Nihonkai Mathematical Journal

Interpolation problem for $\ell^1$ and an $F$-space

Takahiko Nakazi

Full-text: Open access

Abstract

Let $B$ be an $F$-space and $B^\ast _1$ the unit ball of the dual space. A sequence $(\phi _n)$ in $B^\ast _1$ is called $\ell ^1$-interpolating if for every sequence $(w_n)$ in $\ell ^1$ there exists an element $f$ in $B$ such that $\phi _n(f)=w_n$ for all $n$. In order to study an interpolation problem for $\ell ^1$, we introduce two quantities $\rho _n$ and ${\prod_{k\ne n}}\sigma (\phi _n,\phi _k)$. For arbitrary Banach space, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n}\rho _n>0$. Moreover, when a Banach space has a predual, we show that if ${\inf_n\prod_{k\ne n}}\sigma(\phi_n, \phi_k)>0$ then $(\phi_n)$ is an $\ell^1$-interpolating sequence. When $(\phi _n)$ is embeded in the open unit disc in the complex plane, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n \prod_{k\ne n}}\sigma (\phi _n,\phi _k)>0$, for a Hardy space $H^p(D)(1\leq p\leq \infty )$ and the Smirnov class $N_+(D)$.

Article information

Source
Nihonkai Math. J., Volume 19, Number 2 (2008), 75-83.

Dates
First available in Project Euclid: 18 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1363634621

Mathematical Reviews number (MathSciNet)
MR2490130

Zentralblatt MATH identifier
1179.32001

Subjects
Primary: 32A35: Hp-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15] 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]

Keywords
Interpolation $\ell^1$ $F$-space Hardy space Smirnov class

Citation

Nakazi, Takahiko. Interpolation problem for $\ell^1$ and an $F$-space. Nihonkai Math. J. 19 (2008), no. 2, 75--83. https://projecteuclid.org/euclid.nihmj/1363634621


Export citation

References

  • L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930.
  • A. Hartmann, X. Massaneda, A. Nicolau and P. Thomas, Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants, J. Funct. Anal. 217 (2004), 1–37.
  • O. Hatori, The Shapiro-Shields theorem on finite connected domains, Surikaisekikenky$\mbox{\=u}$sho K$\mbox{\=o}$ky$\mbox{\=u}$roku 1049 (1998), 21–29 (in Japanese).
  • V. Kabaila, Interpolation sequnces for the $H_p$ classes in the case $p<1$, Litovsk. Mat. Sb. 3 (1) (1963), 141–147 (in Russian).
  • H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, 199, Springer-Verlag, New York, 2000.
  • T. Nakazi, Interpolation problem for $\ell ^1$ and a uniform algebra, J. Austral. Math. Soc. 72 (2002), 1–11.
  • H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532.
  • A. K. Snyder, Sequence spaces and interpolation problems for analytic functions, Studia Math. 39 (1971), 137–153.
  • N. Yanagihara, Interpolation theorems for the class $N^+$, Illinois J. Math. 18 (1974), 427–435.