Unconditionally convergent multipliers and Bessel sequences (Q2013085)

Let \(H\) be a separable Hilbert space. A bounded linear operator \(M:H\rightarrow H\) is a multiplier if there exist a scalar sequence \(\{m_{k}\}\) and \(\{x_{k}\},\{y_{k}\}\subset H\) such that \[ Mf=\sum_{k=1}^{\infty }m_{k}(f\cdot y_{k})x_{k} \] for every \(f\in H\). The multiplier \(M\) is unconditionally convergent if the series is unconditionally converging for every \(f\). A sequence \(\{z_{k}\}\subset H\) is a Bessel sequence if there exists \(B>0\) such that \[ \sum_{k=1}^{\infty }\left| f\cdot z_{k}\right| ^{2}\leq B\left\| f\right\| ^{2} \] for \(f\in H\). The authors show that if \(\{u_{k}\}\subset H\) is unconditionally convergent, then there exist \(\{a_{k}\}\in l^{2}\) and a Bessel sequence \(\{b_{k}\}\) such that \(u_{k}=a_{k}b_{k}\). The authors then use this result to address a conjecture of \textit{D. T. Stoeva} and \textit{P. Balazs} [J. Math. Anal. Appl. 399, No. 1, 252--259 (2013; Zbl 1259.42024)] on unconditionally convergent multipliers. The conjecture is that, if \(M\) is an unconditionally convergent multiplier, then there exist scalar sequences \( \{a_{k}\},\{b_{k}\}\) such that \(m_{k}=a_{k}\overline{b_{k}}\) and \( \{a_{k}x_{k}\},\{b_{k}y_{k}\}\) are Bessel sequences. The authors show that the conjecture holds in some special cases.