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Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Comparing Two Generalized Nevanlinna-Pick
Theorems
Rachael M. Norton
University of Iowa
INFAS
April 30, 2016
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Outline
1
Motivation
2
Background
3
Generalized Nevanlinna-Pick Theorem
4
Comparison
5
Future Work
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Classical Nevanlinna-Pick Theorem
Let D = {z ∈ C | |z| < 1}.
Let H ∞ (D) = {f : D → C | f is bounded and analytic}.
Theorem (Pick 1915)
Given N distinct points z1 , . . . , zN ∈ D and N points
λ1 , . . . , λN ∈ C, there exists f ∈ H ∞ (D) such that ||f || ≤ 1 and
f (zi ) = λi , i = 1, . . . , N,
if and only if the Pick matrix
h
iN
1−λi λj
1−zi zj i,j=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Classical Nevanlinna-Pick Theorem
Let D = {z ∈ C | |z| < 1}.
Let H ∞ (D) = {f : D → C | f is bounded and analytic}.
Theorem (Pick 1915)
Given N distinct points z1 , . . . , zN ∈ D and N points
λ1 , . . . , λN ∈ C, there exists f ∈ H ∞ (D) such that ||f || ≤ 1 and
f (zi ) = λi , i = 1, . . . , N,
if and only if the Pick matrix
h
iN
1−λi λj
1−zi zj i,j=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Definitions
W ∗ -algebra
A W ∗ -algebra M is a C ∗ -algebra that is a dual space.
W ∗ -correspondence
A W ∗ -correspondence E over a W ∗ -algebra M is a self-dual right
Hilbert C ∗ -module over M equipped with a normal
∗-homomorphism ϕ : M → L (E ) that gives the left action of M
on E .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Examples of W ∗ -correspondences
M=E =C
a · c · b = acb
hc, di = cd
M = C, E = CN




c1
ac1 b
 


a ·  ...  · b =  ... 
c
acN b
 N  
* c1
d1 +
P
 ..   .. 
ci di
 . , .  =
cN
dN
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Examples Cont.
G = (G 0 , G 1 , r , s), M = C (G 0 ), E = C (G 1 )
(a · ξ · b)(e) =
Xa(r (e))ξ(e)b(s(e))
hξ, ηi(v ) =
ξ(e)η(e)
s(e)=v
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
W ∗ -Correspondence Setting
Given
M, a W∗ -algebra
E , a W∗ -correspondence over M
define
L
⊗k ,
the Fock space F (E ) to be the direct sum ∞
k=0 E
⊗0
where E = M, viewed as a bimodule over itself
the von Neumann algebra of bounded operators L (F (E ))
on the Fock space of E
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Operators on the Fock Space F (E )
Define the left action operator ϕ∞ : M → L (F (E )) by


a

 ϕ(a)


ϕ∞ (a) = 
(2)

ϕ (a)


..
.
where ϕ(k) (a) : E ⊗k → E ⊗k is given by
ϕ(k) (a)(ξ1 ⊗ ξ2 ⊗ . . . ξk ) = (ϕ(a)ξ1 ) ⊗ ξ2 ⊗ . . . ξk .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Operators on the Fock Space F (E ) Cont.
For ξ ∈ E , define the left creation operator Tξ : F (E ) → F (E )
by Tξ (η) = ξ ⊗ η. As a matrix,


0
T (1)

0
 ξ



Tξ = 
(2)

Tξ
0


.. ..
.
.
(k)
where Tξ
: E ⊗k−1 → E ⊗k is given by
(k)
Tξ (η1 ⊗ . . . ⊗ ηk−1 ) = ξ ⊗ η1 ⊗ . . . ⊗ ηk−1 .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Subalgebras of L (F (E ))
Tensor Algebra of E
The tensor algebra of E , denoted T+ (E ), is defined to be the
norm-closed subalgebra of L (F (E )) generated by
{ϕ∞ (a) | a ∈ M} and {Tξ | ξ ∈ E }.
Hardy Algebra of E
The Hardy algebra of E , denoted H ∞ (E ), is defined to be the
ultraweak closure of T+ (E ) in L (F (E )).
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
The σ-dual E σ
Given
(M, E )
σ : M → B(H), a faithful, normal representation of M on a
Hilbert space H,
define
the induced representation σ E : L (E ) → B(E ⊗σ H) by
σ E (T ) = T ⊗ IH
σ-dual
E σ := {η ∈ B(H, E ⊗σ H) | ησ(a) = σ E ◦ ϕ(a)η for all a ∈ M}
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
The σ-dual E σ
Given
(M, E )
σ : M → B(H), a faithful, normal representation of M on a
Hilbert space H,
define
the induced representation σ E : L (E ) → B(E ⊗σ H) by
σ E (T ) = T ⊗ IH
σ-dual
E σ := {η ∈ B(H, E ⊗σ H) | ησ(a) = σ E ◦ ϕ(a)η for all a ∈ M}
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
E σ Cont.
E σ := {η ∈ B(H, E ⊗σ H) | ησ(a) = σ E ◦ ϕ(a)η for all a ∈ M}
E σ is a W ∗ -correspondence over σ(M)0 . For a, b ∈ σ(M)0 and
η, ξ ∈ E σ , define
a · η · b := (IE ⊗ a)ηb
hη, ξi := η ∗ ξ
We define H ∞ (E σ ) analogously.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
E σ Cont.
E σ := {η ∈ B(H, E ⊗σ H) | ησ(a) = σ E ◦ ϕ(a)η for all a ∈ M}
E σ is a W ∗ -correspondence over σ(M)0 . For a, b ∈ σ(M)0 and
η, ξ ∈ E σ , define
a · η · b := (IE ⊗ a)ηb
hη, ξi := η ∗ ξ
We define H ∞ (E σ ) analogously.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Algebra of Upper Triangular Toeplitz Operators
UT (E , H, σ)
Define UT (E , H, σ) to be the algebra of upper triangular
“Toeplitz” operators T ∈ L (F (E ) ⊗σ H) such that


T00
T01
T02
T03
···
 0 IE ⊗ T00 IE ⊗ T01
IE ⊗ T02 · · ·


T = 0

0
I
⊗
T
I
⊗
T
·
·
·
⊗2
⊗2
01
00
E
E


..
..
..
..
..
.
.
.
.
.
T0j (ϕ(j) (a) ⊗ IH ) = σ(a)T0j for all a ∈ M.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
UT (E , H, σ)∗ = ρ(H ∞ (E σ ))
Define U : F (E σ ) ⊗ι H → F (E ) ⊗σ H by
U(η1 ⊗ · · · ⊗ ηk ⊗ h) = (IE ⊗k−1 ⊗ η1 ) · · · (IE ⊗ ηk−1 )ηk h.
Define ρ : H ∞ (E σ ) → B(F (E ) ⊗σ H) by
ρ(X ) = U(X ⊗ IH )U ∗ .
Then
UT (E , H, σ)∗ = ρ(H ∞ (E σ )).
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
UT (E , H, σ)∗ = ρ(H ∞ (E σ ))
Define U : F (E σ ) ⊗ι H → F (E ) ⊗σ H by
U(η1 ⊗ · · · ⊗ ηk ⊗ h) = (IE ⊗k−1 ⊗ η1 ) · · · (IE ⊗ ηk−1 )ηk h.
Define ρ : H ∞ (E σ ) → B(F (E ) ⊗σ H) by
ρ(X ) = U(X ⊗ IH )U ∗ .
Then
UT (E , H, σ)∗ = {X ⊗ IH | X ∈ H ∞ (E )}0 = ρ(H ∞ (E σ )),
where the second equality is due to Muhly and Solel [4, Theorem
3.9].
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Cauchy Kernel
E σ := {η ∈ B(H, E ⊗σ H) | ησ(a) = σ E ◦ ϕ(a)η for all a ∈ M}
For η ∈ E σ and k ∈ N, define
η (k) ∈ B(H, E ⊗k ⊗σ H) by
η (k) = (IE ⊗k−1 ⊗ η)(IE ⊗k−2 ⊗ η) · · · (IE ⊗ η)η
the Cauchy Kernel C (η) ∈ B(H, F (E ) ⊗σ H) by
C (η) = IH
Rachael M. Norton
η η (2) η (3) · · ·
T
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Point Evaluation
Point Evaluation
For X ∈ H ∞ (E σ ) and η ∈ E σ with kηk < 1, define the point
evaluation X̂ (η) ∈ σ(M)0 by
X̂ (η) = hρ(X )C (0), C (η)i
= C (0)∗ ρ(X )∗ C (η),
where C (0) = IH
0 0 ···
T
Rachael M. Norton
is the Cauchy kernel at 0.
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Observations about Point Evaluation
X̂ (η) = hρ(X )C (0), C (η)i,
Not multiplicative, ie if X , Y ∈ H ∞ (E σ ) and η ∈ E σ with
kηk < 1, then
d (η) 6= X̂ (η)Ŷ (η)
XY
Induces an algebra antihomomorphism from H ∞ (E σ ) into the
completely bounded maps on σ(M)0
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Observations Cont.
Definition
For X ∈ H ∞ (E σ ) and η ∈ E σ with kηk < 1, define ΦηX on σ(M)0
by
ΦηX (a) := hC (η), ρ(ϕσ∞ (a))ρ(X )C (0)i, a ∈ σ(M)0 .
ΦηX is a completely bounded map on σ(M)0 .
The map X 7→ ΦηX is an algebra antihomomorphism on
H ∞ (E σ ), ie ΦηXY = ΦηY ◦ ΦηX .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Generalized Nevanlinna-Pick Theorem
Theorem (N. 2016)
For i = 1, . . . , N, let zi ∈ E σ with kzi k < 1 and Λi ∈ σ(M)0 . There
exists X ∈ H ∞ (E σ ) with kX k ≤ 1 such that
X̂ (zi ) = Λi , i = 1, . . . , N,
if and only if the operator matrix
N
C (zi )∗ (IF (E ) ⊗ (IH − Λ∗i Λj ))C (zj ) i,j=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Corollary I
Let M = C, E = Cd , and σ : M → B(H) be given by σ(a) = aIH .
Then E σ = Rd (B(H)) and σ(M)0 = B(H).
Theorem (Constantinescu and Johnson 2003)
For i = 1, . . . , N, let zi ∈ Rd (B(H)) with kzi k < 1 and Λi ∈ B(H).
There exists X ∈ H ∞ (Rd (B(H))) with kX k ≤ 1 such that
X̂ (zi ) = Λi , i = 1, . . . , N,
if and only if the operator matrix
N
C (zi )∗ (IF (E ) ⊗ (IH − Λ∗i Λj ))C (zj ) i,j=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Corollary II
Let M = E = C and σ : M → B(C) be given by σ(a) = a.
Then E σ = C and σ(M)0 = C.
Theorem (Pick 1915)
For i = 1, . . . , N, let zi ∈ D and Λi ∈ C. There exists X ∈ H ∞ (C)
with kX k ≤ 1 such that
X̂ (zi ) = Λi , i = 1, . . . , N,
if and only if the matrix
h
iN
1−Λi Λj
1−zi zj i,j=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Displacement Equation
Given Φ : B(H) → B(H) with kΦk < 1 and B ∈ B(H), define the
displacement equation
(IB(H) − Φ)(A) = B.
Since kΦk < 1, solve for A by computing the Neumann series
A = (IB(H) − Φ)−1 (B)
∞
X
=
Φk (B).
k=0
We are interested in the case when Φ is completely positive. In
this case, (IB(H) − Φ)−1 is completely positive as well.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof of Generalized NP Theorem (N. 2015)
Step 1
 
 ∗
IH
Λ1

 .. 
 .. 
..
 , U =  .  , and V =  . , and form the
.
zN
IH
Λ∗N
displacement equation

z1

Let z = 

(IB(H) − Φz )(A) = UU ∗ − VV ∗ ,
where Φz (A) = z∗ (IE ⊗ A)z, and kΦz k < 1 because kzk < 1.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof Cont.
Step 2
Observe
The Pick matrix
N
A = C (zi )∗ (IF (E ) ⊗ (IH − Λ∗i Λj ))C (zj ) i,j=1
is the unique solution of the displacement equation.
∗
∗
We can rewrite
the Pick matrix as A = U∞ U∞ − V∞ V∞ ,
where U∞ = C (z1 ) · · · C (zN ) and
V∞ = (IF (E ) ⊗ Λ1 )C (z1 ) · · · (IF (E ) ⊗ ΛN )C (zN ) .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof Cont.
Lemma (Step 3)
∗ U − V∗ V
A = U∞
∞
∞ ∞ is positive if and only if there exists
X ∈ H ∞ (E σ ) with kX k ≤ 1 such that ρ(X )∗ U∞ = V∞ .
Step 4
ρ(X )∗ U∞ = V∞ if and only if X̂ (zi ) = Λi for all i.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof of Lemma
Lemma
∗ U − V∗ V
A = U∞
∞
∞ ∞ is positive if and only if there exists
X ∈ H ∞ (E σ ) with kX k ≤ 1 such that ρ(X )∗ U∞ = V∞ .
Proof: (⇒)
A ≥ 0 ⇒ ∃L ∈ σ (N) (M)0 such that A = LL∗
Displacement equation becomes
L∗
∗
(IE ⊗ L∗ )z
L V
= z (IE ⊗ L) U
V∗
U∗
Â∗ Â = B̂ ∗ B̂
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof of Lemma Cont.
X
Douglas’s Lemma ⇒ ∃! partial isometry θ =
Y
that
Z
W
such
 = θB̂
Inn(θ) ⊆ Range(B̂)
kθk = 1
Define

W
0

T =0

0

Y (IE ⊗ Z ) Y (IE ⊗ X )(IE ⊗2 ⊗ Z ) · · ·
IE ⊗ W
IE ⊗ Y (IE ⊗ Z )
· · ·

0
IE ⊗2 ⊗ W
· · ·

..
.
0
0
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof of Lemma Cont.
Then
T ∈ UT (E , H, σ)
kT k ≤ 1 because T is the transfer map of the contractive
time-varying system
x(t)
x(t + 1)
= (IE ⊗t ⊗ θ)
y (t)
u(t)
TU∞ = V∞
Thus there exists X ∈ H ∞ (E σ ) with kX k ≤ 1 such that
T = ρ(X )∗ , and ρ(X )∗ U∞ = V∞ .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof of Lemma Cont.
(⇐) If there exists X ∈ H ∞ (E σ ) such that kX k ≤ 1 and
ρ(X )∗ U∞ = V∞ , then
∗
∗
A = U∞
U∞ − V∞
V∞
∗
∗
U∞ − U∞
ρ(X )ρ(X )∗ U∞
= U∞
∗
= U∞
(I − ρ(X )ρ(X )∗ )U∞
≥ 0
since kX k ≤ 1 and ρ is an isometry.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
M-S Generalized Nevanlinna-Pick Theorem
Theorem (Muhly and Solel 2004)
For i = 1, . . . , N, let zi ∈ E σ with kzi k < 1 and Λi ∈ B(H). There
exists Y ∈ H ∞ (E ) with kY k ≤ 1 such that
Ŷ (z∗i ) = Λi ,
i = 1, . . . , N,
if and only if the map from MN (σ(M)0 ) to MN (B(H)) defined by
N
Φ = (IB(H) − Ad(Λi , Λj )) ◦ (IB(H) − Φzi ,zj )−1 i,j=1
is completely positive.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
M-S Generalized Nevanlinna-Pick Theorem
Theorem (Muhly and Solel 2004)
For i = 1, . . . , N, let zi ∈ E σ with kzi k < 1 and Λi ∈ B(H). There
exists Y ∈ H ∞ (E ) with kY k ≤ 1 such that
Ŷ (z∗i ) = Λi ,
i = 1, . . . , N,
if and only if the map from MN (σ(M)0 ) to MN (B(H)) defined by
N
Φ = (IB(H) − Ad(Λi , Λj )) ◦ (IB(H) − Φzi ,zj )−1 i,j=1
is completely positive.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Comparing the Generalized Nevanlinna-Pick Theorems
Theorem (N. 2016)
For i = 1, . . . , N, let zi ∈ Z(E σ ) with kzi k < 1 and Λi ∈ Z(σ(M)0 ).
The following are equivalent:
1
There exists Y ∈ H ∞ (Z(E )) with kY k ≤ 1 such that
Ŷ (z∗i ) = Λ∗i ,
i = 1, . . . , N
in the sense of Muhly and Solel’s theorem.
2
There exists X ∈ H ∞ (Z(E σ )) with kX k ≤ 1 such that
X̂ (zi ) = Λi ,
i = 1, . . . , N
in the sense of Norton’s theorem.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Definition
Center of a W ∗ -correspondence
If E is a W ∗ -correspondence over a W ∗ -algebra M, then the
center of E , denoted Z(E ), is the collection of points ξ ∈ E such
that a · ξ = ξ · a for all a ∈ M, and Z(E ) is a W ∗ -correspondence
over Z(M).
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
What’s next?
Compare Popescu’s Generalized Nevanlinna-Pick Theorem to
Constantinescu and Johnson’s
Study Popescu’s proof
Use the displacement theorem to prove a commutant lifting
theorem?
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Popescu’s setting
Let S1 , . . . , Sn be the left creation operators on the Fock Space of
Cn . For Φ ∈ H ∞ (Cn ) ⊗ B(H), we can write
X
Φ=
Sα ⊗ Aα , Aα ∈ B(H).
α∈Fn+
Popescu’s Point Evaluation
P
Let Z = (Z1 , . . . , Zn ), where Zi ∈ B(H) and
Zi Zi∗ < rIH ,
0 < r < 1. Define the point evaluation of Φ at Z by
X
Φ(Z) =
Zα Aα .
α∈Fn+
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Popescu’s Generalized Nevanlinna-Pick Theorem
Theorem (Popescu 2003)
For j = 1, . . . , m, let Bj , Cj ∈ B(H) and
Zj = [Zj1 , . . . , Zjn ] : H ⊕n → H, with r (Z) < 1.
There exists Φ ∈ H ∞ (Cn ) ⊗ B(H) such that kΦk ≤ 1 and
[(IF (Cn ) ⊗ Bj )Φ](Zj ) = Cj ,
j = 1, . . . , m
if and only if the operator matrix
hP
im
∞ P
∗ − C C ∗ ](Z )∗
Z
[B
B
PP =
jα
j
j
kα
|α|=k
k=0
k
k
j,k=1
is positive semidefinite.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof via Constantinescu and Johnson’s theorem
Step 1
∗ , . . . , Z ∗ ]. By Constantinescu and Johnson’s
Define Z̃j = [Zj1
jn
theorem, there exists T ∈ UT (Cn , H, σ) such that Bj T (Z̃j ) = Cj∗
if and only if
h
im
PCJ = C (Z̃j )∗ (IF (E ) ⊗ (Bj∗ Bk − Cj∗ Ck ))C (Z̃k )
j,k=1
is positive semidefinite.
Step 2
The Pick matrices are equal.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
Proof Cont.
Step 3
There exists T ∈ UT (Cn , H, σ) with kT k ≤ 1 and such that
Bj T (Z̃j ) = Cj∗ ⇐⇒ there exists Φ ∈ H ∞ (Cn ) ⊗ B(H) has
kΦk ≤ 1 and satisfies [(IF (Cn ) ⊗ Bj )Φ](Zj ) = Cj .
Hint: Φ := (J ⊗ IH )T ∗ (J ⊗ IH ).
Flipping Isomorphism
Define the “flipping” isomorphism J : F (Cn ) → F (Cn ) by
J(ei1 ⊗ · · · ⊗ eik ) = eik ⊗ · · · ⊗ ei1 .
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
References I
J. Agler and J. McCarthy, Pick Interpolation and Hilbert
Function Spaces, American Mathematical Society, Providence,
RI (2002).
T. Constantinescu and J. L. Johnson, A Note on
Noncommutative Interpolation, Canad. Math. Bull. 46 (1)
(2003), 59-70.
R. Douglas, On Majorization, Factorization, and Range
Inclusion of Operators on Hilbert Space, American
Mathematical Society 17 (1966), 413-415.
P. Muhly and B. Solel, Hardy Algebras, W∗ -correspondences
and interpolation theory, Math. Ann. 330 (2004), 353-415.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems
Motivation
Background
Generalized Nevanlinna-Pick Theorem
Comparison
Future Work
References II
P. Muhly and B. Solel, Schur Class Operator Functions and
Automorphisms of Hardy Algebras, Documenta Math. 13
(2008), 365-411.
R. Norton, Comparing Two Generalized Noncommutative
Nevanlinna-Pick Theorems, Complex Analysis and Operator
Theory (2016) 10.1007/s11785-016-0540-9.
G. Popescu, Multivariable Nehari problem and interpolation,
Journal of Functional Analysis 200 (2003), 536-581.
Rachael M. Norton
Comparing Two Generalized Nevanlinna-Pick Theorems