Integr. Equ. Oper. Theory (2018) 90:39
Published online May 30, 2018
Springer International Publishing AG,
part of Springer Nature 2018
and Operator Theory
A Note on the Factorization of Some
Structured Matrix Functions
Ilya M. Spitkovsky
and Anatoly F. Voronin
Abstract. Let G be a block matrix function with one diagonal block A
being positive deﬁnite and the oﬀ diagonal blocks complex conjugates of
each other. Conditions are obtained for G to be factorable (in particular,
with zero partial indices) in terms of the Schur complement of A.
Mathematics Subject Classiﬁcation. 47A68, 30H15, 15A60.
Keywords. Riemann–Hilbert problem, Canonical factorization, Numer-
ical range, Schur complement, Hankel operator.
1. Preliminary Results
Let L be a simple closed curve in the complex plane C. Denote its interior and
exterior domains by D
(∞) respectively. The Riemann–Hilbert
boundary value problem consists in ﬁnding functions φ
analytic in D
(t)+g(t),t∈ L, (1)
imposed on its boundary values. Here G and g are known functions deﬁned
In the vector version of (1), φ
and g are vector functions with say n
entries while G is an n-by-n matrix function.
We are interested in the L
setting of (1). This means that g ∈ L
)andG ∈ L
(L) (all inclusions for vector and matrix
functions here and below are understood entrywise). We are using the nota-
(Ω) for the Smirnov classes in the domain Ω, 0 <p≤∞. See e.g.
or[1, Chapter 6] for the deﬁnition and properties of these classes. Note
in particular that, in the case of L being the unit circle T, E
classical Hardy spaces H
The ﬁrst author was supported in part by Faculty Research funding from the Division of
Science and Mathematics, New York University Abu Dhabi.