A noncommutative version of the Julia-Caratheodory Theorem Serban - PowerPoint PPT Presentation

A noncommutative version of the Julia-Caratheodory Theorem Serban T. Belinschi CNRS – Institut de Mathématiques de Toulouse Free Probability and the Large N Limit, V Berkeley, California 22–26 March 2016 Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 1 / 18

Contents The Julia-Carathéodory Theorem 1 Classical Noncommutative About the proof 2 A norm estimate on the derivative About the proof An example Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 2 / 18

Contents The Julia-Carathéodory Theorem 1 Classical Noncommutative About the proof 2 A norm estimate on the derivative About the proof An example Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 3 / 18

Self-maps of the upper half-plane We let C + = { z ∈ C : ℑ z > 0 } and f : C + → C + be analytic. Theorem (The Julia-Carathéodory Theorem) If α ∈ R is such that ℑ f ( z ) lim inf = c < ∞ , ℑ z z → α then lim f ( z ) = f ( α ) ∈ R , and z − → α ∢ f ( z ) − f ( α ) f ′ ( z ) = c . lim = lim z − α z − → α z − → α ∢ ∢ (Guarantees identification of a Fatou point - P . Mellon) Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 4 / 18

Contents The Julia-Carathéodory Theorem 1 Classical Noncommutative About the proof 2 A norm estimate on the derivative About the proof An example Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 5 / 18

Noncommutative (nc) functions Let M , N be operator spaces. An nc set is a family Ω = (Ω n ) n ∈ N such that Ω n ⊆ M n ( M ) and Ω m ⊕ Ω n ⊆ Ω m + n . Definition (J. L. Taylor - after Kaliuzhnyi-Verbovetskii & Vinnikov) An nc function defined on an nc set Ω is a family f = ( f n ) n ∈ N such that f n : Ω n → M n ( N ) and whenever m , n ∈ N , f m + n ( a ⊕ c ) = f m ( a ) ⊕ f n ( c ) for all a ∈ Ω m , c ∈ Ω n , and 1 Tf n ( c ) T − 1 = f n ( TcT − 1 ) for all c ∈ Ω n , T ∈ GL n ( C ) such that 2 TcT − 1 ∈ Ω n . We restrict ourselves to M = N = A - von Neumann algebra. We let Ω = H + ( A ) , Ω n = H + n ( A ) = { a ∈ M n ( A ): ℑ a := ( a − a ∗ ) / 2 i > 0 } . Fix f n : H + n ( A ) → H + f = ( f n ) n ∈ N , n ( A ) . Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 6 / 18

Derivatives For any a ∈ H + m ( A ) , c ∈ H + n ( A ) , there exists a linear operator ∆ f m , n ( a , c ): M m × n ( A ) → M m × n ( A ) such that �� a �� � f m ( a ) � b ∆ f m , n ( a , c )( b ) f m + n = , b ∈ M m × n ( A ) . 0 c 0 f n ( c ) If m = n , then ∆ f n , n ( a , a ) = f ′ n ( a ) , the Fréchet derivative of f n at a , and ∆ f n , n ( a , c )( a − c ) = f n ( a ) − f n ( c ) . With these notions, we can state: Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 7 / 18

The Julia-Carathéodory Theorem for nc functions Theorem (2015) Let f : H + ( A ) → H + ( A ) be an nc analytic function and let α = α ∗ ∈ A . Assume that for any v ∈ A , v > 0 and any state ϕ : A → C , we have ϕ ( ℑ f 1 ( α + zv )) lim inf < ∞ . ℑ z z → 0 , z ∈ C + Then (i) lim f n ( α ⊗ 1 n + zv ) = f 1 ( α ) ⊗ 1 n ∈ A exists in norm and is z − → 0 ∢ selfadjoint for any n ∈ N , v ∈ M n ( A ) , v > 0 , and ∆ f n , n ( α ⊗ 1 n + zv , α ⊗ 1 n + zv ′ )( b ) exists in the weak operator (ii) lim z − → 0 ∢ topology for any fixed v , v ′ > 0 , b ∈ M n ( A ) . Moreover, if v = v ′ = b > 0 , then the above limit equals the so- limit lim y → 0 ℑ f n ( α ⊗ 1 n + iyv ) / y . Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 8 / 18

The Julia-Carathéodory Theorem for nc functions Important: statement (ii) of the main theorem does NOT mean that f ′ ( α ) = lim y → 0 f ′ ( α + iyv ) exists, in the sense that the limit operator would not depend on v . (Counterexamples from Rudin, Abate, Agler - Tully-Doyle - Young.) However, IF the limit is independent of v , then it is completely positive. There are many results generalizing the Julia-Carathéodory Theorem for functions of several complex variables (Rudin, Abate, Agler - 1 Tully-Doyle - Young); functions on C + with values in spaces of operators (Ky Fan); 2 functions between domains in Banach spaces, operator spaces, 3 operator algebras (Jafari, Włodarczyk, Mackey - Mellon), etc. Beyond its noncommutative nature, the result above seems to be new in the sense that it guarantees the existence of the limits of operators evaluated in any direction b , and it requires, as hypothesis, only a very weak initial condition. Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 9 / 18

The Julia-Carathéodory Theorem for nc functions Important: statement (ii) of the main theorem does NOT mean that f ′ ( α ) = lim y → 0 f ′ ( α + iyv ) exists, in the sense that the limit operator would not depend on v . (Counterexamples from Rudin, Abate, Agler - Tully-Doyle - Young.) However, IF the limit is independent of v , then it is completely positive. There are many results generalizing the Julia-Carathéodory Theorem for functions of several complex variables (Rudin, Abate, Agler - 1 Tully-Doyle - Young); functions on C + with values in spaces of operators (Ky Fan); 2 functions between domains in Banach spaces, operator spaces, 3 operator algebras (Jafari, Włodarczyk, Mackey - Mellon), etc. Beyond its noncommutative nature, the result above seems to be new in the sense that it guarantees the existence of the limits of operators evaluated in any direction b , and it requires, as hypothesis, only a very weak initial condition. Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 9 / 18

Contents The Julia-Carathéodory Theorem 1 Classical Noncommutative About the proof 2 A norm estimate on the derivative About the proof An example Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 10 / 18

Using the definition of the domain Let a , c ∈ H + n ( A ) . Then � a � b ⇒ 4 ℑ a > b ( ℑ c ) − 1 b ∗ ⇐ ⇒ 4 ℑ c > b ∗ ( ℑ a ) − 1 b ℑ > 0 ⇐ 0 c � � ( ℑ a ) − 1 / 2 b ( ℑ c ) − 1 / 2 � ⇐ ⇒ � < 2 . � � � a � ǫ b 2 So given b ∈ M n ( A ) , ℑ > 0 for any 0 < ǫ < � ( ℑ a ) − 1 / 2 b ( ℑ c ) − 1 / 2 � . 0 c Since f maps H + ( A ) into itself and ∆ f ( a , c ) is linear, � < 2 for any such ǫ . Get � � ( ℑ f ( a )) − 1 / 2 ∆ f ( a , c )( b )( ℑ f ( c )) − 1 / 2 � ǫ � � ( ℑ f ( a )) − 1 / 2 ∆ f ( a , c )( b )( ℑ f ( c )) − 1 / 2 � � � ( ℑ a ) − 1 / 2 b ( ℑ c ) − 1 / 2 � � ≤ � , or � � � � 2 � � ∆ f ( a , c )( b )( ℑ f ( c )) − 1 ∆ f ( a , c )( b ) ∗ ≤ � ( ℑ a ) − 1 2 b ( ℑ c ) − 1 · ℑ f ( a ) . � 2 � � Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 11 / 18

Aside (not used in this proof) If A = C , a = c = z , get | f ′ ( z ) | ≤ ℑ f ( z ) / ℑ z , the Schwarz-Pick ineq. It is natural to define � � ( ℑ a ) − 1 / 2 ( a − c )( ℑ c ) − 1 / 2 � � � B + a ∈ H + n ( c , r ) = n ( A ): � ≤ r . � � B + n ( c , r ) is convex, norm-closed, noncommutative; If f ( c ) = c , then f n ( B + n ( c , r )) ⊆ B + n ( c , r ) ; If a ∈ B + n ( c , r ) , then √ √ �    r 2 + 2 + r r 2 + 4 r 2 + 2 + r r 2 + 4  , � a � ≤ �ℜ c � + �ℑ c � + r 2 2 1 ℑ a ≥ 2 + r 2 ℑ c . Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 12 / 18

Aside (not used in this proof) Note similarity with [Agler, Operator theory and the Carathéodory metric ] - description of pseudo-Carathéodory metric on U ⊂ C d as d ( z , w ) = inf sin θ M , θ M being the angle between the eigenvectors of a d -tuple M of commuting 2 × 2 matrices for which the joint spectrum is in U and U is a spectral domain for M . (Thanks to V. Paulsen) Pseudo-Carathéodory metric: if z , w ∈ U , then d ( z , w ) = sup {| f ( z ) − f ( w ) | / | 1 − f ( w ) f ( z ) | : f : U → D holo } . Spectral domain: set containing the joint spectrum of M s.t. Π: H ∞ ( U ) → B ( C 2 ) , Π( h ) = h ( M ) is a contraction. Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 13 / 18

Contents The Julia-Carathéodory Theorem 1 Classical Noncommutative About the proof 2 A norm estimate on the derivative About the proof An example Serban T. Belinschi An nc version of the Julia-Carathéodory Thm 23 - III - 2016 14 / 18