Some theorems of Korovkin type

Tom 155 / 2003

Tomoko Hachiro, Takateru Okayasu Studia Mathematica 155 (2003), 131-143 MSC: Primary 41A36; Secondary 46E15. DOI: 10.4064/sm155-2-3


We take another approach to the well known theorem of Korovkin, in the following situation: $X$, $Y$ are compact Hausdorff spaces, $M$ is a unital subspace of the Banach space $C(X)$ (respectively, $C_{{\mathbb R}}(X)$) of all complex-valued (resp., real-valued) continuous functions on $X$, $S\subset M$ a complex (resp., real) function space on $X$, ${\{\phi_{n}\}}$ a sequence of unital linear contractions from $M$ into $C(Y)$ (resp., $C_{{\mathbb R}}(Y)$), and $\phi_{\infty}$ a linear isometry from $M$ into $C(Y)$ (resp., $C_{{\mathbb R}}(Y)$). We show, under the assumption that ${\mit\Pi}_{N} \subset {\mit\Pi}_{T}$, where ${\mit\Pi}_{N}$ is the Choquet boundary for $N=\mathop{\rm Span} (\bigcup_{1\le n\le \infty}N_n)$, $N_n=\phi_{n}(M)\ (n=1,2,\ldots , \infty)$, and ${\mit\Pi}_{T}$ the Choquet boundary for ${T=\phi_{\infty}(S)}$, that ${\{\phi_{n}(f)\}}$ converges pointwise to $\phi_{\infty}(f)$ for any $f\in M$ provided $\{\phi_{n}(f)\}$ converges pointwise to ${\phi_{\infty}(f)}$ for any $f\in S$; that ${\{\phi_{n}(f)\}} $ converges uniformly on any compact subset of ${{\mit\Pi}_N} $ to $\phi_{\infty}(f)$ for any $f\in M$ provided ${\{\phi_{n}(f)\}} $ converges uniformly to ${\phi_{\infty}(f)}$ for any $f\in S$; and that, in the case where $S$ is a function algebra, $\{\phi_n\}$ norm converges to $\phi_{\infty}$ on $M$ provided ${\{\phi_{n}(f)\}}$ norm converges to $\phi_{\infty}$ on $S$. The proofs are in the spirit of the original one for the theorem of Korovkin.


  • Tomoko HachiroMathematics Division
    Sendai Shirayuri Gakuen High School
    Murasaki-Yama 1-2-1, Izumi-ku
    Sendai 981-3205, Japan
  • Takateru OkayasuDepartment of Mathematical Sciences
    Faculty of Science
    Yamagata University
    Yamagata 990-8560, Japan

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