Some upper bounds for the $$\mathbb {A}$$ A -numerical radius of $$2\times 2$$
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Tusi Mathematical Research Group
ORIGINAL PAPER
Some upper bounds for the A-numerical radius of 2 · 2 block matrices Qingxiang Xu1 • Zhongming Ye1 • Ali Zamani2 Received: 26 May 2020 / Accepted: 19 August 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract
A 0 be the 2 2 diagonal operator matrix determined by a positive 0 A bounded linear operator A on a Hilbert space. For semi-Hilbertian operators X and Y, we first show that n o 0 X 1 max XX ]A þ Y ]A Y A ; X ]A X þ YY ]A A w2A 4 Y 0 1 þ max wA ðXYÞ; wA ðYXÞ ; 2 Let A ¼
where wA ðÞ, k kA and wA ðÞ are the A-numerical radius, A-operator seminorm and A-numerical radius, respectively. We then apply the above inequality to find some upper bounds for the A-numerical radius of certain 2 2 operator matrices. In particular, we obtain some refinements of earlier A-numerical radius inequalities for semi-Hilbertian operators. An upper bound for the A-numerical radius of 2 2 block matrices of semi-Hilbertian space operators is also given. Keywords Positive operator Operator matrix Semi-inner product A-numerical radius Inequality
Mathematics Subject Classification 47A05 46C05 47B65 47A12
Communicated by Miguel Martin. & Ali Zamani [email protected] Qingxiang Xu [email protected] Zhongming Ye [email protected] Extended author information available on the last page of the article
Q. Xu et al.
1 Introduction Throughout this paper, H is a complex Hilbert space whose inner product is denoted by h; i. Let BðHÞ be the set of all bounded linear operators on H. The identity operator on H is denoted simply by I. For every T 2 BðHÞ, let T and RðTÞ be the adjoint operator and the range of T, respectively. Unless otherwise specified, A 2 BðHÞ is positive. Put hx; yiA ¼ hAx; yi for x; y 2 H. It is clear that h; iA : H H ! C is a positive semidefinite sesquilinear form, which yields a seminorm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k kA as kxkA ¼ hAx; xi for x 2 H. Suppose that T 2 BðHÞ. An element S 2 BðHÞ is said to be an A-adjoint operator of T if hTx; yiA ¼ hx; SyiA for every x; y 2 H, which can be interpreted as AS ¼ T A. It follows from [8, Theorem 1] that T admits an A-adjoint operator if and only if RðT AÞ RðAÞ, and in which case there exists a unique operator X 2 BðHÞ such that AX ¼ T A and RðXÞ RðAÞ. This unique operator is usually denoted by T ]A and is called the distinguished Aadjoint operator of T. The set of all operators in BðHÞ admitting A-adjoint is denoted by BA ðHÞ. Note that if T 2 BA ðHÞ, then T ]A 2 BA ðHÞ with ðT ]A Þ]A ¼
] PRðAÞ T ¼ PRðAÞ TPRðAÞ such that ðT ]A Þ]A A ¼ T ]A , where PRðAÞ denotes the orthogonal projection from H onto the closure of RðAÞ. Furthermore, it is easy to verify that for every T; S 2 BA ðHÞ, we have TS 2 BA ðHÞ such that ðTSÞ]A ¼ S]A T ]A . By convention, the set of all operators in BðHÞ admitting A1=2 adjoint is denoted by BA1=2 ðHÞ. It can be shown that BA ðHÞ BA1=2 ðHÞ. If T 2 BA1=2 ðHÞ, then it holds that kTkA :¼
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