Spectral properties of tensor products of linear operators. I

Ichinose, Takashi

doi:10.1090/S0002-9947-1978-0472915-2

Spectral properties of tensor products of linear operators. I
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by Takashi Ichinose

Trans. Amer. Math. Soc. 235 (1978), 75-113

DOI: https://doi.org/10.1090/S0002-9947-1978-0472915-2

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Abstract:

The aim of the present paper is to obtain, for tensor products of linear operators, their essential spectra in the sense of F. E. Browder, F. Wolf and M. Schechter and explicit formulae of their nullity, deficiency and index. The theory applies to $A \otimes I + I \otimes B$ and $A \otimes B$.

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
E. Balslev and T. W. Gamelin, The essential spectrum of a class of ordinary differential operators, Pacific J. Math. 14 (1964), 755–776. MR 171179
Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/61), 22–130. MR 209909, DOI 10.1007/BF01343363
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl. (2) 13 (1960), 185–264. MR 0113146
Bernhard Gramsch and David Lay, Spectral mapping theorems for essential spectra, Math. Ann. 192 (1971), 17–32. MR 291846, DOI 10.1007/BF02052728
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539

Résumé de la théorie métrique des produits tensoriels topologiques

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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
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
Takashi Ichinose, Operational calculus for tensor products of linear operators in Banach spaces, Hokkaido Math. J. 4 (1975), no. 2, 306–334. MR 374958, DOI 10.14492/hokmj/1381758773
Takashi Ichinose, Operators on tensor products of Banach spaces, Trans. Amer. Math. Soc. 170 (1972), 197–219. MR 322553, DOI 10.1090/S0002-9947-1972-0322553-1
M. A. Kaashoek and D. C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc. 169 (1972), 35–47. MR 312299, DOI 10.1090/S0002-9947-1972-0312299-8
Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
David Lay, Characterizations of the essential spectrum of F. E. Browder, Bull. Amer. Math. Soc. 74 (1968), 246–248. MR 221318, DOI 10.1090/S0002-9904-1968-11905-6
Dragan Miličić and Krešimir Veselić, On the boundary of essential spectra, Glasnik Mat. Ser. III 6(26) (1971), 73–78 (English, with Serbo-Croatian summary). MR 298448
Roger D. Nussbaum, Spectral mapping theorems and perturbation theorems for Browder’s essential spectrum, Trans. Amer. Math. Soc. 150 (1970), 445–455. MR 265967, DOI 10.1090/S0002-9947-1970-0265967-9
John Piepenbrink and Peter Rejto, Some singular Schrödinger operators with deficiency indices $(n^{2},\,n^{2})$, Duke Math. J. 41 (1974), 593–605. MR 348254
Michael Reed and Barry Simon, Tensor products of closed operators on Banach spaces, J. Functional Analysis 13 (1973), 107–124. MR 0348538, DOI 10.1016/0022-1236(73)90038-4
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 essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13 (1966), 205–215. MR 188798, DOI 10.1016/0022-247X(66)90085-0
Barry Simon, Quadratic form techniques and the Balslev-Combes theorem, Comm. Math. Phys. 27 (1972), 1–9. MR 321456
Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
Angus E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163 (1966), 18–49. MR 190759, DOI 10.1007/BF02052483
František Wolf, On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228. MR 107750, DOI 10.1002/cpa.3160120202