Measure, Integration & Real Analysis · Problem 10A.9

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · diagonal-operator-properties

Problem

Suppose b1,b2,…b_1, b_2, \ldots is a bounded sequence in F\mathbf{F}. Define a bounded linear map T:ℓ2→ℓ2T: \ell^2 \to \ell^2 by T(a1,a2,…)=(a1b1,a2b2,…).T(a_1, a_2, \ldots) = (a_1 b_1, a_2 b_2, \ldots). (a) Find a formula for T∗T^*. (b) Show that TT is injective if and only if bk≠0b_k \neq 0 for every k∈Z+k \in \mathbf{Z}^+. (c) Show that TT has dense range if and only if bk≠0b_k \neq 0 for every k∈Z+k \in \mathbf{Z}^+. (d) Show that TT has closed range if and only if inf⁡{∣bk∣:k∈Z+ and bk≠0}>0.\inf\{|b_k| : k \in \mathbf{Z}^+ \text{ and } b_k \neq 0\} > 0. (e) Show that TT is invertible if and only if inf⁡{∣bk∣:k∈Z+}>0\inf\{|b_k| : k \in \mathbf{Z}^+\} > 0.

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