{
  "statement": "H1: Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).\n\nProve H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
  "starting_statement": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\).",
  "target_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
  "case_id": "OA-1-H1-to-H3",
  "given_context": {
    "premise": "Let \\(n\\ge1\\), let \\(S\\in M_n(\\mathbb C)\\) be invertible, and set\n\\[\nB=S\\operatorname{diag}(\\beta_1,\\ldots,\\beta_n)S^{-1},\\qquad\nT=S\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_n)S^{-1},\n\\]\nwhere the \\(\\beta_i\\) are pairwise distinct and \\(|\\lambda_i|\\le1\\); the \\(\\lambda_i\\) need not be distinct.\nSuppose \\(H:\\{w\\in\\mathbb C:|w|<1\\}\\to M_n(\\mathbb C)\\) is analytic, \\(H(0)=I\\), \\(\\operatorname{Re}H(w)\\succeq0\\), and\n\\[\nH(w)-(I-wT)^{-1}\\in\\operatorname{alg}(B^*)\\qquad(|w|<1).\n\\]\n\nSet \\(G=S^*S\\), \\(\\tau=1/2\\), and define matrices \\(P,Q,Y\\) by\n\\[\nP_{ij}=\\frac{G_{ij}}{1-\\tau^2\\overline{\\lambda_i}\\lambda_j},\\qquad\nQ_{ij}=\\frac{G_{ij}}{1-\\tau\\overline{\\lambda_i}\\lambda_j},\\qquad Y=Q-P.\n\\]\nThen\n\\[\n4Y-YG^{-1}P-PG^{-1}Y\\succeq0.\n\\]\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, and \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\)."
  },
  "protocol": "single"
}