Table Of ContentLIFTING TO THE SPECTRAL BALL WITH
INTERPOLATION
RAFAEL B. ANDRIST
5 Abstract. Wegivenecessaryandsufficientconditionsforsolving
1 the spectral Nevanlinna–Pick lifting problem. This reduces the
0 spectral Nevanlinna–Pick problem to a jet interpolation problem
2
into the symmetrized polydisc.
b
e
F
2
1. Introduction
1
The spectral ball is the set of square matrices with spectral radius
]
V
less than 1. It appears naturally in Control Theory [BFT89,BFT91],
C
but is also of theoretical interest in Several Complex Variables.
.
h
Definition 1.1. The spectral ball of dimension n ∈ N is defined to be
t
a
m Ω := {A ∈ Mat(n×n; C) : ρ(A) < 1}
n
[
where ρ denotes the spectral radius, i.e. the modulus of the largest
2
eigenvalue.
v
5
4 Note that in dimension n = 1, the spectral ball is just the unit disc.
1 We will assume throughout the paper that n ≥ 2.
7
0 The Nevanlinna–Pick problem is an interpolation problem for holo-
1. morphic functions on the unit disc D. The classical Nevanlinna–Pick
0 problem for holomorphic functions D → D with interpolation in a finite
5
set of points has been solved by Pick [Pic15] and Nevanlinna [Nev20].
1
: The spectral Nevanlinna–Pick problem is the analogue interpolation
v
i problem for holomorphic maps D → Ω :
X n
Given m ∈ N distinct points a ,...,a ∈ D, decide
r 1 m
a whether there is a holomorphic map F: D → Ω such
n
that
F(a ) = A , j = 1,...,m
j j
for given matrices A ,...,A ∈ Ω .
1 m n
2010 Mathematics Subject Classification. 30E05 (primary), and 32Q55, 32M10
(secondary).
Key words and phrases. spectral ball, Nevanlinna-Pick, symmetrized polydisc,
Oka principle, interpolation.
Acknowledgements: TheauthorwouldliketothankFrankKutzschebauch(Uni-
versity of Bern) and Franc Forstneriˇc (University of Ljubljana) for interesting and
helpful discussions.
1
2 RAFAEL B. ANDRIST
It has been studied by many authors, in particular by Agler and Young
for dimension n = 2 and generic interpolation points [AY00,AY04].
Bercovici[Ber03]hasfoundsolutionsforn = 2,andCostara[Cos05]has
found solutions for generic interpolation points in higher dimensions.
In general, the problem is still open.
The spectral ball Ω can also be understood in the following way:
n
Denote by σ ,...,σ : Cn → C the elementary symmetric polynomials
1 n
in n complex variables. Let EV: Mat(n×n; C) → Cn assign to each
matrix a vector of its eigenvalues. Then we denote by π := σ ◦
1 1
EV,...,π := σ ◦ EV the elementary symmetric polynomials in the
n n
eigenvalues. By symmetrizing we avoid any ambiguities of the order of
eigenvalues and obtain a polynomial map π = (π ,...,π ), symmetric
1 n
in the entries of matrices in Mat(n×n; C), actually
n
χ (λ) = λn + (−1)j ·π (A)·λn−j
A X j
j=1
where χ denotes the characteristic polynomial of A.
A
Now we can consider the holomorphic surjection π: Ω → G of the
n n
spectral ball onto the symmetrized polydisc G := (σ ,...,σ )(Dn). A
n 1 n
genericfibre, i.e.afibreaboveabasepointwithnomultipleeigenvalues,
consists exactly of one equivalence class of similar matrices. Thus, a
generic fibre is actually a SL (C)-homogeneous manifold where the
n
group SL (C) acts by conjugation. A singular fibre decomposes into
n
several strata which are SL (C)-homogeneous manifolds as well, but
n
not necessarily connected.
Given this holomorphic surjection, it is natural to consider a weaker
version of the spectral Nevanlinna–Pick problem, which is called the
spectral Nevanlinna–Pick lifting problem:
Given m ∈ N distinct points a ,...,a ∈ D, and a
1 m
holomorphic map f: D → G with f(a ) = π(A ) for
n j j
given matrices A ,...,A ∈ Ω , decide whether there
1 m n
is a holomorphic map F: D → Ω such that
n
F(a ) = A , j = 1,...,m.
j j
i.e. such that the following diagram commutes:
Ω
n
?? ✤
⑧ ✤
F ⑧⑧ ✤✤✤ π
⑧ ✤✤
⑧ f (cid:15)(cid:15)
D // G
n
a ,...,a 7→ π(A ),...,π(A )
1 m 1 m
When this lifting problem is solved, the spectral Nevanlinna–Pick
problem reduces to an interpolation problem D → G . In contrast to
n
the spectral ball, the symmetrized polydisc G is a taut domain, and
n
LIFTING TO THE SPECTRAL BALL WITH INTERPOLATION 3
should be more accessible with techniques from hyperbolic geometry.
Solutions to this lifting problem have been found for dimensions n =
2,3byNikolov, PflugandThomas[NPT11]andrecentlyfordimensions
n = 4,5 by Nikolov, Thomas and Tran [NTT14]. They also provide the
solution to a localised version of the spectral Nevanlinna–Pick lifting
problem:
Proposition 1.2 ([NTT14, Proposition 7]). Let f ∈ O(U,Gn), where
U is a neighborhood of p ∈ D, and let M ∈ Ω . Then there exists a
n
neighborhood U′ ⊂ U of p and F ∈ O(U′,Ω ) such that π ◦ F = f,
n
F(p) = M and F(v) is cyclic for v ∈ U′ \{p} if and only if
dk
dλkχf(v)(λ)(cid:12)(cid:12) = O((v−p)dnj−k(Bj)), 0 ≤ k ≤ nj −1, 1 ≤ j ≤ s
(cid:12)λ=λj
(cid:12) (∗)
where B ∈ Mat(n ×n ; C),...,B ∈ Mat(n ×n ; C) are the Jor-
1 1 1 s s s
dan blocks of M ≃ B ⊕···⊕B and
1 s
d (B ) = min d ∈ N : ∃x ,...x ∈ Cns.t.
ℓ j n 1 d
dimspan Bk1x ,...Bkdx ; k ,...,k ≥ 0 ≥ ℓ
C 1 d 1 d o
(cid:8) (cid:9)
A matrix M ∈ Mat(n×n; C) is called cyclic, if it admits a cyclic
vector x ∈ Cn, i.e. span Mk ·x, k ∈ N = Cn. This is equivalent to
C
(cid:8) (cid:9)
saying that there is only one Jordan block per eigenvalue in the Jordan
block decomposition of M.
Remark 1.3. It is easy to see that the condition (∗) is necessary: it is
a direct consequence of the factorisation π ◦ F = f on the jet level,
explicitly stated using the Jordan block decomposition.
In this paper we prove that the solution of the localised spectral
Nevanlinna–Pick lifting problem implies the global one:
Theorem 1.4. The spectral Nevanlinna–Pick lifting problem can be
solved if and only if it can be solved locally around the interpolation
points which means that (∗) holds for each interpolation point.
As a by-product of the proof we obtain (see Corollary 3.2) that the
spectral ball admits no bounded from above strictly plurisubharmonic
functions, but non-constant bounded holomorphic functions.
4 RAFAEL B. ANDRIST
2. Oka Theory
InthissectionweprovidesomebackgroundinOkatheorythatwillbe
needed in the proof of the main theorem. The goal is to find conditions
for the existence of holomorphic liftings or holomorphic sections. We
first recall the following equivalence:
Remark 2.1. For a holomorphic surjection π: Z → X and a holo-
morphic map f: Y → X, we denote the pull-back of Z under f by
∗
f (Z) = {(y,z) ∈ Y ×Z : f(y) = π(z)}
Together with the natural projections to the first resp. second factor
∗ ∗ ∗
f (π): f (Z) → Y resp. f (Z) → Z we obtain the following commu-
tative diagram
∗
f (Z) // Z
f∗(π) ✤✤✤✤✤ __❄❄❄f∗(F) F ⑧⑧⑧?? ✤✤✤✤✤ π
(cid:15)(cid:15)✤✤ ❄❄ ⑧⑧f (cid:15)(cid:15)✤✤
Y Y // X
We see that the existence of the lifting F is equivalent to the existence
of the lifting f∗(F), y 7→ (y,F(y)). Thus, the lifting problem can
always be reduced to finding a section of f∗(Z) → Y.
The crucial ingredient will be an Oka theorem for branched holo-
morphic maps due to Forstneriˇc. From [For03] we recall the following
definitions:
Definition 2.2. A point z ∈ Z is a branching point of a holomorphic
surjection between complex spaces π: Z → X if π is not submersive
in z. The branching locus of π is denoted by brπ and consists of all
branching points. For maps between complex spaces, also Z and
sing
π−1(X ) are included in brπ by convention.
sing
Definition 2.3. Aholomorphicsurjectionπ: Z → X betweencomplex
spaces is called an elliptic submersion over an open subset V ⊆ X if
reg
(1) π|π−1(V) is a submersion of complex manifolds
(2) each point x ∈ V has an open neighborhood U ⊆ V such that
there exists a holomorphic vector bundle E → π−1(U) together
with a so-called holomorphic dominating spray s: E → π−1(U)
satisfying the following conditions for each z ∈ π−1(U):
(a) s(E ) ⊆ Z
z π(z)
(b) s(O ) = z
z
(c) the derivative ds: T E → T Z maps the subspace E ⊆
0z z z
T E surjectively onto the vertical tangent space kerd π.
0z z
A examination of the proof in [For03] shows that a small variation
of the statement of [For03, Theorem 2.1] is valid with exactly the same
proof:
LIFTING TO THE SPECTRAL BALL WITH INTERPOLATION 5
Theorem 2.4. Let X be a Stein space and π: Z → X be a holomorphic
map of a complex space Z onto X. Assume that Z \brπ is an elliptic
submersion over X. Let X be a (possibly empty) closed subvariety
0
of X. Given a continuous section F: X → Z which is holomorphic
in an open set containing X and such that F(X \ X ) ⊆ Z \ brπ,
0 0
there is for any k ∈ N a homotopy of continuous sections F : X →
t
Z, t ∈ [0,1], such that F = F, for each t ∈ [0,1] the section F is
0 t
holomorphic in a neighborhood of X and tangent to order k along X ,
0 0
and F is holomorphic on X. If F is holomorphic in a neighborhood of
1
K ∪X for some compact, holomorphically convex subset K of X then
0
we can choose F to be holomorphic in a neighborhood of K ∪X and
t 0
to approximate F = F uniformly on K.
0
Remark 2.5. The difference to [For03, Theorem 2.1] is only that we
allow X to be smaller than π(brπ) provided that the section over
0
X \π(brπ) does not hit the branching locus brπ.
0
3. Proof
In case of the spectral Nevanlinna–Pick lifting problem, we have
X = G ,Y = D,Z = Ω , and X := {a ,...,a } is a closed subvariety
n n 0 1 m
inD. WenotethatY isindeedSteinandthatπ isanellipticsubmersion
in the cyclic matrices since the largest strata of all fibres are SL (C)-
n
homogeneous spaces of dimension n, see for example [For11, Example
5.5.13]. We write down the spray explicitly for any U ⊆ G : we
n
choose the bundle E := sl (C) × π−1(U) → π−1(U) with the natural
n
projection, and define the spray s: E → π−1(U) as
(B,A) 7→ exp(B)·A·exp(−B)
The derivative in (0,A), evaluated for a tangent vector C, is then given
by the adjoint representation of sl (C):
n
d s(C) = [A,C]
(0,A)
and the conditions of the elliptic submersion are easily verified.
Now, by remark 2.1 and the fact that the pullback of an elliptic
submersion is still an elliptic submersion (see [For03, Lemma 3.1]) we
translate the problem of finding a lifting F: X → Z with π ◦ F = f
into a problem of finding a section of f∗(π): f∗(Ω ) → D.
n
Due to a lack of reference, we also include the following lemma and
its proof:
Lemma 3.1. Each fibre, the largest stratum of each fibre and also each
connected component of any stratum of each fibre of π: Ω → G is C-
n n
connected.
Proof. Given any two similar matrices B and C, we can connect them
throughfinitelymanyholomorphicimagesofcomplexlines: thereexists
6 RAFAEL B. ANDRIST
a finite number of matrices Q ,...,Q ∈ sl (C) such that
1 s n
C = exp(Q )···exp(Q )·B ·exp(−Q )···exp(−Q ).
s 1 1 s
Byappendingacomplextimeparametert infrontofeachQ weobtain
s s
s lines connecting B to C. This proves that the largest stratum (hence
also a generic fibre) or any connected component of a stratum is C-
connected. For the connectedness of a singular fibre it is now sufficient
to connect the strata. This can be done by introducing parameters in
the off-diagonal entries of the Jordan-block decomposition. (cid:3)
We obtain the following corollary which will however not play any
role in the proof of the main theorem:
Corollary 3.2. The spectral ball Ω ,n ≥ 2, is a union of immersed
n
complex lines. Hence there exists no bounded from above strictly pluri-
subharmonic function on Ω .
n
Proof. Thepull-back ofany(strictly) plurisubharmonic functiontoany
of these complex lines is (strictly) subharmonic, and Liouville’s Theo-
rem applies. (cid:3)
Now we are ready to prove Theorem 1.4:
First we need understand the branching locus of the pull-back of
π: Ω → D by f, f∗(π): f∗(Ω ) → D. Let (z,A) ∈ f∗(Ω ) ⊂ D×Ω ⊂
n n n n
C × Cn2. Assume that (z,A) ∈ f∗(Ω ) is a singularity. This means
n
that the derivative of the defining equations of f∗(Ω ) has not full rank
n
in (z,A),
∃v ∈ Mat(n×1; C),v 6= 0, : (f′(z),−d π)⊺ ·v = 0
A
But then also (d π)⊺ · v = 0 and necessarily A ∈ brπ. Assume that
A
(z,A) ∈ f∗(Ω ) is a branching point of f∗(π) which projects (z,A) to
n
z, i.e. for d f∗(π) = (1,0,...,0)t it holds outside singularities that
(z,A)
∃w ∈ Mat(n×1; C),w 6= 0, : (f′(z),−d π)⊺ ·w = (1,0,...,0)⊺
A
But then again (d π)⊺ · w = 0 and necessarily A ∈ brπ. Hence,
A
∗ ∗
brf (π) which by definition includes also the singularities of f (Ω ),
n
is contained in the pull-back of brπ, and no new branching points are
introduced.
Theorem 2.4 gives the desired section f∗(F) = g , once the existence
1
of g is clear. Proposition 1.2 provides the necessary local liftings in
0
each point a ,...,a which are pulled back by f to local holomorphic
1 m
sections such that they do not hit brf∗(π) except forpossibly theinter-
polation point itself. Choosing the neighborhoods for the local sections
small enough and contractible, we achieve that g is holomorphic in a
0
neighborhood U of X .
0
It remains to show that we can extend g as continuous section over
0
D\U. Since all the components of U are contractible, this is equivalent
LIFTING TO THE SPECTRAL BALL WITH INTERPOLATION 7
to find a continuous section g which interpolates the points. We can
0
connect the finitely many points a ,...,a by arcs such that the union
1 m
γ of these arcs is homeomorphic to an interval, and then contract the
unit disc to γ. Since all the fibres are connected (even C-connected by
Lemma 3.1) there is no topological obstruction to construct a contin-
uous interpolating section γ → Ω .
n
We have proved Theorem 1.4.
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E-mail address: [email protected]
