Helton-Howe trace formula and planar shapes Mihai Putinar - PowerPoint PPT Presentation

Helton-Howe trace formula and planar shapes Mihai Putinar Mathematics Department UC Santa Barbara <mputinar@math.ucsb.edu> UCSB

Trace formula [1] The source Helton, J.William; Howe, Roger E. Integral operators: traces, index, and homology. Proc. Conf. Operator Theory, Dalhousie Univ., Halifax 1973, Lect. Notes Math. 345 , 141-209 (1973). Helton, J.William; Howe, Roger E. Traces of commutators of integral operators. Acta Math. 135, 271-305 (1975). UCSB Bill’s Fest 2010

Trace formula [2] The formula T ∈ L ( H ) with trace-class self-commutator [ T ∗ , T ] ∈ C 1 ( H ) For p ( z, z ) , q ( z, z ) ∈ C [ z, z ] polynomials one defines p ( T, T ∗ ) , q ( T, T ∗ ) using any order of T, T ∗ in the NC monomials. Then trace [ p ( T, T ∗ ) , q ( T, T ∗ )] = u T ( ∂ z p∂ z q − ∂ z q∂ z p ) , where u T ∈ D ′ ( C ) . UCSB Bill’s Fest 2010

Trace formula [3] Example Let S ∈ L ( ℓ 2 ) be the unilateral shift. Then u S = χ D d Area , that is π ( ∂ z p∂ z q − ∂ z q∂ z p ) d Area � trace [ p ( S, S ∗ ) , q ( S, S ∗ )] = . π D Sufficient to verify trace [ S ∗ m S n , S ∗ r S s ] from S ∗ S = I and Se k = e k +1 ... UCSB Bill’s Fest 2010

Trace formula [4] Helton-Howe distribution as spectral invariant u T is functorial in T supp( u T ) = σ ( T ) u T ( ζ ) = − ind( T − ζ ) d Area for ζ / ∈ σ ess ( T ) . π In general u T = g T d Area with g T ∈ L 1 ( C , d Area) (proof involving more π inequalities, scattering theory, singular integrals, as developed by J. Pincus and collaborators). Today g T is known as the principal function of T . The multivariate analog of u T is more involved, less understood but it gave an elegant proof of Atiyah-Singer index formula in the context of pseudo-differential operator calculus. UCSB Bill’s Fest 2010

Trace formula [5] Evolution “ In 1981 I have discovered cyclic cohomology and the spectral sequence relating it to Hochschild cohomology. My original motivation came from the trace formulas of Helton-Howe and Carey-Pincus for operators with trace-class commutators” from A. Connes, Non-Commutative Geometry , Academic Press, 1994, pp. 12. Douglas, R.G.; Voiculescu, Dan On the smoothness of sphere extensions J. Oper. Theory 6, 103-111 (1981). UCSB Bill’s Fest 2010

Trace formula [6] Shade functions There exists a bijective correspondence between functions g ∈ L 1 comp ( C , d Area) , 0 ≤ g ≤ 1 , and irreducible linear operators T ∈ L ( H ) satisfying [ T ∗ , T ] = ξ �· , ξ � . Specifically g = g T , or E ( w, z ) = det(( T − w )( T ∗ − z )( T − w ) − 1 ( T ∗ − z ) − 1 ) = = 1 − � ( T ∗ − z ) − 1 ) ξ, ( T ∗ − w ) − 1 ) ξ � = exp( − 1 g ( ζ )dArea( ζ ) � ( ζ − w )( ζ − z )) . π valid over all C × C and separately continuous there (K. Clancey). UCSB Bill’s Fest 2010

Trace formula [7] Exponential transform Let � z m z n g ( ζ )dArea( ζ ) , a mn = m, n ≤ N, be given (from measurements). Then the series transform N N exp( − 1 b jk X j +1 Y k +1 + O ( X N +2 , Y N +2 ) � a mn X m +1 Y n +1 ) = 1 − � π m,n =0 j,k =0 has coefficients bound by the positivity conditions b jk = � T ∗ ( k +1) ξ, T ∗ ( j +1) ξ � , [ T ∗ , T ] = ξ �· , ξ � . UCSB Bill’s Fest 2010

Trace formula [8] Ramifications Reconstruction of g via a 2D Pad´ e approximation scheme (finite central projection of the matrix attached to T ) Gauss type cubatures for the weight g in matrix form Regularity of free boundaries In the case g = χ Ω( t ) where Ω( t ) are planar domains following the Laplacian Growth dynamics, identification of E ( z, w ) with the Tau function of a completely integrable hierarchy Elimination theory on compact Riemann surfaces, with E ( z, w ) as the correct resultant and univalence criteria for analytic functions UCSB Bill’s Fest 2010

Trace formula [9] Quadrature domains det( b jk ) N j,k =0 = 0 if and only if g = χ Ω where � f ( ζ )dArea( ζ ) = c 1 f ( a 1 ) + ... + c N f ( a n ) = π � f ( T ) ξ, ξ � = π � f ( T 0 ) ξ, ξ � Ω for all analytic functions f ∈ L 1 a (Ω) . In this case the reconstruction algorithm (“ moments to shape”) is exact at rank N . Applications to geometric tomography (joint work with G. Golub et al). UCSB Bill’s Fest 2010

Trace formula [10] Hele-Shaw flows QD (and other classes of algebraic boundaries) are preserved under Hele-Shaw flows: Ω t ⊂ C nested with z = 0 ∈ Ω t boundary velocity V ( ζ ) = ∂ n G Ω t ( ζ, 0) d Ω t z n dA = , n > 0 . � has sequence of conserved quantities: dt UCSB Bill’s Fest 2010

Trace formula [11] Generalized lemniscates, linear pencils and sums of hermitian squares Let Q ( z, z ) be a hermitian polynomial. The following are equivalent: 1. There exists A ∈ L ( C n ) with cyclic vector ξ such that � � ξ �· , ξ � A − z � � Q ( z, z ) = � ; A ∗ − z � � I � 2. There are polynomials Q k ( z ) of degree degQ k = k , such that Q ( z, z ) = | Q N ( z ) | 2 − | Q N − 1 ( z ) | 2 − ... − | Q 1 ( z ) | 2 − | Q 0 ( z ) | 2 . UCSB Bill’s Fest 2010

Trace formula [12] Union of disks If the disks D ( a j , r j ) are mutually disjoint, then the equation of their union is a generalized lemniscate N � � ξ �· , ξ � A − z [ | z − a j | 2 − r 2 � � � Q ( z, z ) = j ] = A ∗ − z � � I � � j =1 In particular the matrix [ Q ( a j , a k )] N j,k =0 is negative definite. But a little more (a four argument kernel) is needed to characterize disjointness. UCSB Bill’s Fest 2010

Trace formula [13] Cauchy-Riemann system Let [ T ∗ , T ] = ξ �· , ξ � acting on the Hilbert space H . For every ϕ ∈ L 2 ( C , dArea) there exists u : C − → H such that the output of the system  ∂u ∂z = Tu ( z ) + ϕ ( z ) ξ   ξ v ( z ) = � u ( z ) , � ξ � �   satisfies � v � 2 , C ≤ � ϕ � 2 , C . UCSB Bill’s Fest 2010

Trace formula [14] Regularity of free boundaries dA ( ζ ) � Sakai’s Theorem. Let Ω ⊂ C be a domain with Area ∂ Ω = 0. If Ω ζ − z extends analytically across ∂ Ω from C � Ω , then ∂ Ω is real analytic. New proof: Let T be associated to Ω via g T = χ Ω and extend ( T ∗ − z ) − 1 ξ analytically across ∂ Ω using the Schwarz function S ( z ) = z + χ Ω ( z ) + 1 dA ( ζ ) � ζ − z . π Ω Then remark that E T ( z, z ) = 1 − � ( T ∗ − z ) − 1 ξ � 2 = 0 along the boundary. UCSB Bill’s Fest 2010

Trace formula [15] Ahlfors-Beurling inequality Let g ∈ L 1 comp ( C , d Area) , 0 ≤ g ≤ � g � ∞ < ∞ . Then � g ( ζ ) dA ( ζ ) | 2 ≤ π � g � 1 � g � ∞ . | ζ − z � g T ( ζ ) dA ( ζ ) = π � ( T ∗ − z ) − 1 ξ, ξ � and g Proof: Let T with g T = � g � ∞ . Then ζ − z � ( T ∗ − z ) − 1 ξ � ≤ 1 everywhere. Helton-Howe formula � ξ � 2 = � g � 1 π . UCSB Bill’s Fest 2010

Trace formula [16] References Gustafsson, Bj¨ orn; Putinar, Mihai, An exponential transform and regularity of free Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26 , boundaries in two dimensions. No.3, 507-543 (1998). Gustafsson, Bj¨ orn; Putinar, Mihai, The exponential transform: A renormalized Riesz potential at critical exponent. Indiana Univ. Math. J. 52 , No. 3, 527-568 (2003). Putinar, Mihai, A renormalized Riesz potential and applications , Neamtu, Marian (ed.) et al., Advances in constructive approximation: Vanderbilt 2003. Proceedings of the international conference, Nashville, TN, USA, May 14–17, 2003. Brentwood, TN: Nashboro Press. Modern Methods in Mathematics, 433-465 (2004). Mineev-Weinstein, Mark; Putinar, Mihai; Teodorescu, Razvan, Random matrices in 2D, Laplacian growth and operator theory , J. Phys. A, Math. Theor. 41, No. 26, Article ID 263001, 74 p. (2008). UCSB Bill’s Fest 2010