Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C > 0 and N is an element of 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 parallel to x(j)parallel to = parallel to f(j)parallel to for all 1 <= j <= N and parallel to Sigma(j=1) epsilon(j) < x, f(j)>:x(j)parallel to <= C parallel to x parallel to, for all x is an element of l(2)(M) and vertical bar epsilon(j)vertical bar = 1. If the frame operator for (f(j))(j=1)(N) has eigenvalues lambda(1) >= ... >= lambda(M) and beta > 0 is such that lambda(1) <= beta M-1 Sigma(M)(j=1) lambda(j) then (f(j))(j=1)(N) has Bessel bound 27 beta C-2. The same holds for (x(j))(j=1)(N). (c) 2023 Elsevier Inc. All rights reserved.