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Prolongations and cyclic vectors. (English) Zbl 1062.30005
Let $$X$$ be a Banach space of analytic functions on the open disc $${\mathbb D}=\{z\in{\mathbb C}: | z| <1\}$$ which is invariant under the backward shift operator $$Bf:=\frac{f-f(0)}{z}$$. A vector $$f\in X$$ is called cyclic if $$\overline{\text{span}}\,\{B^nf:\,n\in{\mathbb Z}_+\}=X$$. The well known Douglas-Shapiro-Shields result says that a vector $$f$$ in the Hardy space $$H^2$$ is non-cyclic if and only if it admits a pseudocontinuation $$\widetilde f$$ to $${\mathbb D}_e\cup\{\infty\}$$, where $${\mathbb D}_e=\{z\in{\mathbb C}: | z| >1\}$$, (see R.G Douglas, H.S. Shapiro, A.L. Shields [Ann. Inst. Fourier, 20, 37–76 (1970; Zbl 0186.45302)]). The latter means that there is a meromorphic function $$\widetilde f$$ of bounded type on $${\mathbb D}_e\cup\{\infty\}$$ such that $$\widetilde f(\zeta)=f(\zeta)$$ a.e. on $${\mathbb T}=\{\zeta:| \zeta| =1\}$$. Another continuation $$\widetilde f_L$$ of $$f$$ to $${\mathbb D}_e$$, given by the formula $$\widetilde f_L(w):=L\left(\frac{f}{z-w}\right)/L\left(\frac{1}{z-w}\right)$$, where $$L$$ is a bounded linear functional on $$X$$ annihilating on $$f$$, is called an $$L$$-prolongation of $$f$$. In some spaces, such as Hardy and Bergman spaces, $$L$$-prolongation $$\widetilde f_L$$ of $$f$$ coincides with its pseudocontinuation and does not depend on the choice of $$L$$, in others, like Dirichlet space, a non cyclic vector $$f$$ may have an $$L$$-prolongation without admitting a pseudocontinuation.
In the present paper the authors investigate compatibility of $$L$$-prolongations with ordinary analytic continuation for a wide class of Banach spaces of analytic functions. In the case when the $$L$$-prolongation $$\widetilde f_L$$ of $$f$$ has an analytic continuation across a boundary point $$\zeta\in{\mathbb T}$$ the following question is studied: Is this continuation equal to $$f$$ on a neighborhood of $$\zeta$$? As was shown by H. S. Shapiro [Ark. Mat. 7, 343–349 (1968; Zbl 0159.41903)] the concept of $$L$$-prolongation is related to the problem of “overconvergence” of rational functions from the $$B$$-invariant subspace $${\mathcal M}$$ of $$X$$ with the spectral synthesis property. The authors give a generalization (and a new proof) of a result from that paper saying that any sequence $$\{f_n\}_{n\geq 1}$$ of finite linear combinations of root vectors from $${\mathcal M}$$ which converges in norm to $$f\in{\mathcal M}$$ converges also to $$\widetilde f_L$$ uniformly on compact subsets of $${\mathbb D}_e\setminus\{a_n\}_{n\geq 1}$$, where $$L\in{\mathcal M}^\perp$$ and $$a_n^{-1}$$ are eigenvalues of $$B| {\mathcal M}$$. The problem of “overconvergence” of rational functions from $$B$$-invariant subspaces with the approximate spectral synthesis property is also examined.
##### MSC:
 30B40 Analytic continuation of functions of one complex variable 47B38 Linear operators on function spaces (general)
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##### References:
 [1] A. Aleman, S. Richter and W. T. Ross, Pseudocontinuations and the backward shift, Indiana Univ. Math. J. 47 1998) No.1, 223–276. MR 2000i:47009 · Zbl 0907.46019 [2] A. Aleman and W. T. Ross, The backward shift on weighted Bergman spaces, Michigan Math. J. 43 1996) No.2, 291–319. MR 97i:47053 · Zbl 0873.32003 [3] A. Beurling, Sur les fonctions limites quasi analytiques des fractions rationnelles, Proc. Eighth Scand. Math. Congress, Lund (1935), 199–210. · Zbl 0012.02303 [4] A. Borichev, On convolution equations with restrictions on supports, Algebra i Analiz 14 2002) No.2, 1–10. · Zbl 1040.47025 [5] Carleman, L’Intégrale de Fourier et Questions que s’y Rattachent, Publications Scientifiques de l’Institut Mittag-Leffler, vol. 1, Uppsala, 1944. MR 7,248c · Zbl 0060.25504 [6] J. A. Cima and W. T. Ross, The Backward Shift on the Hardy Space, rican Mathematical Society, Providence, RI 2000. MR 2002f:47068 · Zbl 0952.47029 [7] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math. 96 (1956), 1–66. MR 17,1228a · Zbl 0071.11302 [8] Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3 (1957), 429–440. MR 19,408c · Zbl 0078.09301 [9] Y. Domar, On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 1975) No.3, 203–224. MR 52 #3885 · Zbl 0272.46030 [10] R. G. Douglas, H. S. Shapiro and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970) No. fasc. 1, 37–76. MR 42 #5088 · Zbl 0186.45302 [11] P. L. Duren, Theory of Hp Spaces, demic Press, New York 1970. MR 42 #3552 [12] J. Esterle, Toeplitz operators on weighted Hardy spaces, Algebra i Analiz 14 2002) No.2, 92–116. · Zbl 1040.47018 [13] M. B. Gribov and N. K. Nikol’skiľ, Invariant subspaces and rational approximation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 103–114, 320, Investigations on linear operators and the theory of functions, IX. MR 81f:47006 [14] H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993), 1–9. MR 94k:30092 [15] P. Koosis, Introduction to Hp Spaces, second ed., bridge University Press, Cambridge 1998. MR 2000b:30052 · Zbl 1024.30001 [16] B. I. Korenblum, Closed ideals of the ring An, Funkcional. Anal. i Priložzen. 6 1972) No.3, 38–52. MR 48 #2776 [17] A. L. Matheson, Closed ideals in rings of analytic functions satisfying a Lipschitz condition, in: Banach Spaces of Analytic Functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976), Springer, Berlin, 1977, pp. 67–72. Lecture Notes in Math., Vol. 604. MR 57 #3864 [18] N. K. Nikol’skiľ, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986. MR 87i:47042 [19] N. K. Nikol’skiľ, V. P. Havin and S. V. Hrušsčcëv (eds.), Issledovaniya po lineinym operatoram i teorii funktsii, ”Nauka” Leningrad. Otdel., Leningrad, 1978, 99 nereshennykh zadach lineinogo i kompleksnogo analiza. [99 unsolved problems in linear and complex analysis], Compiled and edited by N. K. Nikol’skiľ, V. P. Havin and S. V. Hrušsčcev, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 81 (1978). MR 82e:46002 [20] E. Nordgren and P. Rosenthal, A sufficient condition that an operator be cyclic, unpublished. [21] B. Nyman, On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, Thesis, University of Uppsala, 1950. MR 12,108g · Zbl 0037.35401 [22] S. Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304 1987) No.2, 585–616. MR 88m:47056 [23] S. Richter and C. Sundberg, Invariant subspaces of the Dirichlet shift and pseudocontinuations, Trans. Amer. Math. Soc. 341 1994) No.2, 863–879. MR 94d:47026 · Zbl 0816.47037 [24] W. T. Ross and H. S. Shapiro, Generalized Analytic Continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 2003h:30003 · Zbl 1009.30002 [25] H. S. Shapiro, Generalized analytic continuation, Symposia on Theoretical Physics and Mathematics, Vol. 8 (Symposium, Madras, 1967), Plenum, New York, 1968, pp. 151–163. MR 39 #2953 [26] H. S. Shapiro, Overconvergence of sequences of rational functions with sparse poles, Ark. Mat. 7 (1968), 343–349. MR 38 #4658 · Zbl 0159.41903 [27] A. L. Shields, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR 50 #14341 [28] S. Shimorin, Approximate spectral synthesis in the Bergman space, Duke Math. J. 101 (2000) No.1, 1–39. MR 2001d:47015 · Zbl 0982.46021 [29] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147–189. MR 2002c:47018 · Zbl 0974.47014 [30] N. A. Shirokov, Closed ideals of algebras of type $B {$$\backslash$$alpha}_{pq}$ , Izv. Akad. Nauk SSSR Ser. Mat. 46 1982) No.6, 1316–1332, 1344. MR 84b:46061 [31] R. V. Sibilev, A uniqueness theorem for Wolff-Denjoy series, Algebra i Analiz 7 1995) No.1, 170–199. MR 96j:30006 [32] G. D. Taylor, Multipliers on $D_$$\backslash$$alpha$ , Trans. Amer. Math. Soc. 123 (1966), 229–240. MR 34 #6514 [33] G. C. Tumarkin, Description of a class of functions admitting an approximation by fractions with preassigned poles, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 1 1966) No.2, 89–105. MR 34 #6123 [34] J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Plane, Amer. Math. Soc. Coll. Pub. (20), Providence, RI, 1935. · Zbl 0013.05903
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