Schauder bases and norm ideals of compact operators

Holub, J. R.

doi:10.1090/S0002-9939-1973-0317012-2

Schauder bases and norm ideals of compact operators
HTML articles powered by AMS MathViewer

by J. R. Holub PDF

Proc. Amer. Math. Soc. 38 (1973), 343-348 Request permission

Abstract:

This paper studies the relationship between unitarily invariant crossnorms on the tensor product of Hilbert spaces and the corresponding symmetric sequence spaces. It is shown that problems concerning the behavior of norm ideals of compact operators on Hilbert space can often be treated successfully by translating the problem to one concerned with basis theory (and hence, perhaps, to one more transparent). For example, Schatten has asked for necessary and sufficient conditions that the conjugate space of a minimal norm ideal again be a minimal norm ideal. Using the ideas developed here we give such a characterization.

References

C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281–1286. MR 147881
J. R. Holub, Compactness in topological tensor products and operator spaces, Proc. Amer. Math. Soc. 36 (1972), 398–406 (1973). MR 326458, DOI 10.1090/S0002-9939-1972-0326458-7

A representation theorem for topological tensor products, with applications

J. R. Holub, Hilbertian, Besselian, and semi-shrinking bases, Studia Math. 37 (1970/71), 203–211. MR 303262, DOI 10.4064/sm-37-3-203-211
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112
Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399
M. Zippin, A remark on bases and reflexivity in Banach spaces, Israel J. Math. 6 (1968), 74–79. MR 236677, DOI 10.1007/BF02771607