Positivity (2013) 17:1115–1122
On operator factorization of linear relations
Adrian Sandovici · Zoltán Sebestyén
Received: 15 August 2012 / Accepted: 12 January 2013 / Published online: 29 January 2013
© Springer Basel 2013
Abstract Assume that A and B are two linear relations in certain linear spaces.
The main objective of this note is to present two characterizations of the existence
of two linear operators C and T such that A = BC and A = TB, respectively.
These factorizations extend and complete some known results due to R.G. Douglas,
Z. Sebestyén and D. Popovici.
Keywords Linear space · linear operator · linear relation · Douglas factorization
Mathematics Subject Classiﬁcation (2000) 47A06 · 47A05
The concept of a linear relation in a linear space generalizes the concept of a (single-
valued) linear operator to that of a multi-valued operator. This notion goes back at
least to R Arens who gave a systematic treatment in . Since then, it has proved
useful in different areas and it has been studied in various speciﬁc contexts, cf. .
R.G. Douglas proposed in  two conditions on a given pair (A, B) of bounded linear
operators acting on a Hilbert space H which are equivalent to the existence of a bounded
linear operator T on H such that the factorization A = BT holds true (see also for
The work of Adrian Sandovici was supported by AM-POSDRU, project POSDRU/89/1.5/S/49944.
A. Sandovici (
Department of Sciences, University “Al. I. Cuza”, Lasc˘ar Catargi 54, 700107 Ia¸si, Romania
Department of Applied Analysis and Computational Mathematics,
Eötvös Loránd University of Budapest, Pázmány sétány 1/C, Budapest 1117, Hungary