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A sequence of distinct vectors $\{g_1,g_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that $$A\|g\|^2\leq\sum_{n=1}^\infty |(g,g_n)|^2\leq B \|g\|^2.$$ for each $g\in H$. Prove that $\{g_n\}$ is a frame if and only if the so called frame operator $$Sg=\sum_n (g,g_n) g_n$$ is continuous and continuously invertible on its range.

Any suggestions to approach a proof, please?

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  • $\begingroup$ What is $g$ in your question? $\endgroup$ – mathcounterexamples.net Jun 9 '15 at 8:51
  • $\begingroup$ An element of H. I've edited my post. $\endgroup$ – Mark Jun 9 '15 at 8:52
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For general normed linear spaces $X,Y$ there is the following characterisation of normed space isomorphisms:

The operator $T : X \to Y$ is a normed space isomorphism if, and only if, there exists constants $A,B > 0$ such that, for all $x \in X$, $$ A \| x \|_X \leq \| T x \|_Y \leq B\| x \|_X. $$

Note that a normed space isomorphism is, by definition, a continuous mapping with continuous inverse on its range. Do you see how to proceed from here?

EDIT: Here are, as requested, some steps:

  • Note that the frame inequality can be written in terms of the frame operator as $$ A \|x\|^2_{\mathcal{H}} \leq \langle S x, x \rangle_{\mathcal{H}} \leq B \|x\|^2_{\mathcal{H}}. $$

  • The frame operator $S$ is positive, and hence an application of Cauchy-Schwarz gives $$ \langle Sx , x \rangle \leq \| Sx \| \|x\|. $$

Do not hestitate to ask more questions if you think you need more help.

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  • $\begingroup$ Thank you, but I don't how to proceed from here. Could you write some steps, please? $\endgroup$ – Mark May 25 '16 at 16:43

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