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An Operator Approach to Extremal Problems on Hardy and Bergman Spaces


Abstract

S. Abbott and B. Hanson developed an operator-theoretic approach to solve some extremal problems. We give a different proof of a theorem of S. Abbott and B. Hanson in the case when the corresponding operator is unbounded. We apply our theorem to the classical Kolmogorov and Szegö infimum problems. We also consider Kolmogorov and Szegö type infima, when integration over the unit circle is replaced by integration over the unit disk.

С. Аббот і Б. Хенсон розвинули теоретико-операторний підхід до розв’язанні деяких екстремальних задач. Ми даємо нове доведення теореми С. Аббота і Б. Хенсона для випадку, коли відповідний оператор необмежений. Теорема застосовується для класичних задач Колмогорова і Сеге про інфімум. Також розглянуті задачі Колмогорова і Сеге про інфімум для випадку, коли інтегрування ведеться не по колу, а по кругу.

Key words: Unbounded operator, Kolmogorov infimum, Szegö infimum, Hardy space, Bergman space.


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Article Information

TitleAn Operator Approach to Extremal Problems on Hardy and Bergman Spaces
SourceMethods Funct. Anal. Topology, Vol. 27 (2021), no. 2, 142-150
DOI10.31392/MFAT-npu26_2.2021.02
MathSciNet MR4296403
MilestonesReceived 26/07/2020; Revised 12/03/2021
CopyrightThe Author(s) 2021 (CC BY-SA)

Authors Information

Miron B. Bekker
Department of Mathematics the University of Pittsburgh at Johnstown, 450 Schoolhouse Rd, Johnstown PA 15904

Joseph A. Cima
Department of Mathematics The University of North Carolina at Chapel Hill, CB 3250, 329 Phillips Hall, Chapel Hill, NC 27599


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Citation Example

Miron B. Bekker and Joseph A. Cima, An Operator Approach to Extremal Problems on Hardy and Bergman Spaces, Methods Funct. Anal. Topology 27 (2021), no. 2, 142-150.


BibTex

@article {MFAT1560,
    AUTHOR = {Miron B. Bekker and Joseph A. Cima},
     TITLE = {An Operator Approach to  Extremal Problems on Hardy and Bergman Spaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {27},
      YEAR = {2021},
    NUMBER = {2},
     PAGES = {142-150},
      ISSN = {1029-3531},
  MRNUMBER = {MR4296403},
       DOI = {10.31392/MFAT-npu26_2.2021.02},
       URL = {http://mfat.imath.kiev.ua/article/?id=1560},
}


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