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August 2019 Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications
Ali Zamani
Ann. Funct. Anal. 10(3): 433-445 (August 2019). DOI: 10.1215/20088752-2019-0001

Abstract

In the following we generalize the concept of Birkhoff–James orthogonality of operators on a Hilbert space when a semi-inner product is considered. More precisely, for linear operators T and S on a complex Hilbert space H, a new relation TABS is defined if T and S are bounded with respect to the seminorm induced by a positive operator A satisfying T+γSATA for all γC. We extend a theorem due to Bhatia and Šemrl by proving that TABS if and only if there exists a sequence of A-unit vectors {xn} in H such that lim n+TxnA=TA and lim n+Txn,SxnA=0. In addition, we give some A-distance formulas. Particularly, we prove

inf γCT+γSA=sup {|Tx,yA|;xA=yA=1,Sx,yA=0}. Some other related results are also discussed.

Citation

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Ali Zamani. "Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications." Ann. Funct. Anal. 10 (3) 433 - 445, August 2019. https://doi.org/10.1215/20088752-2019-0001

Information

Received: 15 October 2018; Accepted: 3 January 2019; Published: August 2019
First available in Project Euclid: 6 August 2019

zbMATH: 07089129
MathSciNet: MR3989187
Digital Object Identifier: 10.1215/20088752-2019-0001

Subjects:
Primary: 46C05
Secondary: 47B65, 47L05

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.10 • No. 3 • August 2019
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