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 $\kappa>0$ so that for all $C,\beta>0$ 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 $|x_j|=|f_j|$ for all $1\leq j\leq N$ and $$\Big|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big|\leq C|x|, \qquad\textrm{ for all $x\in \ell_2^M$ and $|\varepsilon_j|=1$}. $$ If the frame operator for $(f_j){j=1}^N$ has eigenvalues $\lambda_1\geq...\geq\lambda_M$ and $\lambda_1\leq \beta M^{-1}\sum{j=1}^M\lambda_j$ then $(f_j){j=1}^N$ has Bessel bound $\kappa \beta^2 C$. The same holds for $(x_j){j=1}^N$.