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In this paper, we derive an interesting result that the mixed-type reverse order lawshold if and only if the bounded linear operator productsis invariant, whereis taken, for instance, as,,,,as well as. Moreover, the invariance property ofis related to the reverse order law, and we point out that such an invariance property occurs if and only ifandare left orandare right invertible operators, oris left andis right invertible operator, where,andare nonzero regular operators. [ABSTRACT FROM PUBLISHER]

Mitochondria targeting complexes are widely utilized as photosensitizers in photodynamic therapy. However, the mechanisms by which they regulate reactive oxygen species (ROS) production at the molecular level and their influence on intracellular mitochondrial signaling and ultrastructures remain rarely studied. Herein, we present two terpyridyl Zn(II) complexes with different side alkyl chain lengths (Zn-2C and Zn-6C) that lead to low and high ROS productivities in vitro , respectively. Both complexes could enter live cells effectively with minimal dark toxicity and accumulate preferably in the mitochondria. We also demonstrated that Zn-6C, with more efficient ROS productivity, could significantly downregulate the caspase signaling pathway but showed no evident influence on mitochondrial membrane proteins. We also highlighted and compared the mitochondrial ultrastructural variations during such a process by stimulated emission depletion (STED) super-resolution nanoscopy.