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By V. P. Havin, N. K. Nikol’skii

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Additional info for Complex Analysis and Spectral Theory: Seminar, Leningrad 1979/80

Example text

O, ~)). 3. Let ~ be a function analytic in ~ \ ~ and T g ~ L~ (T), P<'~ and T ~ e ~ s~t + . Then '" p~-4- (T), P<~ ' P-~:[- a. ~

1(b) that this subspace is cyclic. It is generated by the function ~_ % . 2 below. Let X be an invariant subspace of ~ < 6 ) . Denote by ~ the set of all rotation-invariant functlonals on X . t is clear that X0~ is a subspace of the Banach space X ~ . ~4~ Xo* ~< ~ if ~ c [ ~ + o o ) . 1o Let ~ be a compact abelian group and let be an invariant subspace of L~ ~ ~) . Suppose that < d~TF5 X t . Then the invariant subspace, generated by a collection of functions { ~ K ~ ~ in ~ does not coincide with X.

8 fall for non-locally holomorphlc spaces. §3. 2 that (_A~+) scalar of all does n o t depend on ~ . 47 (T ~) Le~ B P ~'1]) ~-~-C d e n o t e t h e s e t Of a l l holomorphlc functions such that ~f t~(~)t <+CO . 7 and the induction prove the following in , it is not hard to • Then THEOREM 3 . 2 . ,~-~ ~ ( A I / P ' ~ ct r uch that f ~-~- for ~ ~ ~ ~+~ (~r ~ ) . ,o~ ~ t i c ~ H[(T'5 6Ui+ )~¢ deZi~es a linear oonti=uo~ functional on by t h e f o r m u l a (3)• I' 1 ~ o o ~ . Let ~ e ( A ~ " ) . Chec~ that the f o = ~ (3) ~,fines a linear continuous functlon-] on H~ ( T ~) • Note t h a t Hence 48 It remains to note that 1, ~' ~-~r__~a~-~)'1 fo .

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