By Bacco M., Mocellin V.

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11 ) r are satisfied. The following assertion is also true. 5. Let f be a simple closed curve. Necessary and sufficient for the function cp (E Lp(r, p)) to belong to the subspace Lt (r, p) is the validity of the conditions jCP(t)tndt=O (n=0,1,2, ... ). 3. The function ~'P(z) has to be expanded into a series in a neighbourhood of infinity. 4. SINGULAR PROJECTIONS be a closed nonsimple curve. We define the subspace o L;(r,p) := L; (r,p) L; (r, p) by setting • + span{l} , where span {I} is the linear hull of the function 'f'(t) == 1 .

1. Let £ll £2 (£1 ~ £2) be two subspaces of a Banach space B with dim £d £1 < 00 . If the linear manifold £ satisfies the condition then £ is a subspace. 3 CHAPTER 2. ONE-SIDED INVERTIBLE OPERATORS Linear operators. Notations and simplest classes Let B1 and B2 be Banach spaces. By L(B 1, B2) we denote the Banach space of all linear bounded operators mapping B1 into B2 . As norm in L(B1, B2) we use the operator norm. In the following, the space L(B, B) will be denoted by L(B) . It is a Banach algebra.

The following theorems yield criteria for the membership of a function f (E Lp(f,p)) o to the subspaces Lt(f,p) and L; (f,p). Let Ft, ... , F;t; and F1-,· Fr , respectively. 3. f ( cp(T) dT . T-ex. ) n = 0 (J = 1, ... , kj ) be satisfied, where exj are certain points from Fj- . n=I,2, ... 8) 60 CHAPTER 2. ONE-SIDED INVERTIBLE OPERATORS Proof. Let cp E Lp(f,p) . The function ~'P(z) is holomorphic in Fr:, and in a neighbourhood of the points z = aj it can be expanded into the series ~'P(z) = L: oo (z - aj)k 27n.