Operators on tensor products of Banach spaces

Ichinose, Takashi

doi:10.1090/S0002-9947-1972-0322553-1

Operators on tensor products of Banach spaces
HTML articles powered by AMS MathViewer

by Takashi Ichinose PDF

Trans. Amer. Math. Soc. 170 (1972), 197-219 Request permission

Abstract:

The present paper is a study of operators on tensor products of Banach spaces with the notion of maximal extensions introduced by G. Köthe such that the closure of a closable operator is its unique maximal extension. For a class of such operators the spectral mapping theorem is established. The results apply to the operator $A \otimes I + I \otimes B$ and give a new meaning to the method of separation of variables.

References

E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys. 22 (1971), 280–294. MR 345552
Erik Balslev, Spectral theory of Schrödinger operators of many-body systems with permutation and rotation symmetries, Ann. Physics 73 (1972), 49–107. MR 332037, DOI 10.1016/0003-4916(72)90314-4
Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
Robert Carroll, Problems in linked operators. I, Math. Ann. 151 (1963), 272–282. MR 166629, DOI 10.1007/BF01398238
Heinz Otto Cordes, Über die Spektralzerlegung von hypermaximalen Operatoren, die durch Separation der Variablen zerfallen. I, II, Math. Ann. 128 (1954), 257–289; 373–411 (1955) (German). MR 66562, DOI 10.1007/BF01360138
B. Friedman, An abstract formulation of the method of separation of variables, Proceedings of the conference on differential equations (dedicated to A. Weinstein), University of Maryland Book Store, College Park, Md., 1956, pp. 209–226. MR 0082638
Jesús Gil de Lamadrid, Uniform cross norms and tensor products of Banach algebras, Duke Math. J. 32 (1965), 359–368. MR 190714
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Takashi Ichinose, On the spectra of tensor products of linear operators in Banach spaces, J. Reine Angew. Math. 244 (1970), 119–153. MR 278096, DOI 10.1515/crll.1970.244.119

Operational calculus for tensor product of linear operators in Banach spaces

Tosio Kato, Note on fractional powers of linear operators, Proc. Japan Acad. 36 (1960), 94–96. MR 121666
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473

Topologische lineare Räume

33

Gottfried Köthe, General linear transformations of locally convex spaces, Math. Ann. 159 (1965), 309–328. MR 192347, DOI 10.1007/BF01362768

The theory of matrices

Lidia Maurin and Krzysztof Maurin, Spektraltheorie separierbarer Operatoren, Studia Math. 23 (1963), 1–29 (German). MR 155184, DOI 10.4064/sm-23-1-1-29
V. P. Mihaĭlov, The first boundary-value problem for a class of hypoelliptic equations, Mat. Sb. (N.S.) 63 (105) (1964), 238–264 (Russian). MR 0163066
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
Martin Schechter, On the spectra of operators on tensor products, J. Functional Analysis 4 (1969), 95–99. MR 0257782, DOI 10.1016/0022-1236(69)90024-x
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824