Does the invertibility of the frame operator for a set $E=(e_j)_{j \in J}$ imply that $E$ is a frame?

by Muzi   Last Updated June 23, 2018 08:20 AM

I'm reading a book about Time-Frequency Analysis and I have a question regarding the property weather or not a set is a frame for a Hilbert space:

Let $H$ be a Hilbert space and $E= (e_j)_{j \in J} \subset H$ a subset of elements in $H$ ($J$ countable). We define the associated frame operator $S$ via

$$ S:H \to H, \ \ Sf=\sum_{j \in J} \langle f,e_j\rangle e_j $$

Now assume that $S$ is a bounded operator then it is well known that $S$ is a positive operator. My question is: If we further assume that $S$ is invertible, can we conclude that an estimate of the form

$$ A||f||^2 ≤ \langle Sf,f \rangle \leq B ||f||^2 $$

holds for $0 < A \leq B ?$



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