ABSTRACT
K may be calculated by means of the formulas (14.50) - (14.52).
To illustrate the hypotheses of the last theorem, let us consider Lebesgue spaces as an elementary example.
Example 14.6. Suppose that the operator (14.3) acts in the space LP1 = LP1 (T), and the operator (14.4) acts in the space LP2 = LP2 (S). We know that [Lp1 - LP2 ] ~ [Lp1 -+ LP2 ] for 1 :$ PI :$ 1'2 < 00, and(Lp1 -+ LP2 ] ~ [Lp1 - L.rJ for 1 ~ 1'2 ~ PI < <X> (see Subsection 12.2). Putting t +*= 1'2 +*=1 as usual, the four conditions of Theorem 14.10 read then (a) 1'2 ~ PI, T 1-+ 1I/(.,T)II~ E L'l; (b) 1'2 :$ PI, S 1-+ IIm(.!, ')IILo E LP2 ; (c) PI ~ 1'2, t 1-+ II /(t")IILts E LP1 ; (d) PI ~ 1'2, q 1-+ IIm(·,q)IILPlI E Ln' Under these assumptions, the equalities (14.26), (14.27) and (14.36) hold, where the operator K is considered in the space [Lp1 - LP2 ] in the cases (a) and (c), and in the space [Lp1 -+ LP2 ] in the cases (b) and (d). In addition, for any ). as in (14.49), the defect numbers and the index of the operator K - >"1 may be calculated by means of the formulas (14.50) - (14.52).
