Measure, Integration & Real Analysis · Problem 10B.14

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · approximate-eigenvalues-normal

Problem

Suppose TT is a normal operator on a Hilbert space and α∈F\alpha \in \mathbf{F}. Prove that α∈sp(T)\alpha \in \mathrm{sp}(T) if and only if α\alpha is an approximate eigenvalue of TT. [A number α∈F\alpha \in \mathbf{F} is called an approximate eigenvalue of a bounded operator TT on a Hilbert space VV if inf⁡{∥(T−αI)f∥:f∈V and ∥f∥=1}=0.\inf\{\|(T - \alpha I)f\| : f \in V \text{ and } \|f\| = 1\} = 0.]

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