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8

Part of 2001 Miklós Schweitzer

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Miklós Schweitzer 2001 Problem 8

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2/12/2017
Let HH be a complex Hilbert space. The bounded linear operator AA is called positive if ⟨Ax,x⟩≥0\langle Ax, x\rangle \geq 0 for all x∈Hx\in H. Let A\sqrt A be the positive square root of AA, i.e. the uniquely determined positive operator satisfying (A)2=A(\sqrt{A})^2=A. On the set of positive operators we introduce the A∘B=ABBA\circ B=\sqrt A B\sqrt B operation. Prove that for a given pair A,BA, B of positive operators the identity (A∘B)∘C=A∘(B∘C)(A\circ B)\circ C=A\circ (B\circ C) holds for all positive operator CC if and only if AB=BAAB=BA.
Miklos SchweitzerFunctional Analysislinear algebra