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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∗  = 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