Generalized numerical radius inequalities of operators in Hilbert spaces
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Tusi Mathematical Research Group
ORIGINAL PAPER
Generalized numerical radius inequalities of operators in Hilbert spaces Kais Feki1,2 Received: 29 February 2020 / Accepted: 16 July 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract This paper is concerned with linear operators on a complex Hilbert space H, which are bounded with respect to the seminorm induced by a positive operator A on H. In particular, several inequalities involving the A-numerical radius of operators are established. Keywords Positive operator Semi-inner product Numerical radius
Mathematics Subject Classification 46C05 47A05 47B65
1 Introduction and preliminaries Let BðHÞ denote the C -algebra of all bounded linear operators acting on a nontrivial complex Hilbert space ðH; h j iÞ and I stand for the identity operator on H. If S is a linear subspace of H, then S denotes its closure in the norm topology of H. An operator T 2 BðHÞ is called positive if hTx j xi 0 for all x 2 H. If dim H ¼ d, we identify BðHÞ with the matrix algebra Md of all d d complex matrices. For the sequel, we will write T 0 if T is a positive operator. Moreover, the square root of every positive operator T will be denoted by T 1=2 . In addition, by an operator we mean a bounded linear operator. The range of every operator T is denoted by RðTÞ, Communicated by Miguel Martin. & Kais Feki [email protected]; [email protected] 1
Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Monastir, Tunisia
2
Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Department of Mathematics, University of Sfax, Faculty of Sciences of Sfax, Soukra Road, km 3.5, BP 1171, 3018 Sfax, Tunisia
K. Feki
its null space by N ðTÞ and T is the adjoint of T. Let A 0. Then, the semi-inner product induced by A is given by h j iA : H H ! C; ðx; yÞ ! hx j yiA :¼ hAx j yi: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The seminorm induced by h j iA is given by kxkA :¼ hx j xiA ¼ kA1=2 xk for all x 2 H. One can verify that kxkA ¼ 0 if and only if x 2 N ðAÞ. Thus, the semi-Hilbert space ðH; k kA Þ is a normed space if and only if A is injective. Moreover, it can be seen that ðH; k kA Þ is complete if and only if A has closed range in H. Recently, the theory of operators acting on semi-Hilbert spaces received considerable attention (e.g., see [2–4, 6, 7, 16, 17, 23, 24], and their references). The A-numerical range of T 2 BðHÞ is defined in [6] as WA ðTÞ ¼ hTx j xiA ; x 2 H; kxkA ¼ 1 : This concept is studied in [6]. In particular, it was shown that WA ðTÞ is a convex subset of C. Moreover, the supremum modulus of WA ðTÞ is called the A-numerical radius of T and it is denoted by xA ðTÞ. More precisely, we have xA ðTÞ ¼ sup hTx j xiA ; x 2 H; kxkA ¼ 1 : Obviously, if A ¼ I, we obtain the classical numerical radius of Hilbert space operators which received considerable attention in recent years by many researchers. We refer the reader to [5, 11, 18–20]) and the references therein.
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