By Pierre Schapira

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2) /"/· g da = 0 (all g in 9Π), INVARIANT SUBSPACES IN Lp 27 is a simply invariant subspace of Lp, and must have the form q*Hp by what has just been proved. 2) ; when r = oo we have to add that 9TC is star-closed in order to make the conclusion. That is, 9Π is χ · # Ή Γ , and this is the form 9TC was asserted to have. Now the theorem is proved for all values of p. It was a little tedious to introduce an argument about conjugate functions, but really some information beyond Theorem 5 was necessary in the proof.

Lax's extension of Beurling's theorem gave the form of simply invariant subspaces which are contained in H2 [14]. More generally we have: Theorem 7. Each simply invariant subspace of L2 has the form Q'H2, where q is a measurable function on the line such that \ q(x) \ = 1 almost everywhere, q is unique up to a constant factor. We study a simply invariant subspace 9ΪΙ of L2. Set 9fïl\ = S\$K, and let P\ be the orthogonal projection of L2 on 3Πλ. The hypothesis 40 LECTURE V that 2ΠΖ is simply invariant means that P\ decreases as λ increases, and indeed in the strict sense.

6) //·χ^Μ = 0 (fc = 0, 1, · · . ) for each / in 9ÏÏ. A second important theorem of Riesz, this time of F. and M. Riesz (see Hoffman [12]), assets that a measure v is necessarily absolutely continuous with respect to Lebesgue measure if it vanishes as a linear functional on xk for k = 0, 1, · · · . 6) for each / in 9ΪΙ. Write μ = μα + μ8 with μα absolutely continuous and μ8 singular. 7) ' /άμ8 = 0 ( / in 9ÎI, μ in 91). Let Kf be the set of points on the circle where / is zero. Kf is closed because / is continuous, and has measure zero because / is analytic (unless/ is the null function).