SLIDE 1 Transfer function realization for inner functions
Joseph A. Ball
Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
HeltonFest UCSD October 3, 2010
SLIDE 2 The classical Schur class
Definition
S ∈ S(U, Y) if S : D →
holo L(U, Y) with S(z) ≤ 1 for z ∈ D.
Then:
◮ S ∈ S(U, Y) if and only if there exists unitary/contractive
U = A B
C D
- : X ⊕ U → X ⊕ Y for some auxiliary X so that
S(z) = D + zC(I − zA)−1B.
◮ Assume that U = Y: S is inner (boundary-value function
S(ζ) on T is unitary-valued) ⇐ ⇒ can take U unitary with A and A∗ stable (powers go to zero strongly).
◮ S is rational ⇐
⇒ also dim X < ∞.
History
Circuit theory: Youla, Newcombe, Belevitch, Wohlers Operator theory: Livˇ sic, Sz.-Nagy-Foias, de Branges-Rovnyak Scattering theory: Lax-Phillips Interconnections: Helton
SLIDE 3
d-variable generalizations
Given S : Dd →
holo L(U, Y): S = S(z) where z = (z1, . . . , zd) ∈ Dd
Distinguish:
◮ S ∈ Sd(U, Y): S(z) ≤ 1 for z ∈ Dd. ◮ S ∈ SAd(U, Y): S(T) ≤ 1 for T = (T1, . . . , Td)
commuting contractions on some H.
Then:
SAd(U, Y) ⊂ Sd(U, Y) Consequence of Ando dilation theorem: SA2(U, Y) = S2(U, Y) Varopoulas: von Neumann ≤ fails for d ≥ 3 Parrot: Sz.-Nagy/Ando Dilation Theorem fails for d > 2 = ⇒ SAd(U, Y) = Sd(U, Y) for d > 2.
SLIDE 4 d-variable Schur-Agler class
Agler(1990), B.-Trent (1998), Agler-McCarthy (1999): Then S ∈ SAd(U, Y) ⇐ ⇒
◮ S has an Agler decomposition: For some positive kernels
Kk(z, w) on Dd: I − S(z)S(w)∗ = d
k=1(1 − zkwk)Kk(z, w) ◮ There is a unitary U =
A B
C D
k=1Xk
U
k=1Xk
Y
that S(z) = D + C(I − Zd(z)A)−1Zd(z)B where Zd(z) = z1IX1 ...
zdIXd
Polydisk inner funtion:
S : Dd →
holo L(U, U), limr↑1 S(rζ) unitary for a.e. ζ ∈ Td.
SLIDE 5 Realization theory for rational inner Schur-Agler class
Then:
S rational inner Schur-Agler class ⇐ ⇒
◮ Finite-dim. Agler decomposition:
I − S(z)S(w)∗ = d
k=1(1 − zkwk)Kk(z, w) with Kk pos.
kernel with dim H(Kk) < ∞
◮ S(z) = D + C(I − Zd(z)A)−1Zd(z)B with
U = A B
C D
k=1Xk
Cm
k=1Xk
Cm
Ingredients:
◮ Cole-Wermer (1999):
p(z)p(w)∗ − p(z) p(w)∗ = d
k=1(1 − zkwk)K ′ k(z, w) with
p = polynomial, p = reverse polynomial = ⇒ K ′
k also a
polynomial matrix (finite sum of squares)
◮ B.-Bolotnikov (2009): Agler decomposition −
→ canonical functional model realization: Xk = H(Kk).
SLIDE 6 Bidisk rational inner functions
d = 2 = ⇒ SA2(Cm) = S2(Cm). Thus S rational inner ⇐ ⇒ S has a realization S(z) = D + [ C1 C2 ] In1−z1A11
−z1A12 −z2A21 In2−z2A22
−1
z1B1 z2B2
- where U is a unitary matrix of the form
U =
A21 A22 B2 C1 C2 D
X2 Cm
X2 Cm
- and dim Xk < ∞ for k = 1, 2.
Solution 1: Cole-Wermer/B.-Bolotnikov as above with extra ingredient: for d = 2 (scalar-valued case), one can obtain sum-of-squares/Agler decomposition p(z)p(w)∗ − p(z) p(w)∗ = 2
k=1(1 − zkwk)hk(z)hk(w)∗ directly
(without appeal to Ando-Agler theory): Knese (arXiv preprint) Solution 2 (B.-Sadosky-Vinnikov (2005)): Any S ∈ Sd(U, Y) can be realized as the scattering function of a Lax-Phillips d-evolution scattering system. When d = 2, one can identify unitary colligation matrix U = A B
C D
- leading to realization of S from
scattering geometry; when S is rational, in addition one can verify that dim X1 < ∞, dim X2 < ∞.
SLIDE 7 Engineering/Circuit Theory Solution:
Youla (1966) & Kummert (1989): Given a 2-var. DLBM (discrete lossless bounded matrix = bidisk rational inner function) S:
Step 1:
Write S(z) = S′
11(z1) + S′ 12(z1)(z−1 2 Imn2 − S′ 22(z1))−1S′ 21(z1) where
S′(z1) = S′
11 S′ 12
S′
21 S′ 22
- (z1) is obtained explicitly from S(z1, z2) as
follows:
◮ Write det S = σ g
g poly with no zeros in D2)
◮ Write S = P/
g, P(z) = P0(z1) + P1(z1)z2 + · · · + Pn2(z1)zn2
2 ,
- g(z) = a0(z1) + a1(z1)z2 + · · · + an2(z1)zn2
2 . ◮ Set ui = ai a0 , Ni = Pi a0 for 0 ≤ i ≤ n2, Ai = Ni − N0ui for
1 ≤ i ≤ n2.
◮ Then S′ 11 = N0, S′ 12 = [ 0 ··· 0 Im ],
S′
21 =
An2
. . .
A2 A1
, S′
22 =
0 0 ··· 0 −un2Im Im 0 ··· 0 −un2−1Im
. . . . . . . . . . . .
0 0 ··· Im −u1Im
does the job
SLIDE 8 Step 2 of Kummert:
Construct T(z1) so that S′′(z1) =
0 T(z1)
0 T(z1)
1-var DLBM as follows: T constructed as solution of spectral factorization/sum-of-squares problem K = T −1 Ir 0
0 0
T −1 where K =
u0Im u1Im ··· un2−1Im u0Im ··· un2−2Im
...
u0Im
. . . . . . ...
un2−2Im ··· u0Im
−
N0 N1 ··· Nn2−1 N0 ··· Nn2−2
... . . .
N0
. . . . . . ...
Nn2−2 ···
≥ 0.
SLIDE 9 Kummert solution continued
Write S′′(z1) = S′′
11 S′′ 12
S′′
21 S′′ 22
22(z1) has size r × r. Then
S′′ is DLBM and S(z1, z2) = S′′
11(z1) + S′′ 12(z1)(z−1 2 Ir − S′′ 22(z1))−1S′′ 21(z1)
S′′(z1) is 1-var. DLBR so, by 1-variable theory, has a unitary finite-dimensional realization S′′
11 S′′ 12
S′′
21 S′′ 22
D21 D22
C2
1 In1 − A)−1 [ B1 B2 ] .
with U :=
B1 B2 C1 D11 D12 C2 D21 D22
S(z1, z2) = D + C(I − Z2(z)A)−1Z2(z)B where A B C D
A B2 B1 C2 D22 D21 C1 D12 D11 is a unitary colligation matrix. Arveson-Stinespring theory = ⇒ new proof of Ando’s theorem.
SLIDE 10 Nevanlinna-Agler class of the right half plane C+
Nevanlinna class over the poly-right half plane (C+)d: H : (C+)d so that H(z) + H(z)∗ ≥ 0 for z ∈ (C+)d. Nevanlinna-Agler class over the right half plane C+: H : (C+)d so that H(T) + H(T)∗ ≥ 0 for T = (T1, . . . , Td) d-tuple of strictly accretive operators on some H Agler (1990) (after Cayley transform change of variable): H ∈ NA(U) ⇐ ⇒ H(z) + H(w)∗ = d
k=1(zk + wk)Kk(z, w) for
some positive kernels Kk. 1-variable realization theory: more complicated: unbounded
- perators required in general, singularities at infinity:
Staffans/Weiss (well-posed systems); Arlinski-Hassi-Tsekanovskii Circuit theory interest: Rational Cayley inner functions: H rational in N(Cm) and boundary-value function satisfies H(z) + H(z)∗ = 0 for z ∈ (iR)d. Koga (1968) asserted: H rational, d-variable Cayley inner, no poles at ∞ = ⇒ H(z) = D + B∗(Zd(z) − A)−1B with D = −D∗, A = −A∗ = ⇒ H ∈ NA(Cm) = ⇒ incorrect!
SLIDE 11
Koga’s gap
Koga’s Factorization Lemma (1968):
F = F(x, . . . , xm) is an n × n matrix polynomial of degree 2 in each xi with F(x1, . . . , xm) ≥ 0 (pos-semidef) for xi’s real ⇒ F can be factored as F(x) = M(x)M(x)∗ with M an n × q matrix polynomial in x1, . . . , xm with real coefficients. The proof uses:
Koga’s Sum-of-Squares Lemma
f = f (x1, . . . , xm) = polynomial over R, f quadratic in each variable, f ≥ 0 for real xi ⇒ f = n
i=1 h2 i with each hi linear in
each variable, n ≤ 2m.
Choi’s counterexample (1975) (publicized by N.K. Bose)
f (x1, x2, x3, y1, y2, y3) = x2
1y2 1 + x2 2y2 2 + x2 3y2 3
−2(x1x2y1y2 + x2x3y2y3 + x3x1y3y1) + 2(x2
1y2 2 + x2 2y2 3 + x2 3y2 1 ).
SLIDE 12
Connections with Hilbert problem
Hilbert’s problem
Given p = p(x1, . . . , xm) polynomial of even degree p with p(x1, . . . , xm) ≥ 0 for xi’s real, find real polynomials hi so that f =
i h2 i .
Hilbert’s result
Given the number of variables m and the even degree p, this is possible in general only for three cases:
◮ p = 2, m arbitrary: dehomogenize to reduce to the affine
1-variable case
◮ p arbitrary, m = 2: Sylvester inertia theorem ◮ p = 4, m = 3: Bernd Sturmfels talk
Connection of Koga sum-of-squares lemma
This does not prove Koga lemma wrong: the Koga quadratic-in-each-variable hypothesis does not include a whole Hilbert (m, p) class.
SLIDE 13 Why Choi was interested
Biquadratic forms
F(x; y) = αjkpqxjxkypyq, (j ≤ k, p ≤ q)
Connection with positive maps on matrices
Sn = symmetric n × n matrices
◮ Φ: Sm
→
pos lin Sn one-to-one correspondence with positive
semidefinite biquadratic form F(x, y) = ytΦ(x · xt)y
◮ Moreover, F(x, y) = f (x, y)2 (f = bilinear map in x, y) ⇔
Φ(A) = V tAV is a congruence map
◮ Sum-of-squares decomposition F(x, y) = k f 2 k ⇔ Φ = sum
- f congruence maps = completely positive map
◮ Choi counterexample also says: there exists Φ: S3 → S3
which is positive linear but not completely positive.
SLIDE 14
Bessmertny˘ ı–Kaliuzhnyi-Verbovetskyi classes BP(U) and B(U)
Bessmertny˘ ı positive functions BP(U)
H : ∪λ∈T (λC+)d →
holo L(U) such that
(BP1) H(z) + H(z)∗ ≥ 0 for s ∈ Cd
+,
(BP2) H(z) + H(z)∗ ≤ 0 for s ∈ (−C+)d, (BP3) H(z) + H(z)∗ ≥ 0 for s ∈ (iC+)d, (BP4) H(z) + H(z)∗ ≤ 0 for s ∈ (−iC+)d.
Bessmertny˘ ı class B(U)
H : ∪λ∈T (λC+)d →
holo L(U) such that there are positive kernels
K1, . . . , Kd on ∪λ∈T(λC+)d so that [H(z) + H(w)∗ = d
k=1(zk + wk)Kk(z, w) and
H(z) − H(w)∗ = d
k=1(zk − wk)Kk(z, w)] or, equivalently,
H(z) = d
k=1 zkKk(z, w).
SLIDE 15 Realization theory for B(U):
Bessmertny˘ ı (1982)–Kaliuzhnyi-Verbovetskyi (2004):
H ∈ B(U) ⇐ ⇒ H(z) = a(z) − b(z)d(z)−1c(z) where
c(z) d(z)
- = A1z1 + · · · + Adzd with Ak = A∗
k ≥ 0 for
k = 1, . . . , d.
d = 2:
Then B(U) = BP(U).
Equivalent characterization of BP(U):
(BP1) H(z) + H(z)∗ ≥ 0 for z ∈ Cd
=
(BP2’) H(λz1, . . . , λzd) = λH(z1, . . . , zd) for λ ∈ C \ {0} (BP3’) H(z) = H(z)∗ = H(z)t In particular, for d = 1 BP = {zA: A = A∗ ≥ 0}.
Conjecture:
H ∈ HAd(U) ⇐ ⇒ H(z) = a(z) − b(z)d(z)−1c(z) with
c(z) d(z)
- = A0 + z1A1 + · · · + zdAd with A0 = −A∗
0 and
Ak = A∗
k ≥ 0 for k = 1, . . . , d.
SLIDE 16 Special case of conjecture:
Koga’s realization:
H(z) = D + [ B∗
1 B∗ 2 ]
z1In1
z2In2
A21 A22
−1
B1 B2
D = −D∗, A = −A∗ can be written as H(z) = a(z) − b(z)d(z)−1c(z) where
c(z) d(z)
A0 = D −B∗
1
−B∗
2
B1 −A11 −A12 B2 −A22 −A22 , A1 = In1 , A2 = In2 satisfying A0 = −A∗
0, A1 = A∗ 1 ≥ 0, A2 = A∗ 2 ≥ 0.
True for general d for rational case subject to growth condition at infinity (work in progress)
SLIDE 17
Summary
Dichotomy between d ≤ 2 and d > 2 (d = number of variables):
◮ Dilation theory ◮ Realization of polydisk inner functions ◮ Sums-of-squares problems ◮ Positive linear vs. completely positive maps on symmetric
matrices
SLIDE 18 Selected references:
J.A. Ball, Multidimensional circuit synthesis and multivariable dilation theory, Multidimensional Systems and Signal Processing, to appear. B.J. Cole and J. Wermer, Ando’s theorem and sums of squares, Indiana University Mathematics Journal 48 (1999), No. 3, 767–791. D.S. Kalyuzhny˘ ı-Verbovetzki˘ ı, On the Bessmertny˘ ı class of homogeneous positive holomorphic functions of several variables, Current Trends in Operator Theory and its Applications, OT 149, Birkh¨ auser, Basel, 2004.
- T. Koga, Synthesis of finite passive n-ports with prescribed positive
real matrices of several variables, IEEE Transactions on Circuit Theory CT-15 (1968) NO. 1, 2–23.
- G. Knese, Polynomials with no zeros on the bidisk, arXiv.
- A. Kummert, Synthesis of two-dimensional lossless m-ports with
prescribed scattering matrix, Circuits Systems Signal Processing 8
