The Closed Extensions of a Closed Operator

  • Christoph FischbacherEmail author


Given a densely defined and closed (but not necessarily symmetric) operator A acting on a complex Hilbert space \({\mathcal {H}}\), we establish a one-to-one correspondence between its closed extensions and subspaces \({\mathfrak {M}}\subset {\mathcal {D}}(A^*)\), that are closed with respect to the graph norm of \(A^*\) and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of \(A^*\). After this, we will express our results using the language of Gel’fand triples generalizing the well-known results for the selfadjoint case.


Closed extensions Gelfand triples Symmetric and selfadjoint operators 



I am very grateful to Sergii Kuzhel for valuable feedback on this manuscript. I would also like to thank him, Yury Arlinskiĭ and the referee for providing me with useful references. Parts of this work have been done during my PhD studies at the University of Kent in Canterbury, UK (cf. [11, Chapter 4]), and I would also like to thank my thesis advisors Sergey Naboko and Ian Wood for support and guidance. Finally, I express my appreciation to the UK Engineering and Physical Sciences Research Council (Doctoral Training Grant Ref. EP/K50306X/1) and the School of Mathematics, Statistics and Actuarial Science at the University of Kent for a Ph.D. studentship.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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