Vincent's NT Word Studies
26. To the most excellent (tw kratistw). "His excellency": an official title. Compare ch. xxiv. 3; xxvi. 25.Greeting (cairein). See on ch. xv. 23.
Robertson's NT Word Studies
23:26 {Most excellent} (kratistwi). See on Lu 1:3 to Theophilus though not in #Ac 1:1. It is usual in addressing men of rank as here, like our "Your Excellency" in #24:3 and Paul uses it to Festus in #26:25. {Greeting} (cairein). Absolute infinitive with independent or absolute nominative (klaudios lusias) as is used in letters (#Ac 15:23; Jas 1:1) and in countless papyri (Robertson, _Grammar_, p. 1092).