Transfer function realization for inner functions on the bidisk - PowerPoint PPT Presentation

Transfer function realization for inner functions on the bidisk Joseph A. Ball Department of Mathematics, Virginia Tech, Blacksburg, VA, USA HeltonFest UCSD October 3, 2010

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 � A B � U = : X ⊕ U → X ⊕ Y for some auxiliary X so that C D S ( z ) = D + zC ( I − zA ) − 1 B . ◮ 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

d -variable generalizations Given S : D d → holo L ( U , Y ): S = S ( z ) where z = ( z 1 , . . . , z d ) ∈ D d Distinguish: ◮ S ∈ S d ( U , Y ): � S ( z ) � ≤ 1 for z ∈ D d . ◮ S ∈ SA d ( U , Y ): � S ( T ) � ≤ 1 for T = ( T 1 , . . . , T d ) commuting contractions on some H . Then: SA d ( U , Y ) ⊂ S d ( U , Y ) Consequence of Ando dilation theorem: SA 2 ( U , Y ) = S 2 ( U , Y ) Varopoulas: von Neumann ≤ fails for d ≥ 3 ⇒ Parrot: Sz.-Nagy/Ando Dilation Theorem fails for d > 2 = SA d ( U , Y ) � = S d ( U , Y ) for d > 2.

d -variable Schur-Agler class Agler(1990), B.-Trent (1998), Agler-McCarthy (1999): Then S ∈ SA d ( U , Y ) ⇐ ⇒ ◮ S has an Agler decomposition : For some positive kernels K k ( z , w ) on D d : I − S ( z ) S ( w ) ∗ = � d k =1 (1 − z k w k ) K k ( z , w ) � � � � � A B � ⊕ d ⊕ d k =1 X k ◮ There is a unitary U = k =1 X k : → so C D Y U that S ( z ) = D + C ( I − Z d ( z ) A ) − 1 Z d ( z ) B where � z 1 I X 1 � ... Z d ( z ) = . z d I X d Polydisk inner funtion: S : D d → holo L ( U , U ), lim r ↑ 1 S ( r ζ ) unitary for a.e. ζ ∈ T d .

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 − z k w k ) K k ( z , w ) with K k pos. kernel with dim H ( K k ) < ∞ ◮ S ( z ) = D + C ( I − Z d ( z ) A ) − 1 Z d ( z ) B with � � � � � A B � ⊕ d ⊕ d k =1 X k k =1 X k U = : → unitary, dim X k < ∞ . C D C m C m Ingredients: ◮ Cole-Wermer (1999): p ( w ) ∗ = � d p ( z ) p ( w ) ∗ − � k =1 (1 − z k w k ) K ′ p ( z ) � k ( z , w ) with ⇒ K ′ p = polynomial, � p = reverse polynomial = k also a polynomial matrix (finite sum of squares) ◮ B.-Bolotnikov (2009): Agler decomposition − → canonical functional model realization: X k = H ( K k ).

Bidisk rational inner functions ⇒ SA 2 ( C m ) = S 2 ( C m ). Thus S rational inner ⇐ d = 2 = ⇒ S has �� I n 1 − z 1 A 11 �� − 1 � � − z 1 A 12 z 1 B 1 a realization S ( z ) = D + [ C 1 C 2 ] − z 2 A 21 I n 2 − z 2 A 22 z 2 B 2 where U is a unitary matrix of the form � � � � � � A 11 A 12 B 1 X 1 X 1 U = : → and dim X k < ∞ for k = 1 , 2. A 21 A 22 B 2 X 2 X 2 C m C m C 1 C 2 D 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 ( w ) ∗ = � 2 p ( z ) p ( w ) ∗ − � k =1 (1 − z k w k ) h k ( z ) h k ( w ) ∗ directly p ( z ) � (without appeal to Ando-Agler theory): Knese (arXiv preprint) Solution 2 (B.-Sadosky-Vinnikov (2005)): Any S ∈ S d ( U , Y ) can be realized as the scattering function of a Lax-Phillips d-evolution scattering system . When d = 2, one can identify unitary � A B � colligation matrix U = leading to realization of S from C D scattering geometry ; when S is rational, in addition one can verify that dim X 1 < ∞ , dim X 2 < ∞ .

Engineering/Circuit Theory Solution: Youla (1966) & Kummert (1989): Given a 2-var. DLBM ( discrete lossless bounded matrix = bidisk rational inner function ) S : Step 1: 12 ( z 1 )( z − 1 Write S ( z ) = S ′ 11 ( z 1 ) + S ′ 2 I mn 2 − S ′ 22 ( z 1 )) − 1 S ′ 21 ( z 1 ) where � S ′ � 11 S ′ S ′ ( z 1 ) = 12 ( z 1 ) is obtained explicitly from S ( z 1 , z 2 ) as S ′ 21 S ′ 22 follows: ◮ Write det S = σ g g poly with no zeros in D 2 ) g ( | σ | = 1, � � g , P ( z ) = P 0 ( z 1 ) + P 1 ( z 1 ) z 2 + · · · + P n 2 ( z 1 ) z n 2 ◮ Write S = P / � 2 , g ( z ) = a 0 ( z 1 ) + a 1 ( z 1 ) z 2 + · · · + a n 2 ( z 1 ) z n 2 � 2 . ◮ Set u i = a i a 0 , N i = P i a 0 for 0 ≤ i ≤ n 2 , A i = N i − N 0 u i for 1 ≤ i ≤ n 2 . ◮ Then S ′ 11 = N 0 , S ′ 12 = [ 0 ··· 0 I m ],     0 0 ··· 0 A n 2 − u n 2 I m . I m 0 ··· 0 − u n 2 − 1 I m .   , S ′   does the job S ′ 21 = 22 = . . . . . . . . . . . . . A 2 A 1 0 0 ··· I m − u 1 I m

Step 2 of Kummert: � � � � I m 0 I m 0 Construct T ( z 1 ) so that S ′′ ( z 1 ) = S ′ ( z 1 ) is 0 T ( z 1 ) 0 T ( z 1 ) 1-var DLBM as follows: T constructed as solution of spectral factorization/sum-of-squares problem K = T − 1 � I r 0 � � T − 1 where 0 0     u 0 I m u 1 I m ··· u n 2 − 1 I m � u 0 I m u 1 I m � u 0 I m � u 0 I m ··· u n 2 − 2 I m     K = . . ... ... . . . . � u n 2 − 1 I m � u n 2 − 2 I m ··· � u 0 I m u 0 I m     � N 0 N 1 ··· N n 2 − 1 N 0 � � N 0 ··· N n 2 − 2  N 1 N 0    −  ≥ 0 .  . . . ... ... . . . . . . N n 2 − 1 � � � N 0 N n 2 − 2 ··· N 0

Kummert solution continued � S ′′ � 11 S ′′ Write S ′′ ( z 1 ) = ( z 1 ) where S ′′ 22 ( z 1 ) has size r × r . Then 12 S ′′ 21 S ′′ 22 S ′′ is DLBM and 12 ( z 1 )( z − 1 22 ( z 1 )) − 1 S ′′ S ( z 1 , z 2 ) = S ′′ 11 ( z 1 ) + S ′′ 2 I r − S ′′ 21 ( z 1 ) S ′′ ( z 1 ) is 1-var. DLBR so, by 1-variable theory, has a unitary finite-dimensional realization � S ′′ � � � � � 11 S ′′ 1 I n 1 − A ) − 1 [ B 1 B 2 ] . D 11 D 12 C 1 ( z − 1 12 ( z 1 ) = + S ′′ 21 S ′′ D 21 D 22 C 2 22 � � A B 1 B 2 with U := unitary matrix. Then C 1 D 11 D 12 C 2 D 21 D 22 S ( z 1 , z 2 ) = D + C ( I − Z 2 ( z ) A ) − 1 Z 2 ( z ) B where   � A � A B 2 B 1 B   is a unitary colligation matrix. = C 2 D 22 D 21 C D C 1 D 12 D 11 ⇒ new proof of Ando’s theorem. Arveson-Stinespring theory =

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 . H : ( C + ) d so Nevanlinna-Agler class over the right half plane C + : that H ( T ) + H ( T ) ∗ ≥ 0 for T = ( T 1 , . . . , T d ) d -tuple of strictly accretive operators on some H Agler (1990) (after Cayley transform change of variable): ⇒ H ( z ) + H ( w ) ∗ = � d H ∈ NA ( U ) ⇐ k =1 ( z k + w k ) K k ( z , w ) for some positive kernels K k . 1-variable realization theory: more complicated: unbounded operators 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 ( C m ) and boundary-value function satisfies H ( z ) + H ( z ) ∗ = 0 for z ∈ ( i R ) d . Koga (1968) asserted: H rational, d-variable Cayley inner, no poles ⇒ H ( z ) = D + B ∗ ( Z d ( z ) − A ) − 1 B with D = − D ∗ , at ∞ = A = − A ∗ = ⇒ H ∈ NA ( C m ) = ⇒ incorrect!

Koga’s gap Koga’s Factorization Lemma (1968): F = F ( x , . . . , x m ) is an n × n matrix polynomial of degree 2 in each x i with F ( x 1 , . . . , x m ) ≥ 0 (pos-semidef) for x i ’s real ⇒ F can be factored as F ( x ) = M ( x ) M ( x ) ∗ with M an n × q matrix polynomial in x 1 , . . . , x m with real coefficients. The proof uses: Koga’s Sum-of-Squares Lemma f = f ( x 1 , . . . , x m ) = polynomial over R , f quadratic in each variable, f ≥ 0 for real x i ⇒ f = � n i =1 h 2 i with each h i linear in each variable, n ≤ 2 m . Choi’s counterexample (1975) (publicized by N.K. Bose) f ( x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 − 2( x 1 x 2 y 1 y 2 + x 2 x 3 y 2 y 3 + x 3 x 1 y 3 y 1 ) + 2( x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 ) .

Connections with Hilbert problem Hilbert’s problem Given p = p ( x 1 , . . . , x m ) polynomial of even degree p with p ( x 1 , . . . , x m ) ≥ 0 for x i ’s real, find real polynomials h i so that f = � i h 2 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.