Relative boundedness and relative compactness for linear operators in Banach spaces

If and are linear operators acting between Banach spaces, we show that compactness of relative to does not in general imply that has -bound zero. We do, however, give conditions under which the above implication is valid.

     

Heterolytic activation of hydrogen using frustrated Lewis pairs containing tris(2,2',2'-perfluorobiphenyl)borane

The extremely sterically hindered borane tris(2,2',2'-perfluorobiphenyl)borane (PBB) has been structurally characterised. In combination with bulky nitrogen bases, it forms the 'frustrated Lewis pairs' (FLPs) PBB/2,2,6,6-tetramethylpiperidine (TMP) (1), PBB/1,4-diazobicyclo[2.2.2]-octane (DABCO) (2) and PBB/2,6-lutidine (lut) (3). These novel, unquenched acid-base pairs have been shown to effect facile room temperature heterolytic cleavage of dihydrogen to form the ammonium borate salts [2,2,6,6-Me(4)C(5)H(6)NH(2)][HB(C(12)F(9))(3)] (4) and [N(C(2)H(4))(3)NH][HB(C(12)F(9))(3)] (5), and lutidinium borate [2,6-Me(2)C(5)H(3)NH][HB(C(12)F(9))(3)] (6). Although these reactions are equilibria, the reverse reaction and release of hydrogen gas was not apparent at temperatures up to 120 °C. The relative Lewis acidity of PBB has been determined using the Gutmann-Beckett method.