frames of a hilbert space

**krooooochan** · April 2nd 2011, 11:36 AM

Hi! I came across this problem while I was reading a supplementary material for our course on hilbert spaces. Can anyone help me with this? Thank you!

given an orthonormal basis { $\phi_{n}$ } of a hilbert space H and a one-to-one and onto continuous linear transformation T from H to H, prove that {T $\phi_{n}$ } is a frame for H.

**Opalg** · April 2nd 2011, 11:25 PM

Originally Posted by krooooochan

Hi! I came across this problem while I was reading a supplementary material for our course on hilbert spaces. Can anyone help me with this? Thank you!

given an orthonormal basis { $\phi_{n}$ } of a hilbert space H and a one-to-one and onto continuous linear transformation T from H to H, prove that {T $\phi_{n}$ } is a frame for H.

For x in H, $\sum_n|\langle x,T\phi_n\rangle|^2 = \sum_n|\langle T^*x,\phi_n\rangle|^2 = \|T^*x\|^2 \leqslant \|T\|^2\|x\|^2.$ That gives you one of the two inequalities that you need for a frame. The other one comes from the Banach inversion theorem, which says that T (and therefore also T*) must have a bounded inverse. It follows that $\|x\| = \|(T^*)^{-1}T^*x\| \leqslant \|(T^*)^{-1}\|\|T^*x\|$ and hence $\|T^*x\|^2 \geqslant \|T^{-1}\|^{-2}\|x\|^2.$

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