FA25

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Let H=l^{2}\, and e_{n}=(0,...,0,1,0,...)\, where 1\, is in the n\,th position. Let \{a_{n}\}\, be a sequence of complex numbers.

(a) Show that Te_{n}=a_{n}e_{n}(n=1,2,...)\, defines a bounded linear operator on H\, if and only if {\mathrm  {sup}}\left\{|a_{n}|:n=1,2,...\right\}<\infty \,. In this case, find the norm of T\,.

(b) Find the necessary and sufficient condition for T\, to be bounded invertible (i.e., the inverse exists and is bounded).