Journal of Operator Theory

Volume 83, Issue 1, Winter 2020 pp. 229-250.

Simultaneous extension of two bounded operators between Hilbert spaces

Authors: Marko S. Djiki\'c (1), Jovana Nikolov Radenkovi\'c (2)
Author institution:(1) Faculty of Sciences and Mathematics, University of Ni\v s, Ni\v s, 18000, Serbia
(2) Faculty of Sciences and Mathematics, University of Ni\v s, Ni\v s, 18000, Serbia

Summary: The paper is concerned with the following question: if $A$ and $B$ are two bounded operators between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, and $\mathcal{M}$ and $\mathcal{N}$ are two closed subspaces in $\mathcal{H}$, when will there exist a bounded operator $C:\mathcal{H}\to\mathcal{K}$ which coincides with $A$ on $\mathcal{M}$ and with $B$ on $\mathcal{N}$ simultaneously? Besides answering this and some related questions, we also wish to emphasize the role played by the class of so-called semiclosed operators and the unbounded Moore--Penrose inverse in this work. Finally, we will relate our results to several well-known concepts, such as the operator equation $XA=B$ and the theorem of Douglas, Halmos' two projections theorem, and Drazin's star partial order.

DOI: http://dx.doi.org/10.7900/jot.2018oct09.2212
Keywords: bounded extension, semiclosed operators, quotient operators, operator ranges, unbounded projections, coherent pairs

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