SOME NEW OBSERVATIONS ON INTERPOLATION
IN THE SPECTRAL UNIT BALL
GAUTAM BHARALI
Abstract. We present several results associated to a holomorphic-interpolation
problem for the spectral unit ball Ωn, n ≥ 2. We begin by showing that a known
necessary condition for the existence of a O(D; Ωn)-interpolant (D here being the
unit disc in C), given that the matricial data are non-derogatory, is not sufficient.
We provide next a new necessary condition for the solvability of the two-point
interpolation problem – one which is not restricted only to non-derogatory data,
and which incorporates the Jordan structure of the prescribed data. We then use
some of the ideas used in deducing the latter result to prove a Schwarz-type lemma
for holomorphic self-maps of Ωn, n ≥ 2.
1. Introduction and Statement of Results
The interpolation problem referred to in the title, and which links the assorted
results of this paper, is the following (D here will denote the open unit disc centered
at 0 ∈ C):
(*) Given M distinct points ζ1, . . . , ζM ∈ D and matrices W1, . . . ,WM in the spec-
tral unit ball Ωn := {W ∈Mn(C) : r(W ) < 1}, find conditions on {ζ1, . . . , ζM}
and {W1, . . . ,WM} such that there exists a holomorphic map F : D −→ Ωn
satisfying F (ζj) =Wj, j = 1, . . . ,M .
In the above statement, r(W ) denotes the spectral radius of the n × n matrix W .
Under a very slight simplification – i.e. that the interpolant F in (*) is required to
satisfy supζ∈D r(F (ζ)) < 1 – the paper [5] provides a characterisation of the interpo-
lation data ((ζ1,W1), . . . , (ζM ,WM )) that admit an interpolant of the type described.
However, this characterisation involves a non-trivial search over a region in Cn
Thus, there is interest in finding alternative characterisations that either: a) circum-
vent the need to perform a search; or b) reduce the dimension of the search-region. In
this regard, a new idea idea was introduced by Agler & Young in the paper [1]. This
idea was further developed over several works – notably in [2], in the papers [7] and
[8] by Costara, and in David Ogle’s thesis [13]. It can be summarised in two steps as
follows:
• If the matrices W1, . . . ,WM are all non-derogatory, then (*) is equivalent to
an interpolation problem in the symmetrized polydisc Gn, n ≥ 2, which is
defined as
Gn :=
(s1, . . . , sn) ∈ C
n : all the roots of zn +
(−1)jsjz
n−j = 0 lie in D
1991 Mathematics Subject Classification. Primary: 30E05, 47A56; Secondary: 32F45.
Key words and phrases. Complex geometry, Carathéodory metric, minimial polynomial, Schwarz
lemma, spectral radius, spectral unit ball.
This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV.
To appear in Integral Eqns. Operator Theory.
http://arxiv.org/abs/0704.1966v2
2 GAUTAM BHARALI
• The Gn-interpolation problem is shown to share certain aspects of the clas-
sical Nevanlinna-Pick problems, either by establishing conditions for a von
Neumann inequality for Gn – note that Gn is compact – or through function
theory.
It would be useful, at this stage, to recall the following
Definition 1.1. A matrix A ∈ Mn(C) is said to be non-derogatory if the geometric
multiplicity of each eigenvalue of A is 1 (regardless of its algebraic multiplicity). The
matrix A being non-derogatory is equivalent to A being similar to the companion
matrix of its characteristic polynomial – i.e., if zn +
j=1 sjz
n−j is the characteristic
polynomial then
A is non-derogatory ⇐⇒ A is similar to
0 −sn
1 0 −sn−1
. . .
. . .
0 1 −s1
The Agler-Young papers treat the case n = 2, while the last two works cited above
consider the higher-dimensional problem. The reader is referred to [2] for a proof
of the equivalence of (*), given non-derogatory matricial data, and the appropriate
Gn-interpolation problem. The similarity condition given in Definition 1.1 is central
to establishing this equivalence.
Before presenting the first result of this paper, we need to examine what is known
about (*) from the perspective of the Gn-interpolation problem. Since we would like
to focus on the matricial interpolation problem, we will paraphrase the results from
[13] and [8] in the language of non-derogatory matrices. Given an n × n complex
matrix W , let its characteristic polynomial χW (z) = zn+
j=1(−1)
jsj(W )z
n−j , and
define the rational function
f(z;W ) :=
j=1 jsj(W )(−1)
jzj−1
j=0 (n− j)sj(W )(−1)
Then, the most general statement that is known about (*) is:
Result 1.2 (paraphrased from [13] and [8]). Let ζ1, . . . , ζM be M distinct points
in D and let W1, . . . ,WM ∈ Ωn be non-derogatory matrices. If there exists a map
F ∈ O(D,Ωn) such that F (ζj) =Wj , j = 1, . . . ,M , then the matrices
(1.1)
1− f(z;Wj)f(z;Wk)
1− ζjζk
j,k=1
≥ 0 for each z ∈ D.
Here, and elsewhere in this paper, given two complex domains X and Y , O(X;Y )
will denote the class of all holomorphic maps from X into Y .
Remark 1.3. The matrices in (1.1) may appear different from those in [13, Corollary
5.2.2], but the latter are, in fact, ∗-congruent to the matrices above.
Even though Result 1.2 provides only a necessary condition, (1.1) is more tractable
for small values of M than the Bercovici-Foias-Tannenbaum condition. Its viability
as a sufficient condition, at least for small M , has been discussed in both [13] and
[8]. This is reasonable because the latter condition is sufficient when n = 2 and
M = 2 (and the given matrices are, of course, non-derogatory); see [4]. Given all
these developments, it seems appropriate to begin with the following:
INTERPOLATION IN THE SPECTRAL UNIT BALL 3
Observation 1.4. When n ≥ 3, the condition (1.1) is not sufficient for the existence
of a O(D; Ωn)-interpolant for the prescribed data ((ζ1,W1), . . . , (ζM ,WM )), where each
Wj ∈ Ωn, j = 1, . . . ,M , is non-derogatory.
The above observation relies on ideas from complex geometry; specifically – esti-
mates for invariant metrics on the symmetrized polydisc Gn, n ≥ 3. Our argument
follows from a recent study [11] of the Carathéodory metric on Gn, n ≥ 3. This
argument is presented in the next section.
Observation 1.4 takes us back to the drawing board when it comes to realising goals
of the type (a) or (b) (as in the opening paragraph) to determine whether a O(D; Ωn)-
interpolant exists for a given data-set. Thus, new conditions that are inequivalent to
(1.1) are desirable for the same reasons as those offered in [2] and [3]. To wit: all
extant approaches to implementing the Bercovici-Foias-Tannenbaum solution of (*)
are computational, and rely upon various search algorithms. Rigorous analytical
results, even if they only indicate when a data-set ((ζ1,W1), . . . , (ζM ,WM )) does not
admit an O(D; Ωn) interpolant – i.e. necessary conditions – provide tests of existing
algorithms/software and illustrate the complexities of (*). We will say more about
this; but first – notations for our next result. Given z1, z2 ∈ D, the pseudohyperbolic
distance between these points, written MD(z1, z2), is defined as:
MD(z1, z2) :=
z1 − z2
1− z2z1
∣∣∣∣ ∀z1, z2 ∈ D
We can now state our next result.
Theorem 1.5. Let F ∈ O(D; Ωn), n ≥ 2, and let ζ1, ζ2 ∈ D. Write Wj = F (ζj), and
σ(Wj) := the set of eigenvalues of Wj , j = 1, 2
(i.e. elements of σ(Wj) are not repeated according to multiplicity).
If λ ∈ σ(Wj), then let m(λ) denote the multiplicity of λ as a zero of the minimal
polynomial of Wj. Then:
(1.2)
 maxµ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ), max
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
Referring back to our previous paragraph: one could ask whether Theorem 1.5 is
able to highlight any complexities of (*) that Result 1.2 misses. There are two parts
to the answer:
1) The Jordan structure of the data-set ((ζ1,W1), (ζ2,W2)): Several well-known
examples from [6] and [2] reveal that the existence of a O(D; Ωn)-interpolant,
n ≥ 2, is sensitive to the Jordan structure of the matrices W1, . . . ,WM . How-
ever, to the best of our knowledge, there are no results in the literature to
date that incorporate information on the Jordan structures or the minimal
polynomials of W1, . . . ,WM . In contrast, the following example shows that
information on minimal polynomials is vital – i.e. that with the correct infor-
mation about the minimal polynomials of F (ζ1) and F (ζ2), condition (1.2) is
sharp.
4 GAUTAM BHARALI
Example 1.6. For n ≥ 3 and d = 2, . . . , n − 1, define the holomorphic map
Fd : D −→ Ωn by
Fd(ζ) :=

1 0 0
. . .
. . .
... 0
0 ζIn−d

, ζ ∈ D,
where In−d denotes the identity matrix of dimension n− d for 1 < d < n. Let
ζ1 = 0 and ζ2 = ζ. One easily computes – in the notation of Theorem 1.5 –
that:
µ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ) = |ζ|,
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
m(µ) = |ζ|2,
where the first equality holds because W1 is nilpotent of order d. So, (1.2) is
satisfied as an equality for the given choice of ζ1 and ζ2 – which is what was
meant above by saying that (1.2) is sharp. �
2) Comparison with (1.1): Theorem 1.5 would not be effective in testing any
of the existing algorithms used in the implementation of the Bercovici-Foias-
Tannenbaum solution to (*) if (1.1) were a universally stronger necessary
condition than (1.2). However, (1.1) is devised with non-derogatory data in
mind, whereas no simple interpolation condition was hitherto known for pairs
of arbitrary matrices in Ωn. Hence, by choosing any one of W1 and W2 to
be derogatory, one would like to examine how (1.1) and (1.2) compare. This
leads to our next observation.
Observation 1.7. For each n ≥ 3, we can find a data-set ((ζ1,W1), (ζ2,W2)) for
which (1.2) implies that it cannot admit any O(D; Ωn)-interpolant, whereas (1.1) pro-
vides no information.
An example pertinent to this observation is presented at the end of Section 3. As for
Theorem 1.5, it may be viewed as a Schwarz lemma for mappings between D and the
spectral unit ball. Note that the inequality (1.2) is preserved under automorphisms
of D and under the “obvious” automorphisms of Ωn (the full automorphism group
Aut(Ωn), n ≥ 2, is not known). The proof of Theorem 1.5 is presented in Section 3.
The key new idea in the proof of Theorem 1.5 – i.e. to focus on the minimal
polynomial of certain crucial matrices that lie in the range of F – pays off in obtaining
a result that is somewhat removed from the our main theme. The result in question
is a generalisation of the following theorem of Ransford and White [14, Theorem 2]:
(1.3) G ∈ O(Ωn; Ωn) and G(0) = 0 =⇒ r(G(X)) ≤ r(X) ∀X ∈ Ωn.
One would like to generalise (1.3) in the way the Schwarz-Pick lemma generalises the
Schwarz lemma for D – i.e. by formulating an inequality that is valid without assuming
that the holomorphic mapping in question has a fixed point. This generalisation is
as follows:
INTERPOLATION IN THE SPECTRAL UNIT BALL 5
Theorem 1.8. Let G ∈ O(Ωn; Ωn), n ≥ 2, and define dG := the degree of the minimal
polynomial of G(0). Then:
(1.4) r(G(X)) ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
∀X ∈ Ωn.
Furthermore, the inequality (1.4) is sharp in the sense that there exists a non-empty
set Sn ⊂ Ωn such that given any A ∈ Sn and d = 1, . . . , n, we can find a G
A,d ∈
O(Ωn; Ωn) such that
dGA,d = d, and
r(GA,d(A)) =
r(A)1/d + r(GA,d(0))
1 + r(GA,d(0))r(A)1/d
.(1.5)
2. A Discussion of Observation 1.4
We begin this discussion with a couple of definitions from complex geometry. Given
a domain Ω ⊂ Cn, the Carathéodory pseudodistance between two points z1, z2 ∈ Ω is
defined as
cΩ(z1, z2) := sup {pD(f(z1), f(z2)) : f ∈ O(Ω;D)} ,
where pD is the Poincaré distance on D (and pD is given by pD(ζ1, ζ2) = tanh
−1(MD(ζ1, ζ2))
for ζ1, ζ2 ∈ D). In the same setting, the Lempert functional on Ω× Ω, is defined as
(2.1) κ̃Ω(z1, z2)
:= inf {pD(ζ1, ζ2) : ∃ψ ∈ O(D; Ω) and ζ1, ζ2 ∈ D such that ψ(ζj) = zj, j = 1, 2.} .
It is not hard to show that the set on the right-hand side above is non-empty. The
reader is referred to Chapter III of [10] for details. Next, we examine a few techni-
cal objects. For the remainder of this section, S = (s1, . . . , sn) will denote a point in
Cn, n ≥ 2. For z ∈ D define the rational map fn(z;S) := (s̃1(z;S), . . . , s̃n−1(z;S)), n ≥
2, by
s̃j(z;S) :=
(n− j)sj − z(j + 1)sj+1
n− zs1
, S ∈ Cn s.t. n− zs1 6= 0,
j = 1, . . . , (n− 1).
Next, define
F (Z; ·) := f2(z1; ·) ◦ · · · ◦ fn(zn−1; ·) ∀Z = (z1, . . . , zn−1) ∈ D
where the second argument varies through that region in Cn where the right-hand
side above is defined. The connection of these objects with our earlier discussions is
established via
f(z;S) :=
j=1 jsj(−1)
jzj−1
j=0 (n− j)sj(−1)
, z ∈ D,
and S varies through that region in Cn where the right-hand side above is defined.
Note the resemblance of f(z;S) to f(z;W ) defined earlier. From Theorem 3.5 of [8],
we excerpt:
Result 2.1. Let S = (s1, . . . , sn) denote a point in C
n. Then:
1) f(z; ·) = F (z, . . . , z; ·) ∀z ∈ D, wherever defined.
2) S ∈ Gn if and only if supz∈D |f(z;S)| < 1, n ≥ 2.
6 GAUTAM BHARALI
3) If S ∈ Gn, n ≥ 2, then
|f(z;S)| = sup
|F (Z;S)|.
For convenience, let us refer to the Carathéodory pseudodistance on Gn, n ≥ 2, by
cn. Next, define – here we refer to Section 2 of [11] – the following distance function
on Gn
(2.2) pn(S, T ) := max
Z∈(∂D)n−1
pD(F (Z;S), F (Z;T )) ∀S, T ∈ Gn.
This is the distance function – whose properties have been studied in [11] – we
shall exploit to support Observation 1.4. The well-definedness of the right-hand
side above follows from parts (2) and (3) of Result 2.1 above. Furthermore, since
F (Z;S), F (Z;T ) ∈ D for each Z ∈ D
whenever S, T ∈ Gn, n ≥ 2, it follows
simply from the definition that
(2.3) cn(S, T ) ≥ pn(S, T ) ∀S, T ∈ Gn.
Since we have now adopted certain notations from [11], we must make the following
Note. We have opted to rely on the notation of [8]. This leads to a slight dis-
crepancy between our definition of pn in (2.2) and that in [11]. This discrepancy is
easily reconciled by the observation that F (·;S) used here and in [8] will have to be
read as F (·;−s1, s2, . . . , (−1)
nsn) in [11]. This is harmless because S ∈ Gn ⇐⇒
(−s1, s2, . . . , (−1)
nsn) ∈ Gn.
Let us now refer back to the condition (1.1) with M = 2. An easy calculation
involving 2× 2 matrices reveals that
When M = 2, (1.1) ⇐⇒ sup
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ ≤
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
If W1 is nilpotent of order n (recall that all matrices occuring in (1.1) are non-
derogatory), then f(·;W1) ≡ 0. Of course,W2 ∈ Ωn implies that (s1(W2), . . . , sn(W2)) ∈
Gn. By part (2) of Result 2.1, f(z;W2) ∈ D ∀z ∈ D. This leads to the following key
fact:
(2.4) When M = 2 and W1 is nilpotent of order n,
(1.1) ⇐⇒ sup
tanh−1|f(z;W2)| = sup
pD(0, f(z;W2)) ≤ pD(ζ1, ζ2).
We now appeal to Proposition 2 in [11], i.e. pn(0, ·) 6= cn(0, ·) for each n ≥ 3. Let
us now fix n ≥ 3. Let S0 ∈ Gn \ {0} be such that cn(0, S0) > pn(0, S0). Let ε0 > 0
be such that cn(0, S0) = pn(0, S0) + 2ε0. Let us write S0 = (s0,1, . . . , s0,n) and choose
two matrices W1,W2 ∈ Ωn as follows:
W1 = a nilpotent of order n, W2 =
0 (−1)n−1s0,n
1 0 (−1)n−2s0,n−1
. . .
. . .
0 1 s0,1
INTERPOLATION IN THE SPECTRAL UNIT BALL 7
i.e. W2 is the companion matrix of the polynomial z
j=1(−1)
js0,jz
n−j . We
emphasize the following facts that follow from this choice of W1 and W2
f(·,W1) = f(·; 0, . . . , 0) ≡ 0, f(·,W2) = f(·;S0),(2.5)
W1 and W2 are, by construction, non-derogatory.
The relations in (2.5) are cases of a general correspondence between matrices in Ωn
and points in Gn , given by the surjective, holomorphic map Πn : Ωn −→ Gn, where
Πn(W ) := (s1(W ), . . . , sn(W )),
and sj(W ), j = 1, . . . n, are as defined in the beginning of this article.
Let us pick two distinct points ζ1, ζ2 ∈ D such that
(2.6) pD(ζ1, ζ2)− ε0 < pn(0, S0) ≤ pD(ζ1, ζ2).
Assume, now, that (1.1) is a sufficient condition for the existence of a O(D; Ωn)-
interpolant. Then, in view of the choices of W1,W2, the second inequality in (2.6),
and (2.5) we get
(2.7) sup
tanh−1|f(z;W2)| = sup
tanh−1|f(z;W2)| = pn(0, S0) ≤ pD(ζ1, ζ2).
The first equality in (2.7) is a consequence of part (2) of Result 2.1: since S0 ∈ Gn,
the rational function f(·;W2) = f(·;S0) ∈ O(D)
C(D), whence the equality follows
from the Maximum Modulus Theorem. But now, owing to the equivalence (2.4), the
estimate (2.7) implies, by assumption, that there exists an interpolant F ∈ O(D; Ωn)
such that F (ζj) =Wj, j = 1, 2. Then, Πn ◦F : D −→ Gn satisfies Πn ◦F (ζ1) = 0 and
Πn ◦ F (ζ2) = S0. Then, by the definition of the Lempert functional (for convenience,
we denote the Lempert functional of Gn by κ̃n)
κ̃n(0, S0) ≤ pD(ζ1, ζ2) < pn(0, S0) + ε0 (from (2.6), 1st part)
< cn(0, S0). (by definition of ε0)
But, for any domain Ω, the Carathéodory pseudodistance and the Lempert function
always satisfy cΩ ≤ κ̃Ω. Hence, we have just obtained a contradiction. Hence our
assumption that (1.1) is sufficient for the existence of an O(D,Ωn)-interpolation, for
n ≥ 3, must be false.
3. The Proof of Theorem 1.5
The proofs in this section depend crucially on a theorem by Vesentini. The result
is as follows:
Result 3.1 (Vesentini, [15]). Let A be a complex, unital Banach algebra and let r(x)
denote the spectral radius of any element x ∈ A. Let f ∈ O(D;A). Then, the function
ζ 7−→ r(f(ζ)) is subharmonic on D.
The following result is the key lemma of this section. The proof of Theorem 1.5
is reduced to a simple application of this lemma. The structure of this proof is
reminiscent of [12, Theorem 1.1]. This stems from the manner in which Vesentini’s
theorem is used. The essence of the trick below goes back to Globevnik [9]. The
reader will notice that Theorem 1.5 specialises to Globevnik’s Schwarz lemma when
W1 = 0.
8 GAUTAM BHARALI
Lemma 3.2. Let F ∈ O(D; Ωn). For each λ ∈ σ(F (0)), define m(λ) :=the multiplic-
ity of λ as a zero of the minimal polynomial of F (0). Define the Blaschke product
B(ζ) :=
λ∈σ(F (0))
ζ − λ
1− λζ
)m(λ)
, ζ ∈ D.
Then |B(µ)| ≤ |ζ| ∀µ ∈ σ(F (ζ)).
Proof. The Blaschke product B induces a matrix function B̃ on Ωn: for any matrix
A ∈ Ωn, we set
B̃(A) :=
λ∈σ(F (0))
(I− λA)−m(λ)(A− λI)m(λ),
which is well-defined on Ωn because whenever λ 6= 0,
(I− λA) = λ(I/λ−A) ∈ GL(n,C).
Furthermore, since ζ 7−→ (ζ − λ)/(1− λζ), |λ| < 1, has a power-series expansion that
converges uniformly on compact subsets of D, it follows from standard arguments
(3.1) σ(B̃(A)) = {B(µ) : µ ∈ σ(A)} for any A ∈ Ωn.
By the definition of the minimal polynomial, B̃ ◦F (0) = 0. Since B̃ ◦F (0) = 0, there
exists a holomorphic map Φ ∈ O(D;Mn(C)) such that B̃ ◦ F (ζ) = ζΦ(ζ). Note that
(3.2) σ(B̃ ◦ F (ζ)) = σ(ζΦ(ζ)) = ζσ(Φ(ζ)) ∀ζ ∈ D.
Since σ(B̃ ◦ F (ζ)) ⊂ D, the above equations give us:
(3.3) r(Φ(ζ)) < 1/R ∀ζ : |ζ| = R, R ∈ (0, 1).
Taking A =Mn(C) in Vesentini’s theorem, we see that ζ 7−→ r(Φ(ζ)) is subharmonic
on the unit disc. Applying the Maximum Principle to (3.3) and taking limits as
R −→ 1−, we get
(3.4) r(Φ(ζ)) ≤ 1 ∀ζ ∈ D.
In view of (3.1), (3.2) and (3.4), we get
|B(µ)| ≤ |ζ|r(Φ(ζ)) ≤ |ζ| ∀µ ∈ σ(F (ζ)).
We are now in a position to provide
3.3. The proof of Theorem 1.5. Define the disc automorphisms
Mj(ζ) :=
ζ − ζj
1− ζjζ
, j = 1, 2,
and write Φj = F ◦M
j , j = 1, 2. Note that Φ1(0) = W1. For λ ∈ σ(W1), let m(λ)
be as stated in the theorem. Define the Blaschke product
B1(ζ) :=
λ∈σ(W1)
ζ − λ
1− λζ
)m(λ)
, ζ ∈ D.
INTERPOLATION IN THE SPECTRAL UNIT BALL 9
Applying Lemma 3.2, we get
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ = |M1(ζ2)| ≥
λ∈σ(W1)
1− λµ
λ∈σ(W1)
MD(µ, λ)
m(λ) ∀µ ∈ σ(Φ1(M1(ζ2))) = σ(W2).(3.5)
Now, swapping the roles of ζ1 and ζ2 and applying the same argument to
B2(ζ) :=
µ∈σ(W2)
ζ − µ
1− µζ
)m(µ)
, ζ ∈ D,
we get
(3.6)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ ≥
µ∈σ(W2)
MD(λ, µ)
m(µ) ∀λ ∈ σ(W1).
Combining (3.5) and (3.6), we get
 maxµ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ), max
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
We conclude this section with an example.
Example 3.4. An illustration of Observation 1.7
We begin by pointing out that the phenomenon below is expected for n = 2. We
want to consider n > 2 and show that there is no interpolant for the following data, but
that this cannot be inferred from (1.1). First the matricial data: let n = 2m, m ≥ 2,
and let
W1 = any block-diagonal matrix with two m×m-blocks that
are each nilpotent of order m.(3.7)
Next, for an α ∈ D, α 6= 0, let
W2 = the companion matrix of the polynomial (z
2m − αzm).
Note that, by construction, W2 is non-derogatory. We have the characteristic poly-
nomials χW1(z) = zm and χW2(z) = z2m − αzm. Hence
f(·;W1) ≡ 0, f(z;W2) =
−mαzm−1
2m−mαzm
We recall, from Section 2, the following equivalent form of (1.1):
(3.8) When M = 2, (1.1) ⇐⇒ sup
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ ≤
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
Since, clearly, f(·;W2) ∈ O(D)
C(D), by the Maximum Modulus Theorem
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ = supz∈∂D
|2m−mαzm|
2m−m|α|
< |α|.(3.9)
10 GAUTAM BHARALI
Observe that σ(W1) = {0} and σ(W2) = {0, |α|
1/mei(2πj+Arg(α))/m, j = 1, . . . ,m}.
Therefore,
µ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ) = |α|,
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
m(µ) = 0.
We set ζ1 = 0 and pick ζ2 ∈ D in such a way that
(3.10)
2m−m|α|
< |ζ2| =
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ < |α|.
Such a choice of ζ2 is made possible by the inequality (3.9). In view of the last
calculation above, we see that the data-set ((W1, ζ1), (W2, ζ2)) constructed violates
the inequality (1.2). Thus, there is no O(D,Ω2m)-interpolant for this data-set. In
contrast, since the equivalent form (3.8) of (1.1) is satisfied, the latter does not yield
any information about the existence of a O(D,Ω2m)-interpolant. �
4. The Proof of Theorem 1.8
In order to prove Theorem 1.8, we shall need the following elementary
Lemma 4.1. Given a fractional-linear transformation T (z) := (az + b)/(cz + d), if
T (∂D) ⋐ C, then T (∂D) is a circle with
centre(T (∂D)) =
bd− ac
|d|2 − |c|2
, radius(T (∂D)) =
|ad− bc|
||d|2 − |c|2|
We are now in a position to present
4.2. The proof of Theorem 1.8. Let G ∈ O(Ωn; Ωn) and let λ1, . . . , λs be the
distinct eigenvalues of G(0). Define m(j) :=the multiplicity of the factor (λ− λj) in
the minimal polynomial of G(0). Define the Blaschke product
BG(ζ) :=
ζ − λj
1− λjζ
)m(j)
, ζ ∈ D.
BG induces the following matrix function which, by a mild abuse of notation, we shall
also denote as BG
BG(Y ) :=
(I− λjY )
−m(j)(Y − λjI)
m(j) ∀Y ∈ Ωn,
which is well-defined on Ωn precisely as explained in the proof of Lemma 3.2. Once
again, owing to the analyticity of BG on Ωn,
σ(BG(Y )) = {BG(λ) : λ ∈ σ(Y )} ∀Y ∈ Ωn,
whence BG : Ωn −→ Ωn. Therefore, if we define
H(X) := BG ◦G(X) ∀X ∈ Ωn,
INTERPOLATION IN THE SPECTRAL UNIT BALL 11
then H ∈ O(Ωn; Ωn) and, by construction, H(0) = 0. By the Ransford-White result,
r(H(X)) ≤ r(X), or, more precisely
µ∈σ(G(X))
µ− λj
1− λjµ
 ≤ r(X) ∀X ∈ Ωn.
In particular:
µ∈σ(G(X))
distM(µ;σ(G(0)))
≤ r(X) ∀X ∈ Ωn,
where, for any compactK  D and µ ∈ D, we define distM(µ;K) := minζ∈K
∣∣(µ− ζ)(1− ζµ)−1
For the moment, let us fixX ∈ Ωn. For each µ ∈ σ(G(X)), let λ
(µ) be an eigenvalue of
G(0) such that
∣∣∣(µ− λ(µ))(1 − λ(µ)µ)−1
∣∣∣ = distM(µ;σ(G(0))). Now fix µ ∈ σ(G(X)).
The above inequality leads to
(4.1)
∣∣∣∣∣
µ− λ(µ)
1− λ(µ)µ
∣∣∣∣∣ ≤ r(X)
1/dG .
Applying Lemma 4.1 to the Möbius transformation
T (z) =
|µ|z − λ(µ)
1− λ(µ)|µ|z
we deduce that ∣∣∣∣∣
ζ − λ(µ)
1− λ(µ)ζ
∣∣∣∣∣ ≥
||µ| − |λ(µ)||
1− |µ||λ(µ)|
∀ζ : |ζ| = |µ|.
Applying the above fact to (4.1), we get
|µ| − |λ(µ)|
1− |µ||λ(µ)|
≤ r(X)1/dG
⇒ |µ| ≤
r(X)1/dG + |λ(µ)|
1 + |λ(µ)|r(X)1/dG
, µ ∈ σ(G(X)).(4.2)
Note that the function
t 7−→
r(X)1/dG + t
1 + r(X)1/dGt
, t ≥ 0,
is an increasing function on [0,∞). Combining this fact with (4.2), we get
|µ| ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
which holds ∀µ ∈ σ(G(X)), while the right-hand side is independent of µ. Since this
is true for any arbitrary X ∈ Ωn, we conclude that
r(G(X)) ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
∀X ∈ Ωn.
In order to prove the sharpness of (1.2), let us fix an n ≥ 2, and define
Sn := {A ∈ Ωn : A has a single eigenvalue of multiplicity n}.
12 GAUTAM BHARALI
Pick any d = 1, . . . , n, and define
Md(X) :=


[tr(X)/n], if d = 1,

0 tr(X)/n
1 0 0
. . .
. . .

, if d ≥ 2,
and, for the chosen d, define G(d) by the following block-diagonal matrix
(d)(Y ) :=
Md(X)
tr(X)
∀X ∈ Ωn.
For our purposes GA,d = G(d) for each A ∈ Sn; i.e., the equality (1.5) will will hold
with the same function for each A ∈ Sn. To see this, note that
• r(G(d)(X)) = |tr(X)/n|1/d; and
• G(d)(0) is nilpotent of degree d, whence dG(d) = d.
Therefore,
r(A)1/d + r(G(d)(0))
1 + r(G(d)(0))r(A)1/d
= r(A)1/d = r(G(d)(A)) ∀A ∈ Sn,
which establishes (1.5) �
References
[1] J. Agler and N.J. Young, A commutant lifting theorem for a domain in C2 and spectral interpo-
lation, J. Funct. Anal. 161 (1999), 452-477.
[2] J. Agler and N.J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations
Operator Theory 37 (2000), 375-385.
[3] J. Agler and N.J. Young, The two-by-two spectral Nevanlinna-Pick problem, Trans. Amer. Math.
Soc. 356 (2004), 573-585.
[4] J.Agler, N.J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14
(2004), 375-403.
[5] H. Bercovici, C. Foias and A. Tannenbaum, Spectral variants of the Nevanlinna-Pick interpo-
lation problem in Signal Processing, Scattering and Operator Theory, and Numerical Methods
(Amsterdam, 1989), 23-45, Progr. Systems Control Theory 5, Birkhuser Boston, Boston, MA,
1990.
[6] H. Bercovici, C. Foias, A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer.
Math. Soc. 325 (1991), 741-763.
[7] C. Costara, The 2 × 2 spectral Nevanlinna-Pick problem, J. London Math. Soc. (2) 71 (2005),
684-702.
[8] C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), 23-55.
[9] J. Globevnik, Schwarz’s lemma for the spectral radius, Rev. Roumaine Math. Pures Appl. 19
(1974), 1009-1012.
[10] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter
Expositions in Mathematics no. 9, Walter de Gruyter & Co., Berlin, 1993.
[11] N. Nikolov, P. Pflug, P.J. Thomas and W. Zwonek, Estimates of the Carathéodory metric on the
symmetrized polydisc, arXiv preprint arXiv:math.CV/0608496.
[12] A. Nokrane and T. Ransford, Schwarz’s lemma for algebroid multifunctions, Complex Variables
Theory Appl. 45 (2001), 183–196.
[13] D.Ogle, Operator and Function Theory of the Symmetrized Polydisc, Thesis (1999),
http://www.maths.leeds.ac.uk/ nicholas/.
[14] T.J. Ransford and M.C. White, Holomorphic self-maps of the spectral unit ball, Bull. London
Math. Soc. 23 (1991), 256-262.
INTERPOLATION IN THE SPECTRAL UNIT BALL 13
[15] E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital. (4) 1 1968,
427-429.
Department of Mathematics, Indian Institute of Science, Bangalore – 560 012
E-mail address: bharali@math.iisc.ernet.in
	1. Introduction and Statement of Results
	2. A Discussion of Observation ??
	3. The Proof of Theorem ??
	4. The Proof of Theorem ??
	References

  We present several results associated to a holomorphic-interpolation problem
for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known
necessary condition for the existence of a
$\mathcal{O}(D;\Omega_n)$-interpolant (D here being the unit disc in the
complex plane), given that the matricial data are non-derogatory, is not
sufficient. We provide next a new necessary condition for the solvability of
the two-point interpolation problem -- one which is not restricted only to
non-derogatory data, and which incorporates the Jordan structure of the
prescribed data. We then use some of the ideas used in deducing the latter
result to prove a Schwarz-type lemma for holomorphic self-maps of \Omega_n,
n\geq 2.

Introduction and Statement of Results
The interpolation problem referred to in the title, and which links the assorted
results of this paper, is the following (D here will denote the open unit disc centered
at 0 ∈ C):
(*) Given M distinct points ζ1, . . . , ζM ∈ D and matrices W1, . . . ,WM in the spec-
tral unit ball Ωn := {W ∈Mn(C) : r(W ) < 1}, find conditions on {ζ1, . . . , ζM}
and {W1, . . . ,WM} such that there exists a holomorphic map F : D −→ Ωn
satisfying F (ζj) =Wj, j = 1, . . . ,M .
In the above statement, r(W ) denotes the spectral radius of the n × n matrix W .
Under a very slight simplification – i.e. that the interpolant F in (*) is required to
satisfy supζ∈D r(F (ζ)) < 1 – the paper [5] provides a characterisation of the interpo-
lation data ((ζ1,W1), . . . , (ζM ,WM )) that admit an interpolant of the type described.
However, this characterisation involves a non-trivial search over a region in Cn
Thus, there is interest in finding alternative characterisations that either: a) circum-
vent the need to perform a search; or b) reduce the dimension of the search-region. In
this regard, a new idea idea was introduced by Agler & Young in the paper [1]. This
idea was further developed over several works – notably in [2], in the papers [7] and
[8] by Costara, and in David Ogle’s thesis [13]. It can be summarised in two steps as
follows:
• If the matrices W1, . . . ,WM are all non-derogatory, then (*) is equivalent to
an interpolation problem in the symmetrized polydisc Gn, n ≥ 2, which is
defined as
Gn :=
(s1, . . . , sn) ∈ C
n : all the roots of zn +
(−1)jsjz
n−j = 0 lie in D
1991 Mathematics Subject Classification. Primary: 30E05, 47A56; Secondary: 32F45.
Key words and phrases. Complex geometry, Carathéodory metric, minimial polynomial, Schwarz
lemma, spectral radius, spectral unit ball.
This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV.
To appear in Integral Eqns. Operator Theory.
http://arxiv.org/abs/0704.1966v2
2 GAUTAM BHARALI
• The Gn-interpolation problem is shown to share certain aspects of the clas-
sical Nevanlinna-Pick problems, either by establishing conditions for a von
Neumann inequality for Gn – note that Gn is compact – or through function
theory.
It would be useful, at this stage, to recall the following
Definition 1.1. A matrix A ∈ Mn(C) is said to be non-derogatory if the geometric
multiplicity of each eigenvalue of A is 1 (regardless of its algebraic multiplicity). The
matrix A being non-derogatory is equivalent to A being similar to the companion
matrix of its characteristic polynomial – i.e., if zn +
j=1 sjz
n−j is the characteristic
polynomial then
A is non-derogatory ⇐⇒ A is similar to
0 −sn
1 0 −sn−1
. . .
. . .
0 1 −s1
The Agler-Young papers treat the case n = 2, while the last two works cited above
consider the higher-dimensional problem. The reader is referred to [2] for a proof
of the equivalence of (*), given non-derogatory matricial data, and the appropriate
Gn-interpolation problem. The similarity condition given in Definition 1.1 is central
to establishing this equivalence.
Before presenting the first result of this paper, we need to examine what is known
about (*) from the perspective of the Gn-interpolation problem. Since we would like
to focus on the matricial interpolation problem, we will paraphrase the results from
[13] and [8] in the language of non-derogatory matrices. Given an n × n complex
matrix W , let its characteristic polynomial χW (z) = zn+
j=1(−1)
jsj(W )z
n−j , and
define the rational function
f(z;W ) :=
j=1 jsj(W )(−1)
jzj−1
j=0 (n− j)sj(W )(−1)
Then, the most general statement that is known about (*) is:
Result 1.2 (paraphrased from [13] and [8]). Let ζ1, . . . , ζM be M distinct points
in D and let W1, . . . ,WM ∈ Ωn be non-derogatory matrices. If there exists a map
F ∈ O(D,Ωn) such that F (ζj) =Wj , j = 1, . . . ,M , then the matrices
(1.1)
1− f(z;Wj)f(z;Wk)
1− ζjζk
j,k=1
≥ 0 for each z ∈ D.
Here, and elsewhere in this paper, given two complex domains X and Y , O(X;Y )
will denote the class of all holomorphic maps from X into Y .
Remark 1.3. The matrices in (1.1) may appear different from those in [13, Corollary
5.2.2], but the latter are, in fact, ∗-congruent to the matrices above.
Even though Result 1.2 provides only a necessary condition, (1.1) is more tractable
for small values of M than the Bercovici-Foias-Tannenbaum condition. Its viability
as a sufficient condition, at least for small M , has been discussed in both [13] and
[8]. This is reasonable because the latter condition is sufficient when n = 2 and
M = 2 (and the given matrices are, of course, non-derogatory); see [4]. Given all
these developments, it seems appropriate to begin with the following:
INTERPOLATION IN THE SPECTRAL UNIT BALL 3
Observation 1.4. When n ≥ 3, the condition (1.1) is not sufficient for the existence
of a O(D; Ωn)-interpolant for the prescribed data ((ζ1,W1), . . . , (ζM ,WM )), where each
Wj ∈ Ωn, j = 1, . . . ,M , is non-derogatory.
The above observation relies on ideas from complex geometry; specifically – esti-
mates for invariant metrics on the symmetrized polydisc Gn, n ≥ 3. Our argument
follows from a recent study [11] of the Carathéodory metric on Gn, n ≥ 3. This
argument is presented in the next section.
Observation 1.4 takes us back to the drawing board when it comes to realising goals
of the type (a) or (b) (as in the opening paragraph) to determine whether a O(D; Ωn)-
interpolant exists for a given data-set. Thus, new conditions that are inequivalent to
(1.1) are desirable for the same reasons as those offered in [2] and [3]. To wit: all
extant approaches to implementing the Bercovici-Foias-Tannenbaum solution of (*)
are computational, and rely upon various search algorithms. Rigorous analytical
results, even if they only indicate when a data-set ((ζ1,W1), . . . , (ζM ,WM )) does not
admit an O(D; Ωn) interpolant – i.e. necessary conditions – provide tests of existing
algorithms/software and illustrate the complexities of (*). We will say more about
this; but first – notations for our next result. Given z1, z2 ∈ D, the pseudohyperbolic
distance between these points, written MD(z1, z2), is defined as:
MD(z1, z2) :=
z1 − z2
1− z2z1
∣∣∣∣ ∀z1, z2 ∈ D
We can now state our next result.
Theorem 1.5. Let F ∈ O(D; Ωn), n ≥ 2, and let ζ1, ζ2 ∈ D. Write Wj = F (ζj), and
σ(Wj) := the set of eigenvalues of Wj , j = 1, 2
(i.e. elements of σ(Wj) are not repeated according to multiplicity).
If λ ∈ σ(Wj), then let m(λ) denote the multiplicity of λ as a zero of the minimal
polynomial of Wj. Then:
(1.2)
 maxµ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ), max
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
Referring back to our previous paragraph: one could ask whether Theorem 1.5 is
able to highlight any complexities of (*) that Result 1.2 misses. There are two parts
to the answer:
1) The Jordan structure of the data-set ((ζ1,W1), (ζ2,W2)): Several well-known
examples from [6] and [2] reveal that the existence of a O(D; Ωn)-interpolant,
n ≥ 2, is sensitive to the Jordan structure of the matrices W1, . . . ,WM . How-
ever, to the best of our knowledge, there are no results in the literature to
date that incorporate information on the Jordan structures or the minimal
polynomials of W1, . . . ,WM . In contrast, the following example shows that
information on minimal polynomials is vital – i.e. that with the correct infor-
mation about the minimal polynomials of F (ζ1) and F (ζ2), condition (1.2) is
sharp.
4 GAUTAM BHARALI
Example 1.6. For n ≥ 3 and d = 2, . . . , n − 1, define the holomorphic map
Fd : D −→ Ωn by
Fd(ζ) :=

1 0 0
. . .
. . .
... 0
0 ζIn−d

, ζ ∈ D,
where In−d denotes the identity matrix of dimension n− d for 1 < d < n. Let
ζ1 = 0 and ζ2 = ζ. One easily computes – in the notation of Theorem 1.5 –
that:
µ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ) = |ζ|,
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
m(µ) = |ζ|2,
where the first equality holds because W1 is nilpotent of order d. So, (1.2) is
satisfied as an equality for the given choice of ζ1 and ζ2 – which is what was
meant above by saying that (1.2) is sharp. �
2) Comparison with (1.1): Theorem 1.5 would not be effective in testing any
of the existing algorithms used in the implementation of the Bercovici-Foias-
Tannenbaum solution to (*) if (1.1) were a universally stronger necessary
condition than (1.2). However, (1.1) is devised with non-derogatory data in
mind, whereas no simple interpolation condition was hitherto known for pairs
of arbitrary matrices in Ωn. Hence, by choosing any one of W1 and W2 to
be derogatory, one would like to examine how (1.1) and (1.2) compare. This
leads to our next observation.
Observation 1.7. For each n ≥ 3, we can find a data-set ((ζ1,W1), (ζ2,W2)) for
which (1.2) implies that it cannot admit any O(D; Ωn)-interpolant, whereas (1.1) pro-
vides no information.
An example pertinent to this observation is presented at the end of Section 3. As for
Theorem 1.5, it may be viewed as a Schwarz lemma for mappings between D and the
spectral unit ball. Note that the inequality (1.2) is preserved under automorphisms
of D and under the “obvious” automorphisms of Ωn (the full automorphism group
Aut(Ωn), n ≥ 2, is not known). The proof of Theorem 1.5 is presented in Section 3.
The key new idea in the proof of Theorem 1.5 – i.e. to focus on the minimal
polynomial of certain crucial matrices that lie in the range of F – pays off in obtaining
a result that is somewhat removed from the our main theme. The result in question
is a generalisation of the following theorem of Ransford and White [14, Theorem 2]:
(1.3) G ∈ O(Ωn; Ωn) and G(0) = 0 =⇒ r(G(X)) ≤ r(X) ∀X ∈ Ωn.
One would like to generalise (1.3) in the way the Schwarz-Pick lemma generalises the
Schwarz lemma for D – i.e. by formulating an inequality that is valid without assuming
that the holomorphic mapping in question has a fixed point. This generalisation is
as follows:
INTERPOLATION IN THE SPECTRAL UNIT BALL 5
Theorem 1.8. Let G ∈ O(Ωn; Ωn), n ≥ 2, and define dG := the degree of the minimal
polynomial of G(0). Then:
(1.4) r(G(X)) ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
∀X ∈ Ωn.
Furthermore, the inequality (1.4) is sharp in the sense that there exists a non-empty
set Sn ⊂ Ωn such that given any A ∈ Sn and d = 1, . . . , n, we can find a G
A,d ∈
O(Ωn; Ωn) such that
dGA,d = d, and
r(GA,d(A)) =
r(A)1/d + r(GA,d(0))
1 + r(GA,d(0))r(A)1/d
.(1.5)
2. A Discussion of Observation 1.4
We begin this discussion with a couple of definitions from complex geometry. Given
a domain Ω ⊂ Cn, the Carathéodory pseudodistance between two points z1, z2 ∈ Ω is
defined as
cΩ(z1, z2) := sup {pD(f(z1), f(z2)) : f ∈ O(Ω;D)} ,
where pD is the Poincaré distance on D (and pD is given by pD(ζ1, ζ2) = tanh
−1(MD(ζ1, ζ2))
for ζ1, ζ2 ∈ D). In the same setting, the Lempert functional on Ω× Ω, is defined as
(2.1) κ̃Ω(z1, z2)
:= inf {pD(ζ1, ζ2) : ∃ψ ∈ O(D; Ω) and ζ1, ζ2 ∈ D such that ψ(ζj) = zj, j = 1, 2.} .
It is not hard to show that the set on the right-hand side above is non-empty. The
reader is referred to Chapter III of [10] for details. Next, we examine a few techni-
cal objects. For the remainder of this section, S = (s1, . . . , sn) will denote a point in
Cn, n ≥ 2. For z ∈ D define the rational map fn(z;S) := (s̃1(z;S), . . . , s̃n−1(z;S)), n ≥
2, by
s̃j(z;S) :=
(n− j)sj − z(j + 1)sj+1
n− zs1
, S ∈ Cn s.t. n− zs1 6= 0,
j = 1, . . . , (n− 1).
Next, define
F (Z; ·) := f2(z1; ·) ◦ · · · ◦ fn(zn−1; ·) ∀Z = (z1, . . . , zn−1) ∈ D
where the second argument varies through that region in Cn where the right-hand
side above is defined. The connection of these objects with our earlier discussions is
established via
f(z;S) :=
j=1 jsj(−1)
jzj−1
j=0 (n− j)sj(−1)
, z ∈ D,
and S varies through that region in Cn where the right-hand side above is defined.
Note the resemblance of f(z;S) to f(z;W ) defined earlier. From Theorem 3.5 of [8],
we excerpt:
Result 2.1. Let S = (s1, . . . , sn) denote a point in C
n. Then:
1) f(z; ·) = F (z, . . . , z; ·) ∀z ∈ D, wherever defined.
2) S ∈ Gn if and only if supz∈D |f(z;S)| < 1, n ≥ 2.
6 GAUTAM BHARALI
3) If S ∈ Gn, n ≥ 2, then
|f(z;S)| = sup
|F (Z;S)|.
For convenience, let us refer to the Carathéodory pseudodistance on Gn, n ≥ 2, by
cn. Next, define – here we refer to Section 2 of [11] – the following distance function
on Gn
(2.2) pn(S, T ) := max
Z∈(∂D)n−1
pD(F (Z;S), F (Z;T )) ∀S, T ∈ Gn.
This is the distance function – whose properties have been studied in [11] – we
shall exploit to support Observation 1.4. The well-definedness of the right-hand
side above follows from parts (2) and (3) of Result 2.1 above. Furthermore, since
F (Z;S), F (Z;T ) ∈ D for each Z ∈ D
whenever S, T ∈ Gn, n ≥ 2, it follows
simply from the definition that
(2.3) cn(S, T ) ≥ pn(S, T ) ∀S, T ∈ Gn.
Since we have now adopted certain notations from [11], we must make the following
Note. We have opted to rely on the notation of [8]. This leads to a slight dis-
crepancy between our definition of pn in (2.2) and that in [11]. This discrepancy is
easily reconciled by the observation that F (·;S) used here and in [8] will have to be
read as F (·;−s1, s2, . . . , (−1)
nsn) in [11]. This is harmless because S ∈ Gn ⇐⇒
(−s1, s2, . . . , (−1)
nsn) ∈ Gn.
Let us now refer back to the condition (1.1) with M = 2. An easy calculation
involving 2× 2 matrices reveals that
When M = 2, (1.1) ⇐⇒ sup
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ ≤
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
If W1 is nilpotent of order n (recall that all matrices occuring in (1.1) are non-
derogatory), then f(·;W1) ≡ 0. Of course,W2 ∈ Ωn implies that (s1(W2), . . . , sn(W2)) ∈
Gn. By part (2) of Result 2.1, f(z;W2) ∈ D ∀z ∈ D. This leads to the following key
fact:
(2.4) When M = 2 and W1 is nilpotent of order n,
(1.1) ⇐⇒ sup
tanh−1|f(z;W2)| = sup
pD(0, f(z;W2)) ≤ pD(ζ1, ζ2).
We now appeal to Proposition 2 in [11], i.e. pn(0, ·) 6= cn(0, ·) for each n ≥ 3. Let
us now fix n ≥ 3. Let S0 ∈ Gn \ {0} be such that cn(0, S0) > pn(0, S0). Let ε0 > 0
be such that cn(0, S0) = pn(0, S0) + 2ε0. Let us write S0 = (s0,1, . . . , s0,n) and choose
two matrices W1,W2 ∈ Ωn as follows:
W1 = a nilpotent of order n, W2 =
0 (−1)n−1s0,n
1 0 (−1)n−2s0,n−1
. . .
. . .
0 1 s0,1
INTERPOLATION IN THE SPECTRAL UNIT BALL 7
i.e. W2 is the companion matrix of the polynomial z
j=1(−1)
js0,jz
n−j . We
emphasize the following facts that follow from this choice of W1 and W2
f(·,W1) = f(·; 0, . . . , 0) ≡ 0, f(·,W2) = f(·;S0),(2.5)
W1 and W2 are, by construction, non-derogatory.
The relations in (2.5) are cases of a general correspondence between matrices in Ωn
and points in Gn , given by the surjective, holomorphic map Πn : Ωn −→ Gn, where
Πn(W ) := (s1(W ), . . . , sn(W )),
and sj(W ), j = 1, . . . n, are as defined in the beginning of this article.
Let us pick two distinct points ζ1, ζ2 ∈ D such that
(2.6) pD(ζ1, ζ2)− ε0 < pn(0, S0) ≤ pD(ζ1, ζ2).
Assume, now, that (1.1) is a sufficient condition for the existence of a O(D; Ωn)-
interpolant. Then, in view of the choices of W1,W2, the second inequality in (2.6),
and (2.5) we get
(2.7) sup
tanh−1|f(z;W2)| = sup
tanh−1|f(z;W2)| = pn(0, S0) ≤ pD(ζ1, ζ2).
The first equality in (2.7) is a consequence of part (2) of Result 2.1: since S0 ∈ Gn,
the rational function f(·;W2) = f(·;S0) ∈ O(D)
C(D), whence the equality follows
from the Maximum Modulus Theorem. But now, owing to the equivalence (2.4), the
estimate (2.7) implies, by assumption, that there exists an interpolant F ∈ O(D; Ωn)
such that F (ζj) =Wj, j = 1, 2. Then, Πn ◦F : D −→ Gn satisfies Πn ◦F (ζ1) = 0 and
Πn ◦ F (ζ2) = S0. Then, by the definition of the Lempert functional (for convenience,
we denote the Lempert functional of Gn by κ̃n)
κ̃n(0, S0) ≤ pD(ζ1, ζ2) < pn(0, S0) + ε0 (from (2.6), 1st part)
< cn(0, S0). (by definition of ε0)
But, for any domain Ω, the Carathéodory pseudodistance and the Lempert function
always satisfy cΩ ≤ κ̃Ω. Hence, we have just obtained a contradiction. Hence our
assumption that (1.1) is sufficient for the existence of an O(D,Ωn)-interpolation, for
n ≥ 3, must be false.
3. The Proof of Theorem 1.5
The proofs in this section depend crucially on a theorem by Vesentini. The result
is as follows:
Result 3.1 (Vesentini, [15]). Let A be a complex, unital Banach algebra and let r(x)
denote the spectral radius of any element x ∈ A. Let f ∈ O(D;A). Then, the function
ζ 7−→ r(f(ζ)) is subharmonic on D.
The following result is the key lemma of this section. The proof of Theorem 1.5
is reduced to a simple application of this lemma. The structure of this proof is
reminiscent of [12, Theorem 1.1]. This stems from the manner in which Vesentini’s
theorem is used. The essence of the trick below goes back to Globevnik [9]. The
reader will notice that Theorem 1.5 specialises to Globevnik’s Schwarz lemma when
W1 = 0.
8 GAUTAM BHARALI
Lemma 3.2. Let F ∈ O(D; Ωn). For each λ ∈ σ(F (0)), define m(λ) :=the multiplic-
ity of λ as a zero of the minimal polynomial of F (0). Define the Blaschke product
B(ζ) :=
λ∈σ(F (0))
ζ − λ
1− λζ
)m(λ)
, ζ ∈ D.
Then |B(µ)| ≤ |ζ| ∀µ ∈ σ(F (ζ)).
Proof. The Blaschke product B induces a matrix function B̃ on Ωn: for any matrix
A ∈ Ωn, we set
B̃(A) :=
λ∈σ(F (0))
(I− λA)−m(λ)(A− λI)m(λ),
which is well-defined on Ωn because whenever λ 6= 0,
(I− λA) = λ(I/λ−A) ∈ GL(n,C).
Furthermore, since ζ 7−→ (ζ − λ)/(1− λζ), |λ| < 1, has a power-series expansion that
converges uniformly on compact subsets of D, it follows from standard arguments
(3.1) σ(B̃(A)) = {B(µ) : µ ∈ σ(A)} for any A ∈ Ωn.
By the definition of the minimal polynomial, B̃ ◦F (0) = 0. Since B̃ ◦F (0) = 0, there
exists a holomorphic map Φ ∈ O(D;Mn(C)) such that B̃ ◦ F (ζ) = ζΦ(ζ). Note that
(3.2) σ(B̃ ◦ F (ζ)) = σ(ζΦ(ζ)) = ζσ(Φ(ζ)) ∀ζ ∈ D.
Since σ(B̃ ◦ F (ζ)) ⊂ D, the above equations give us:
(3.3) r(Φ(ζ)) < 1/R ∀ζ : |ζ| = R, R ∈ (0, 1).
Taking A =Mn(C) in Vesentini’s theorem, we see that ζ 7−→ r(Φ(ζ)) is subharmonic
on the unit disc. Applying the Maximum Principle to (3.3) and taking limits as
R −→ 1−, we get
(3.4) r(Φ(ζ)) ≤ 1 ∀ζ ∈ D.
In view of (3.1), (3.2) and (3.4), we get
|B(µ)| ≤ |ζ|r(Φ(ζ)) ≤ |ζ| ∀µ ∈ σ(F (ζ)).
We are now in a position to provide
3.3. The proof of Theorem 1.5. Define the disc automorphisms
Mj(ζ) :=
ζ − ζj
1− ζjζ
, j = 1, 2,
and write Φj = F ◦M
j , j = 1, 2. Note that Φ1(0) = W1. For λ ∈ σ(W1), let m(λ)
be as stated in the theorem. Define the Blaschke product
B1(ζ) :=
λ∈σ(W1)
ζ − λ
1− λζ
)m(λ)
, ζ ∈ D.
INTERPOLATION IN THE SPECTRAL UNIT BALL 9
Applying Lemma 3.2, we get
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ = |M1(ζ2)| ≥
λ∈σ(W1)
1− λµ
λ∈σ(W1)
MD(µ, λ)
m(λ) ∀µ ∈ σ(Φ1(M1(ζ2))) = σ(W2).(3.5)
Now, swapping the roles of ζ1 and ζ2 and applying the same argument to
B2(ζ) :=
µ∈σ(W2)
ζ − µ
1− µζ
)m(µ)
, ζ ∈ D,
we get
(3.6)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ ≥
µ∈σ(W2)
MD(λ, µ)
m(µ) ∀λ ∈ σ(W1).
Combining (3.5) and (3.6), we get
 maxµ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ), max
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
We conclude this section with an example.
Example 3.4. An illustration of Observation 1.7
We begin by pointing out that the phenomenon below is expected for n = 2. We
want to consider n > 2 and show that there is no interpolant for the following data, but
that this cannot be inferred from (1.1). First the matricial data: let n = 2m, m ≥ 2,
and let
W1 = any block-diagonal matrix with two m×m-blocks that
are each nilpotent of order m.(3.7)
Next, for an α ∈ D, α 6= 0, let
W2 = the companion matrix of the polynomial (z
2m − αzm).
Note that, by construction, W2 is non-derogatory. We have the characteristic poly-
nomials χW1(z) = zm and χW2(z) = z2m − αzm. Hence
f(·;W1) ≡ 0, f(z;W2) =
−mαzm−1
2m−mαzm
We recall, from Section 2, the following equivalent form of (1.1):
(3.8) When M = 2, (1.1) ⇐⇒ sup
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ ≤
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ .
Since, clearly, f(·;W2) ∈ O(D)
C(D), by the Maximum Modulus Theorem
∣∣∣∣∣
f(z;W1)− f(z;W2)
1− f(z;W2)f(z;W1)
∣∣∣∣∣ = supz∈∂D
|2m−mαzm|
2m−m|α|
< |α|.(3.9)
10 GAUTAM BHARALI
Observe that σ(W1) = {0} and σ(W2) = {0, |α|
1/mei(2πj+Arg(α))/m, j = 1, . . . ,m}.
Therefore,
µ∈σ(W2)
λ∈σ(W1)
MD(µ, λ)
m(λ) = |α|,
λ∈σ(W1)
µ∈σ(W2)
MD(λ, µ)
m(µ) = 0.
We set ζ1 = 0 and pick ζ2 ∈ D in such a way that
(3.10)
2m−m|α|
< |ζ2| =
ζ1 − ζ2
1− ζ2ζ1
∣∣∣∣ < |α|.
Such a choice of ζ2 is made possible by the inequality (3.9). In view of the last
calculation above, we see that the data-set ((W1, ζ1), (W2, ζ2)) constructed violates
the inequality (1.2). Thus, there is no O(D,Ω2m)-interpolant for this data-set. In
contrast, since the equivalent form (3.8) of (1.1) is satisfied, the latter does not yield
any information about the existence of a O(D,Ω2m)-interpolant. �
4. The Proof of Theorem 1.8
In order to prove Theorem 1.8, we shall need the following elementary
Lemma 4.1. Given a fractional-linear transformation T (z) := (az + b)/(cz + d), if
T (∂D) ⋐ C, then T (∂D) is a circle with
centre(T (∂D)) =
bd− ac
|d|2 − |c|2
, radius(T (∂D)) =
|ad− bc|
||d|2 − |c|2|
We are now in a position to present
4.2. The proof of Theorem 1.8. Let G ∈ O(Ωn; Ωn) and let λ1, . . . , λs be the
distinct eigenvalues of G(0). Define m(j) :=the multiplicity of the factor (λ− λj) in
the minimal polynomial of G(0). Define the Blaschke product
BG(ζ) :=
ζ − λj
1− λjζ
)m(j)
, ζ ∈ D.
BG induces the following matrix function which, by a mild abuse of notation, we shall
also denote as BG
BG(Y ) :=
(I− λjY )
−m(j)(Y − λjI)
m(j) ∀Y ∈ Ωn,
which is well-defined on Ωn precisely as explained in the proof of Lemma 3.2. Once
again, owing to the analyticity of BG on Ωn,
σ(BG(Y )) = {BG(λ) : λ ∈ σ(Y )} ∀Y ∈ Ωn,
whence BG : Ωn −→ Ωn. Therefore, if we define
H(X) := BG ◦G(X) ∀X ∈ Ωn,
INTERPOLATION IN THE SPECTRAL UNIT BALL 11
then H ∈ O(Ωn; Ωn) and, by construction, H(0) = 0. By the Ransford-White result,
r(H(X)) ≤ r(X), or, more precisely
µ∈σ(G(X))
µ− λj
1− λjµ
 ≤ r(X) ∀X ∈ Ωn.
In particular:
µ∈σ(G(X))
distM(µ;σ(G(0)))
≤ r(X) ∀X ∈ Ωn,
where, for any compactK  D and µ ∈ D, we define distM(µ;K) := minζ∈K
∣∣(µ− ζ)(1− ζµ)−1
For the moment, let us fixX ∈ Ωn. For each µ ∈ σ(G(X)), let λ
(µ) be an eigenvalue of
G(0) such that
∣∣∣(µ− λ(µ))(1 − λ(µ)µ)−1
∣∣∣ = distM(µ;σ(G(0))). Now fix µ ∈ σ(G(X)).
The above inequality leads to
(4.1)
∣∣∣∣∣
µ− λ(µ)
1− λ(µ)µ
∣∣∣∣∣ ≤ r(X)
1/dG .
Applying Lemma 4.1 to the Möbius transformation
T (z) =
|µ|z − λ(µ)
1− λ(µ)|µ|z
we deduce that ∣∣∣∣∣
ζ − λ(µ)
1− λ(µ)ζ
∣∣∣∣∣ ≥
||µ| − |λ(µ)||
1− |µ||λ(µ)|
∀ζ : |ζ| = |µ|.
Applying the above fact to (4.1), we get
|µ| − |λ(µ)|
1− |µ||λ(µ)|
≤ r(X)1/dG
⇒ |µ| ≤
r(X)1/dG + |λ(µ)|
1 + |λ(µ)|r(X)1/dG
, µ ∈ σ(G(X)).(4.2)
Note that the function
t 7−→
r(X)1/dG + t
1 + r(X)1/dGt
, t ≥ 0,
is an increasing function on [0,∞). Combining this fact with (4.2), we get
|µ| ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
which holds ∀µ ∈ σ(G(X)), while the right-hand side is independent of µ. Since this
is true for any arbitrary X ∈ Ωn, we conclude that
r(G(X)) ≤
r(X)1/dG + r(G(0))
1 + r(G(0))r(X)1/dG
∀X ∈ Ωn.
In order to prove the sharpness of (1.2), let us fix an n ≥ 2, and define
Sn := {A ∈ Ωn : A has a single eigenvalue of multiplicity n}.
12 GAUTAM BHARALI
Pick any d = 1, . . . , n, and define
Md(X) :=


[tr(X)/n], if d = 1,

0 tr(X)/n
1 0 0
. . .
. . .

, if d ≥ 2,
and, for the chosen d, define G(d) by the following block-diagonal matrix
(d)(Y ) :=
Md(X)
tr(X)
∀X ∈ Ωn.
For our purposes GA,d = G(d) for each A ∈ Sn; i.e., the equality (1.5) will will hold
with the same function for each A ∈ Sn. To see this, note that
• r(G(d)(X)) = |tr(X)/n|1/d; and
• G(d)(0) is nilpotent of degree d, whence dG(d) = d.
Therefore,
r(A)1/d + r(G(d)(0))
1 + r(G(d)(0))r(A)1/d
= r(A)1/d = r(G(d)(A)) ∀A ∈ Sn,
which establishes (1.5) �
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INTERPOLATION IN THE SPECTRAL UNIT BALL 13
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Department of Mathematics, Indian Institute of Science, Bangalore – 560 012
E-mail address: bharali@math.iisc.ernet.in
	1. Introduction and Statement of Results
	2. A Discussion of Observation ??
	3. The Proof of Theorem ??
	4. The Proof of Theorem ??
	References
