Publication: On the set of bounded linear operators transforming a certain sequence of a Hilbert space into an absolutely summable one
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1980
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North-Holland
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From the text: "Let H be a real, separable Hilbert space, B the set of bounded linear operators on H, and S={an:n∈N} a fixed sequence in H; we set CS={A∈B:∑∞n=1||Aan||<∞}. Obviously CS≠{0}, and it is easy to check that CS is a left ideal. Theorem 1: Let S={an:n∈N} be summable. Then CS contains a noncompletely continuous operator. Theorem 2: Let S={an:n∈N} be such that ∑∞n=1||an|||=∞; then there exists a completely continuous operator C not belonging to CS.''
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Proceedings of the 4th Colloquium on Topology in Budapest, 7-11 Aug. 1978, organized by the Bolyai János Mathematical Society
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J. W. Calkin, Two sided ideals and congruences in the ring of bounded operators in Hilbert spaces, Ann. of Math., 42(2)(1941). 839-873
C. Gohberg - A. Markus, Some relations between eigenvalues and matrix elements of linear operators, Math. Sbornik, 64 (106)(1964), 48-496
M. A. Naimark, Normed rings, Wolters Noordhoff publishing groingen, 1970, the Netherlands
A. Peiczynski, A characterization of Hilbert-Schmidt operators, Studia Mathematica, 28(1967)