Operator-monotone functions and L owner functions of several - PowerPoint PPT Presentation

Operator-monotone functions and L¨ owner functions of several variables Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler and John E. McCarthy Newcastle, March 2015

Abstract A famous theorem of Karl L¨ owner asserts that a real-valued function f on a real interval ( a, b ) acts monotonically on selfadjoint operators if and only if f extends to an analytic function on the upper halfplane Π that maps Π to itself. We prove two generalizations of L¨ owner’s result to several variables. We characterize all rational functions of two variables that are operator-monotone in a rectangle. We give a characterization of functions of d variables that are locally monotone on d -tuples of commuting selfadjoint operators.

Operator-monotone functions Let I be an open interval in R . A function f : I → R is operator-monotone if f ( A ) ≤ f ( B ) whenever A, B are selfadjoint operators such that A ≤ B and the spectra of A, B are contained in I . Examples: f ( x ) = − 1 /x is operator-monotone on (0 , ∞ ) and on ( −∞ , 0). f ( x ) = √ x is operator-monotone on (0 , ∞ ). f ( x ) = x 2 is not operator-monotone on (0 , ∞ ).

The Pick class Let Π = { z ∈ C : Im z > 0 } , the upper halfplane. The Pick class P is the set of holomorphic functions f on Π such that Im f ≥ 0 on Π. √ z, − 1 /z, log z, tan z . Some functions in P : For any open interval I ⊂ R , define the Pick class P ( I ) of I to be the set of restrictions to I of functions f ∈ P that are analytic on I . L¨ owner’s theorem (1934) Let I ⊂ R be an open interval. A real-valued function on I is operator-monotone if and only if f ∈ P ( I ).

The functional calculus Let A 1 , A 2 be commuting n × n Hermitian matrices. By the Spectral Theorem there exists a unitary matrix U and real numbers λ 1 , . . . , λ n , µ 1 , . . . , µ n such that A 1 = U ∗ diag( λ 1 , . . . , λ n ) U, A 2 = U ∗ ( µ 1 , . . . , µ n ) U. Then, for f : R 2 → R , we define a matrix f ( A 1 , A 2 ) by f ( A 1 , A 2 ) = U ∗ diag( f ( λ 1 , µ 1 ) , . . . , f ( λ n , µ n )) U. The n points ( λ 1 , µ 1 ) , . . . , ( λ n , µ n ) ∈ C 2 are called the joint eigenvalues of ( A 1 , A 2 ); the collection of them is called the joint spectrum of ( A 1 , A 2 ).

Local versus global Say that a real-valued C 1 function f on a real interval I is locally operator-monotone if, whenever S ( t ) , 0 ≤ t < 1 , is a C 1 curve of selfadjoint matrices with spectra contained in I , S ′ (0) ≥ 0 ( f ◦ S ) ′ (0) ≥ 0 . ⇒ Then f ∈ C 1 is operator-monotone on I if and only if f is locally operator-monotone on I . Sufficiency follows from � 1 d f ( B ) − f ( A ) = d tf ((1 − t ) A + tB ) d t. 0

Operator-monotonicity in 2 variables Let E be an open set in R 2 . Say that a real-valued func- tion f on E is operator-monotone if f ( A ) ≤ f ( B ) whenever A = ( A 1 , A 2 ) and B = ( B 1 , B 2 ) are commuting pairs of selfadjoint operators such that A 1 ≤ B 1 and A 2 ≤ B 2 and the joint spectra of A and B are contained in E . Say that f ∈ C 1 ( E ) is locally operator-monotone if, when- ever S ( t ) = ( S 1 ( t ) , S 2 ( t )) , 0 ≤ t < 1 , is a C 1 curve of com- muting pairs of selfadjoint matrices with joint spectra con- tained in E , S ′ (0) ≥ 0 ( f ◦ S ) ′ (0) exists and ≥ 0 . ⇒

Local versus global in 2 variables If f is operator-monotone on E then f is locally operator- monotone on E (easy). Does the converse hold? Example �� � � �� 0 0 1 0 A = , , 0 5 0 0 �� � � �� 4 2 2 2 B = , . 2 6 2 4 A and B are commuting pairs of selfadjoint matrices and A ≤ B . There is no commuting pair of selfadjoint matrices lying strictly between A and B . It is unclear whether locally operator-monotone functions are operator-monotone on a general convex open set.

The Pick class in d variables Define the d -variable Pick class P d to be the set of holo- morphic functions F on Π d such that Im F ≥ 0 on Π d . The Pick-Agler class PA d is the set of functions F ∈ P d such that Im F ( T ) ≥ 0 for every d -tuple T of commuting operators having strictly positive imaginary parts. For F ∈ PA d there exist positive analytic kernels A 1 , . . . , A d on Π d such that, for all z, w ∈ Π d , F ( z ) − F ( w ) = ( z 1 − ¯ w 1 ) A 1 ( z, w ) + · · · + ( z d − ¯ w d ) A d ( z, w ) , and conversely.

Cauchy transforms of positive measures Let I ⊂ R be an interval and let µ be a positive measure on R \ I . The Cauchy transform of µ is the function d µ ( s ) µ ( z ) def � ˆ = s − z , R defined for z / ∈ R \ I . ˆ µ is locally operator monotone on I .

Proof: Let S ( t ) = A + tM + o ( t ) for 0 ≤ t < 1 where M ≥ 0. 1 µ ( S (0)) = 1 � ( s − S ( t )) − 1 − ( s − S (0)) − 1 d µ ( s ) t (ˆ µ ( S ( t )) − ˆ t ( s − S ( t )) − 1 S ( t ) − S (0) � ( s − S (0)) − 1 d µ ( s ) = t � ( s − A − tM − o ( t )) − 1 M ( s − A ) − 1 d µ ( s ) = � ( s − A ) − 1 M ( s − A ) − 1 d µ ( s ) . → µ ◦ S ) ′ (0) ≥ 0. Thus (ˆ �

A class of operator-monotone functions Let E be an open rectangle in R d . Let M be a Hilbert space and let P = ( P 1 , . . . , P d ) be a tuple of orthogonal projections on M summing to 1 M . For z ∈ C d let z P denote z 1 P 1 + · · · + z d P d . Let X be a densely defined self-adjoint operator on M such that X − z P is invertible for z ∈ E and let v ∈ M . The function � ( X − z P ) − 1 v, v � F ( z ) = for z ∈ E M is operator-monotone on E . F is a d -variable analogue of the Cauchy transform of a measure with support off E .

Proof - the functional calculus Consider commuting pairs S, T of selfadjoint operators on a Hilbert space H such that S ≤ T and the spectra of S and T lie in E . Write S P = S 1 ⊗ P 1 + S 2 ⊗ P 2 , an operator on H ⊗ M . For the function � ( X − z P ) − 1 v, v � F ( z ) = M we have F ( S ) = R ∗ v ( 1 H ⊗ X − S P ) − 1 R v where R v : H → H ⊗ M is given by R v h = h ⊗ v .

Proof - a difference formula Let ∆ = T − S ≥ 0; then ∆ P ≥ 0. Let Y ( t ) = 1 H ⊗ X − ((1 − t ) S + tT ) P for 0 ≤ t ≤ 1. Then Y ( t ) − 1 exists and d d tY ( t ) − 1 = Y ( t ) − 1 ∆ P Y ( t ) − 1 ≥ 0 . Now v Y (1) − 1 R v − R ∗ v Y (0) − 1 R v F ( T ) − F ( S ) = R ∗ � 1 d = R ∗ d tY ( t ) − 1 d tR v v 0 ≥ 0 . Thus F is operator-monotone on E .

A 2 -variable L¨ owner theorem Let f be a real rational function of 2 variables and let E be an open rectangle in R 2 on which the denominator of f does not vanish. Then f is operator-monotone on E if and only if f ∈ P 2 . The proof consists in showing that f can be approximated by � � ( X − z P ) − 1 v, v functions of the form F ( z ) = M by means of a 2-variable Nevanlinna representation formula. Our proof does not extend to dimension d = 3.

A d -variable Nevanlinna representation Let z 0 ∈ Π d and let F ∈ PA d . For all but countably many automorphisms α of Π there exist a Hilbert space M , a partition P = ( P 1 , . . . , P d ) of M , a selfadjoint operator X on M , a vector v ∈ M and a real number c such that P ( X − z P ) − 1 ( z − z 0 ) P v, v � ( z − z 0 ) ∗ � α ◦ F ◦ α ( z ) = c + � z P v, v � + for all z ∈ Π d . If v is in the domain of X then there is a simpler represen- tation, of the form � ( X − z P ) − 1 v, v � α ◦ F ◦ α ( z ) = c + .

The L¨ owner class in d variables Let E be an open set in R d and let n ≥ 1. The L¨ owner class n ( E ) of E comprises all real-valued C 1 functions f on E L d such that, for every finite set { x 1 , . . . , x n } of distinct points in E , there exist positive n × n matrices A 1 , . . . , A d such that ii = ∂f � A r � for 1 ≤ i ≤ n and 1 ≤ r ≤ d, � ∂x r � x i and d ( x r j − x r i ) A r � f ( x j ) − f ( x i ) = for 1 ≤ i, j ≤ n. ij r =1 owner class L d ( E ) of E is defined to be the intersec- The L¨ tion of L d n ( E ) over all n ≥ 1.

Functions in L d ( E ) are locally operator-monotone Consider a commuting pair S = ( S 1 , S 2 ) of selfadjoint n × n matrices such that σ ( S ) ⊂ E and σ ( S ) consists of simple Let S ( t ) , 0 ≤ t < 1, be a C 1 joint eigenvalues x 1 , . . . , x n . curve of commuting pairs of selfadjoint matrices such that S (0) = S , σ ( S ( t )) ⊂ E for all t and ∆ def = S ′ (0) ≥ 0 . If f satisfies f ( x j ) − f ( x i ) = � 2 r =1 ( x r j − x r i ) A r ij for all i, j as in the definition of L 2 n ( E ), then a calculation shows that ( f ◦ S ) ′ (0) = � ∆ 1 ij A 1 ( i, j ) + ∆ 2 ij A 2 ( i, j ) � ≥ 0 . Hence f is locally operator-monotone.

Locally operator-monotone functions are in L d ( E ) Proof is by a separation argument. Let E be open in R 2 and let f ∈ C 1 ( E ) be locally operator- monotone on E . Fix n ≥ 1 and distinct points x 1 , . . . , x n ∈ E . Let G be the set of real n × n skew-symmetric matrices Γ such that there exists a pair ( A 1 , A 2 ) of real positive n × n matrices that satisfy A r ( i, i ) = ∂f ∂x r ( x i ) , Γ ij = ( x 1 j − x 1 i ) A 1 ( i, j ) + ( x 2 j − x 2 i ) A 2 ( i, j ) for all relevant r, i, j . We claim that Λ def � � = f ( x i ) − f ( x j ) is in G .