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 $0 < A \leq B ?$

Related Questions

Unitary dual of the motion group of $M(n)$, for $n> 2$

Updated January 07, 2018 18:20 PM

Complete positivity vs Positivity.

Updated April 30, 2018 17:20 PM

Density principle in Quasi-Banach Space

Updated September 21, 2017 14:20 PM

Theorem 2.3.2 A First Course in Harmonic Analysis

Updated December 18, 2017 08:20 AM