{
  "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 H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
  "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": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
  "case_id": "OA-1-H1-to-H4",
  "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"
}