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Criterion equivalent to A^n \le B^n for all n

Source: Miklós Schweitzer 2019, Problem 10

December 27, 2019
linear algebra

Problem Statement

Let AA and BB be positive self-adjoint operators on a complex Hilbert space HH. Prove that lim sup⁡n→∞∥Anx∥1/n≤lim sup⁡n→∞∥Bnx∥1/n\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n} holds for every x∈Hx \in H if and only if An≤BnA^n \le B^n for each positive integer nn.
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