Quantitative bounds for unconditional pairs of frames

Abstract

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant $\hat{I}º>0$ so that for all $C,{\hat{I}}^2>0$ and $N\in \mathbb{N}$ the following is true. Let $(x_j{)}_{j=1}^N$ and $(f_j{)}_{j=1}^N$ be sequences in a finite dimensional Hilbert space which satisfy $\Vert x_j\Vert =\Vert f_j\Vert$ for all $1\le j\le N$ and $$\Vert \sum _{j=1}^N{\epsilon }_j⟨x,f_j⟩x_j\Vert \le C\Vert x\Vert ,\text{ for all x∈ℓ2M and ∣εj∣=1}.$$ If the frame operator for $(f_j{)}_{j=1}^N$ has eigenvalues $\hat{I}{»}_1\ge ...\ge \hat{I}{»}_M$ and $\hat{I}{»}_1\le {\hat{I}}^2{M}^{-1}{\sum }_{j=1}^M\hat{I}{»}_j$ then $(f_j{)}_{j=1}^N$ has Bessel bound $κβ^2 C$. The same holds for $(x_j{)}_{j=1}^N$.

19 pages