{
  "statement": "H2: 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\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
  "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\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm.",
  "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
  "case_id": "OA-1-H2-to-H5",
  "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\nThen \\(\\|T\\|\\le2\\).\n\nHere \\(\\operatorname{Re}X=(X+X^*)/2\\), \\(X\\succeq0\\) means Hermitian positive semidefinite, \\(\\operatorname{alg}(B^*)\\) is the unital polynomial algebra generated by \\(B^*\\), and \\(\\|T\\|\\) is its Euclidean operator norm."
  },
  "protocol": "single"
}