and the references therein. We would like to emphasize that it may happen that ωA (T ) = +∞ for some T ∈ B(H). For example, let T be the backward shift operator on the Hilbert space 2 (C) (that is T e1 = 0 and T en = en−1 for all n ≥ 2). Here (en )n≥1 denotes the canonical basis of 2 (C). Consider the positive operator 1 en for all n ≥ 1. Let the sequence (xn )n≥1 A on 2 (C) defined by Aen = n! √ √ (n+1)! be such that xn = √n! e + √2 en+1 for all n. Then one can verify that 2 n xn A = 1 for all n ≥ 1. Hence, √ n + 1 n→+∞ −−−−−→ +∞. ωA (T ) ≥ |T xn | xn A | = 2 The A-Crawford number of an operator T is defined, as in [12], by cA (T )
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