On a theorem of Terzioğlu
On a theorem of Terzioğlu
The theory of compact linear operators acting on a Banach space has a classical core and is familiar tomany. Perhaps less known is the characterization theorem of Terzioğlu for compact maps. This theorem has a numberof important connections that deserves illumination. In this paper we survey Terzioğlu’s characterization theorem forcompact maps and some of its consequences. We also prove a similar characterization theorem for Q-compact maps.
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- [1] Aksoy AG. Q-Compact sets and Q-compact maps. Math Japon 1991; 36: 1-7.
- [2] Aksoy, AG, Almira J. On approximation schemes and compactness. In: Proceedings of the First Conference on
Classical and Functional Analysis; Azuga, Romania; 2014. pp. 5-24.
- [3] Aksoy AG, Nakamura M. The approximation numbers
n(T) and Q-compactness. Math Japon 1986; 31: 827-840.
- [4] Astala K. On measures of non-compactness and ideal variations in Banach spaces. Ann Acad Sci Fenn Ser AI Math
1980; 29: 1-42.
- [5] Brudnyi YuA, Krugljak NYa. On a Family of Approximation Spaces. Studies in the Theory of Function Theory of
Several Real Variables, No. 2. Yaroslavl, USSR: Yaroslavl State University, 1978 (in Russian).
- [6] Butzer PL, Scherer K. Approximation-Prozesse und Interpolations-Methoden. Mannheim, Germany: Biliographisches
Institut, 1968 (in German).
- [7] Diestel J. Sequences and Series in Banach Spaces. Berlin, Germany: Springer-Verlag, 1984.
- [8] Enflo P. A counter example to the approximation property in Banach spaces. Acta Math 1973; 130: 309-317.
- [9] Fernandez DL, Mastylo M, Da Silva EB. Quasi s-numbers and measures of non-compactness of multilinear operators.
Ann Acad Sci-Fi Fenn Math 2013; 38: 805-823.
- [10] Fourie JH. Injective and surjective hulls of classical p-compact operators with applications to unconditionally pcompact
operators. Studia Math 2018; 240: 147-159.
- [11] Hutton CV. On approximation numbers and its adjoint. Math Ann 1974; 210: 277-280.
- [12] Lindenstrauss J. Extension of compact operators. Mem Amer Math Soc 1964; 48: 1-112.
- [13] Lindenstrauss J, Tzafriri L. Classical Banach Spaces I, Sequence Spaces. Berlin, Germany: Springer-Verlag, 1977.
- [14] Nachbin L. A theorem of the Hanh Banach type for linear transformations. T Am Math Soc 1950; 68: 28-46.
- [15] Persson A, Pietsch P. p-Nukleare und p-integrale Abbildungen in Banachrumen. Studia Math 1969; 33: 19-62 (in
German).
- [16] Pietsch P. Approximation spaces. J Approx Theory 1981; 32: 115-134.
- [17] Randtke DJ. Representation theorems for compact operators. P Am Math Soc 1973; 37: 481-485.
- [18] Sofi MA. A note on embedding into product spaces. Czechoslovak Math J 2006; 56: 507-513.
- [19] Terzioğlu T. A characterization of compact linear mappings. Arch Math 1971; 22: 76-78.
- [20] Terzioğlu T. Remarks on compact and infinite-nuclear mappings. Mathematica Balkanica 1972; 2: 251-255.
- [21] Yosida K. Functional Analysis. Berlin, Germany: Springer-Verlag, 1965.