Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 1 / c

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- Solution c

Solution

Choose a unit vector

e

and define the rank-one operator

T h = - ⟨ h, e ⟩ e .

It is bounded, self-adjoint, and Hilbert-Schmidt, with

∥ T ∥_{HS} = 1

. But

⟨ T e, e ⟩ = - 1 < 0,

so it is not positive. Since every covariance operator is positive,

T

is the required counterexample.

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