Orthogonality and norm attainment of operators in semi-Hilbertian spaces
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00104-7 ORIGINAL PAPER
Orthogonality and norm attainment of operators in semi‑Hilbertian spaces Jeet Sen1 · Debmalya Sain2 · Kallol Paul2 Received: 7 August 2020 / Accepted: 4 November 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We study the semi-Hilbertian structure induced by a positive operator A on a Hilbert space ℍ. Restricting our attention to A−bounded positive operators, we characterize the norm attainment set and also investigate the corresponding compactness property. We obtain a complete characterization of the A−Birkhoff–James orthogonality of A−bounded operators under an additional boundedness condition. This extends the finite-dimensional Bhatia-S̆ emrl Theorem verbatim to the infinite-dimensional setting. Keywords Semi-Hilbertian structure · Renorming · Positive operators · A-BirkhoffJames orthogonality · Norm attainment set · Compact operators Mathematics Subject Classification 47C05 · 47L05 · 46B03 · 47A30 · 47B65
1 Introduction The purpose of the paper was to explore the orthogonality and the norm attainment of bounded linear operators in the context of semi-Hilbertian structure induced by positive operators on a Hilbert space. Such a study was initiated by Krein in [10] and it remains an active and productive area of research till date. We refer the readers
Communicated by Jacek Chmielinski. * Kallol Paul [email protected] Jeet Sen [email protected] Debmalya Sain [email protected] 1
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
2
Department of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India
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J. Sen et al.
to [2, 3, 8, 18] and the references therein for more information on this. Let us now mention the relevant notations and the terminologies to be used in the article. We use the symbol ℍ to denote a Hilbert space. Finite-dimensional Hilbert spaces are also known as Euclidean spaces. Unless mentioned specifically, we work with both real and complex Hilbert spaces. The scalar field is denoted by 𝕂, which can be either ℝ or ℂ. The underlying inner product and the corresponding norm on ℍ are denoted by ⟨ , ⟩ and ‖ ⋅ ‖, respectively. In general, inner products on ℍ are defined as positive definite, conjugate symmetric forms which are linear in the first argument. It should be noted that apart from the underlying inner product ⟨ , ⟩ on ℍ, there may be many other inner products defined on ℍ, generating different norms. In order to avoid any confusion, whenever we talk of a topological concept on ℍ, we explicitly mention the norm that generates the corresponding topology. Let Bℍ = {x ∈ ℍ ∶ ‖x‖ ≤ 1} and Sℍ = {x ∈ ℍ ∶ ‖x‖ = 1} be the unit ball and the unit sphere of ℍ, respectively. We use the symbol 𝜃 to denote the zero vector of any Hilbert space other than the scalar fields ℝ and ℂ. For any complex number z, Re(z) and Im(z) denote the real part and the complex part of z, respectively. For any set G ⊂ ℍ, G denot
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