Comparing Two Generalized Nevanlinna-Pick Theorems

Transcription

1 Comparing Two Generalized Nevanlinna-Pick Theorems University of Iowa INFAS April 30, 2016 Comparing Two s

2 Outline Motivation 1 Motivation Comparing Two s

3 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 z 1,..., z N D and N points λ 1,..., λ N C, there exists f H (D) such that f 1 and if and only if the Pick matrix f (z i ) = λ i, i = 1,..., N, is positive semidefinite. [ 1 λi λ j 1 z i z j ] N i,j=1 Comparing Two s

4 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 z 1,..., z N D and N points λ 1,..., λ N C, there exists f H (D) such that f 1 and if and only if the Pick matrix f (z i ) = λ i, i = 1,..., N, is positive semidefinite. [ 1 λi λ j 1 z i z j ] N i,j=1 Comparing Two s

5 Definitions Motivation 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. Comparing Two s

6 Examples of W -correspondences M = E = C a c b = acb c, d = cd M = C, E = C N c 1 a. c N c 1.. b =, d 1.. ac 1 b. ac N b = c i d i c N d N Comparing Two s

7 Examples Cont. Motivation G = (G 0, G 1, r, s), M = C(G 0 ), E = C(G 1 ) (a ξ b)(e) = a(r(e))ξ(e)b(s(e)) ξ, η (v) = ξ(e)η(e) s(e)=v Comparing Two s

8 W -Correspondence Setting Given define M, a W -algebra E, a W -correspondence over M the Fock space F (E) to be the direct sum k=0 E k, where E 0 = M, viewed as a bimodule over itself the von Neumann algebra of bounded operators L (F (E)) on the Fock space of E Comparing Two s

9 Operators on the Fock Space F (E) Define the left action operator ϕ : M L (F (E)) by a ϕ(a) ϕ (a) = ϕ (2) (a)... where ϕ (k) (a) : E k E k is given by ϕ (k) (a)(ξ 1 ξ 2... ξ k ) = (ϕ(a)ξ 1 ) ξ 2... ξ k. Comparing Two s

10 Operators on the Fock Space F (E) Cont. For ξ E, define the left creation operator T ξ : F (E) F (E) by T ξ (η) = ξ η. As a matrix, T ξ = 0 ξ 0 T (2) ξ T (1) where T (k) ξ : E k 1 E k is given by T (k) ξ (η 1... η k 1 ) = ξ η 1... η k 1. Comparing Two s

11 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 {T ξ 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)). Comparing Two s

12 The σ-dual E σ Motivation Given define (M, E) σ : M B(H), a faithful, normal representation of M on a Hilbert space H, the induced representation σ E : L (E) B(E σ H) by σ E (T ) = T I H σ-dual E σ := {η B(H, E σ H) ησ(a) = σ E ϕ(a)η for all a M} Comparing Two s

13 The σ-dual E σ Motivation Given define (M, E) σ : M B(H), a faithful, normal representation of M on a Hilbert space H, the induced representation σ E : L (E) B(E σ H) by σ E (T ) = T I H σ-dual E σ := {η B(H, E σ H) ησ(a) = σ E ϕ(a)η for all a M} Comparing Two s

14 E σ Cont. Motivation 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). For a, b σ(m) and η, ξ E σ, define a η b := (I E a)ηb η, ξ := η ξ We define H (E σ ) analogously. Comparing Two s

15 E σ Cont. Motivation 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). For a, b σ(m) and η, ξ E σ, define a η b := (I E a)ηb η, ξ := η ξ We define H (E σ ) analogously. Comparing Two s

16 Algebra of Upper Triangular Toeplitz Operators U T (E, H, σ) Define U T (E, H, σ) to be the algebra of upper triangular Toeplitz operators T L (F (E) σ H) such that T 00 T 01 T 02 T 03 0 I E T 00 I E T 01 I E T 02 T = 0 0 I E 2 T 00 I E 2 T T 0j (ϕ (j) (a) I H ) = σ(a)t 0j for all a M. Comparing Two s

17 U T (E, H, σ) = ρ(h (E σ )) Define U : F (E σ ) ι H F (E) σ H by U(η 1 η k h) = (I E k 1 η 1 ) (I E η k 1 )η k h. Define ρ : H (E σ ) B(F (E) σ H) by ρ(x ) = U(X I H )U. Then U T (E, H, σ) = ρ(h (E σ )). Comparing Two s

18 U T (E, H, σ) = ρ(h (E σ )) Define U : F (E σ ) ι H F (E) σ H by U(η 1 η k h) = (I E k 1 η 1 ) (I E η k 1 )η k h. Define ρ : H (E σ ) B(F (E) σ H) by Then ρ(x ) = U(X I H )U. U T (E, H, σ) = {X I H X H (E)} = ρ(h (E σ )), where the second equality is due to Muhly and Solel [4, Theorem 3.9]. Comparing Two s

19 Cauchy Kernel Motivation 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) = (I E k 1 η)(i E k 2 η) (I E η)η the Cauchy Kernel C(η) B(H, F (E) σ H) by C(η) = [ I H η η (2) η (3) ]T Comparing Two s

20 Point Evaluation Motivation Point Evaluation For X H (E σ ) and η E σ with η < 1, define the point evaluation ˆX (η) σ(m) by ˆX (η) = ρ(x )C(0), C(η) = C(0) ρ(x ) C(η), where C(0) = [ I H 0 0 ]T is the Cauchy kernel at 0. Comparing Two s

21 Observations about Point Evaluation ˆX (η) = ρ(x )C(0), C(η), Not multiplicative, ie if X, Y H (E σ ) and η E σ with η < 1, then XY (η) ˆX (η)ŷ (η) Induces an algebra antihomomorphism from H (E σ ) into the completely bounded maps on σ(m) Comparing Two s

22 Observations Cont. Definition For X H (E σ ) and η E σ with η < 1, define Φ η X on σ(m) by Φ η X (a) := C(η), ρ(ϕσ (a))ρ(x )C(0), a σ(m). Φ η X is a completely bounded map on σ(m). The map X Φ η X is an algebra antihomomorphism on H (E σ ), ie Φ η XY = Φη Y Φη X. Comparing Two s

23 Theorem (N. 2016) For i = 1,..., N, let z i E σ with z i < 1 and Λ i σ(m). There exists X H (E σ ) with X 1 such that if and only if the operator matrix is positive semidefinite. ˆX (z i ) = Λ i, i = 1,..., N, [ C(zi ) (I F (E) (I H Λ i Λ j))c(z j ) ] N i,j=1 Comparing Two s

24 Corollary I Motivation Let M = C, E = C d, and σ : M B(H) be given by σ(a) = ai H. Then E σ = R d (B(H)) and σ(m) = B(H). Theorem (Constantinescu and Johnson 2003) For i = 1,..., N, let z i R d (B(H)) with z i < 1 and Λ i B(H). There exists X H (R d (B(H))) with X 1 such that if and only if the operator matrix is positive semidefinite. ˆX (z i ) = Λ i, i = 1,..., N, [ C(zi ) (I F (E) (I H Λ i Λ j))c(z j ) ] N i,j=1 Comparing Two s

25 Corollary II Motivation Let M = E = C and σ : M B(C) be given by σ(a) = a. Then E σ = C and σ(m) = C. Theorem (Pick 1915) For i = 1,..., N, let z i D and Λ i C. There exists X H (C) with X 1 such that if and only if the matrix ˆX (z i ) = Λ i, i = 1,..., N, is positive semidefinite. [ 1 Λi Λ j 1 z i z j ] N i,j=1 Comparing Two s

26 Displacement Equation Given Φ : B(H) B(H) with Φ < 1 and B B(H), define the displacement equation (I B(H) Φ)(A) = B. Since Φ < 1, solve for A by computing the Neumann series A = (I B(H) Φ) 1 (B) = Φ k (B). k=0 We are interested in the case when Φ is completely positive. In this case, (I B(H) Φ) 1 is completely positive as well. Comparing Two s

27 Proof of Generalized NP Theorem (N. 2015) Step 1 Let z = z 1... z N displacement equation, U = I Ḥ. I H, and V = Λ 1. (I B(H) Φ z )(A) = UU VV, Λ N, and form the where Φ z (A) = z (I E A)z, and Φ z < 1 because z < 1. Comparing Two s

28 Proof Cont. Motivation Step 2 Observe The Pick matrix A = [ C(z i ) (I F (E) (I H Λ i Λ j))c(z j ) ] N 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(z 1 ) C(z N ) ] and V = [ (I F (E) Λ 1 )C(z 1 ) (I F (E) Λ N )C(z N ) ]. Comparing Two s

29 Proof Cont. Motivation Lemma (Step 3) A = U U V V is positive if and only if there exists X H (E σ ) with X 1 such that ρ(x ) U = V. Step 4 ρ(x ) U = V if and only if ˆX (z i ) = Λ i for all i. Comparing Two s

30 Proof of Lemma Motivation Lemma A = U U V V is positive if and only if there exists X H (E σ ) with X 1 such that ρ(x ) U = V. Proof: ( ) A 0 L σ (N) (M) such that A = LL Displacement equation becomes [ ] [ L ] L V = [ z (I E L) U ] [ (I E L ] )z U V Â Â = ˆB ˆB Comparing Two s

31 Proof of Lemma Cont. [ ] X Z Douglas s Lemma! partial isometry θ = such Y W that  = θ ˆB Inn(θ) Range( ˆB) θ = 1 Define W Y (I E Z) Y (I E X )(I E 2 Z) 0 I E W I E Y (I E Z) T = 0 0 I E 2 W Comparing Two s

32 Proof of Lemma Cont. Then T U T (E, H, σ) T 1 because T is the transfer map of the contractive time-varying system [ ] [ ] x(t) x(t + 1) = (I y(t) E t θ) u(t) TU = V Thus there exists X H (E σ ) with X 1 such that T = ρ(x ), and ρ(x ) U = V. Comparing Two s

33 Proof of Lemma Cont. ( ) If there exists X H (E σ ) such that X 1 and ρ(x ) U = V, then A = U U V V = U U U ρ(x )ρ(x ) U = U (I ρ(x )ρ(x ) )U 0 since X 1 and ρ is an isometry. Comparing Two s

34 M-S Theorem (Muhly and Solel 2004) For i = 1,..., N, let z i E σ with z i < 1 and Λ i B(H). There exists Y H (E) with Y 1 such that Ŷ (z i ) = Λ i, i = 1,..., N, if and only if the map from M N (σ(m) ) to M N (B(H)) defined by Φ = [ (I B(H) Ad(Λ i, Λ j )) (I B(H) Φ zi,z j ) 1] N i,j=1 is completely positive. Comparing Two s

35 M-S Theorem (Muhly and Solel 2004) For i = 1,..., N, let z i E σ with z i < 1 and Λ i B(H). There exists Y H (E) with Y 1 such that Ŷ (z i ) = Λ i, i = 1,..., N, if and only if the map from M N (σ(m) ) to M N (B(H)) defined by Φ = [ (I B(H) Ad(Λ i, Λ j )) (I B(H) Φ zi,z j ) 1] N i,j=1 is completely positive. Comparing Two s

36 Comparing the s Theorem (N. 2016) For i = 1,..., N, let z i Z(E σ ) with z i < 1 and Λ i Z(σ(M) ). The following are equivalent: 1 There exists Y H (Z(E)) with Y 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 X 1 such that ˆX (z i ) = Λ i, i = 1,..., N in the sense of Norton s theorem. Comparing Two s

37 Definition Motivation 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). Comparing Two s

38 What s next? Motivation Compare Popescu s to Constantinescu and Johnson s Study Popescu s proof Use the displacement theorem to prove a commutant lifting theorem? Comparing Two s

39 Popescu s setting Motivation Let S 1,..., S n be the left creation operators on the Fock Space of C n. For Φ H (C n ) B(H), we can write Φ = α F + n S α A α, A α B(H). Popescu s Point Evaluation Let Z = (Z 1,..., Z n ), where Z i B(H) and Z i Zi < ri H, 0 < r < 1. Define the point evaluation of Φ at Z by Φ(Z) = Z α A α. α F + n Comparing Two s

40 Popescu s Theorem (Popescu 2003) For j = 1,..., m, let B j, C j B(H) and Z j = [Z j1,..., Z jn ] : H n H, with r(z) < 1. There exists Φ H (C n ) B(H) such that Φ 1 and [(I F (C n ) B j )Φ](Z j ) = C j, j = 1,..., m if and only if the operator matrix [ P P = k=0 α =k Z jα[b j Bk C jck ](Z kα) ] m is positive semidefinite. j,k=1 Comparing Two s

41 Proof via Constantinescu and Johnson s theorem Step 1 Define Z j = [Zj1,..., Z jn ]. By Constantinescu and Johnson s theorem, there exists T U T (C n, H, σ) such that B j T ( Z j ) = Cj if and only if [ ] m P CJ = C( Z j ) (I F (E) (Bj B k Cj C k))c( Z k ) j,k=1 is positive semidefinite. Step 2 The Pick matrices are equal. Comparing Two s

42 Proof Cont. Motivation Step 3 There exists T U T (C n, H, σ) with T 1 and such that B j T ( Z j ) = Cj there exists Φ H (C n ) B(H) has Φ 1 and satisfies [(I F (C n ) B j )Φ](Z j ) = C j. Hint: Φ := (J I H )T (J I H ). Flipping Isomorphism Define the flipping isomorphism J : F (C n ) F (C n ) by J(e i1 e ik ) = e ik e i1. Comparing Two s

43 References I Motivation 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), R. Douglas, On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space, American Mathematical Society 17 (1966), P. Muhly and B. Solel, Hardy Algebras, W -correspondences and interpolation theory, Math. Ann. 330 (2004), Comparing Two s

44 References II Motivation P. Muhly and B. Solel, Schur Class Operator Functions and Automorphisms of Hardy Algebras, Documenta Math. 13 (2008), R. Norton, Comparing Two Generalized Noncommutative Nevanlinna-Pick Theorems, Complex Analysis and Operator Theory (2016) /s G. Popescu, Multivariable Nehari problem and interpolation, Journal of Functional Analysis 200 (2003), Comparing Two s