Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators

Dyakonov, Konstantin M.

doi:10.1090/S0002-9939-1992-1100649-2

Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators
HTML articles powered by AMS MathViewer

by Konstantin M. Dyakonov PDF

Proc. Amer. Math. Soc. 116 (1992), 1007-1013 Request permission

Abstract:

It is proved that for each inner function $\theta$ there exists an interpolating sequence $\left \{ {{z_n}} \right \}$ in the disk such that ${\sup _n}|\theta ({z_n})| < 1$, but every function $g$ in ${H^\infty }$ with $g({z_n}) = \theta ({z_n})(n = 1,2, \ldots )$ satisfies $||g|{|_\infty } \geq 1$. Some results are obtained concerning interpolation in the star-invariant subspace ${H^2} \ominus \theta {H^2}$. This paper also contains a "geometric" result connected with kernels of Toeplitz operators.

References

Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
K. M. D′yakonov, The geometry of the unit ball in the space $K^1_\theta$, Geometric problems of the theory of functions and sets (Russian), Kalinin. Gos. Univ., Kalinin, 1987, pp. 52–54 (Russian). MR 911715
K. M. D′yakonov, Moduli and arguments of analytic functions from subspaces in $H^p$ that are invariant under the backward shift operator, Sibirsk. Mat. Zh. 31 (1990), no. 6, 64–79 (Russian); English transl., Siberian Math. J. 31 (1990), no. 6, 926–939 (1991). MR 1097956, DOI 10.1007/BF00970058
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
S. V. Hruščëv, N. K. Nikol′skiĭ, and B. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 214–335. MR 643384
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451

Bases of reproducing kernels and exponentials

Knut Øyma, Extremal interpolatory functions in $H^{\infty }$, Proc. Amer. Math. Soc. 64 (1977), no. 2, 272–276. MR 447586, DOI 10.1090/S0002-9939-1977-0447586-6
A. Stray and K. O. Øyma, On interpolating functions with minimal norm, Proc. Amer. Math. Soc. 68 (1978), no. 1, 75–78. MR 457734, DOI 10.1090/S0002-9939-1978-0457734-0