Use "bounded operator" in a sentence

1. Two necessary conditions are given under which an Approximative d-Bessel sequence and Approximative d-frame give rise to a bounded operator with respect to which there is an Approximative …

2. Two necessary conditions are given under which an Approximative X d-Bessel sequence and Approximative X d-frame give rise to a bounded operator with respect to which there is an Approximative K-atomic decomposition