Essential spectra of elementary operators

Fialkow, L. A.

doi:10.1090/S0002-9947-1981-0621980-8

Essential spectra of elementary operators
HTML articles powered by AMS MathViewer

by L. A. Fialkow PDF

Trans. Amer. Math. Soc. 267 (1981), 157-174 Request permission

Abstract:

This paper describes the essential spectrum and index function of the operator $X \to AXB$, where $A$, $B$, and $X$ are Hilbert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.

References

Joel Anderson, John W. Bunce, James A. Deddens, and J. P. Williams, $C^{\ast }$-algebras and derivation ranges, Acta Sci. Math. (Szeged) 40 (1978), no. 3-4, 211–227. MR 515202
Constantin Apostol, Inner derivations with closed range, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 3, 249–265. MR 410459

Some results on nonquasitriangular operators

18

Arlen Brown and Carl Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), 162–166. MR 188786, DOI 10.1090/S0002-9939-1966-0188786-5
Arlen Brown and Carl Pearcy, Compact restrictions of operators, Acta Sci. Math. (Szeged) 32 (1971), 271–282. MR 305119
J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 5790, DOI 10.2307/1968771
Chandler Davis and Peter Rosenthal, Solving linear operator equations, Canadian J. Math. 26 (1974), 1384–1389. MR 355649, DOI 10.4153/CJM-1974-132-6
R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
M. R. Embry and M. Rosenblum, Spectra, tensor products, and linear operator equations, Pacific J. Math. 53 (1974), 95–107. MR 353023
L. Fialkow, A note on the operator $X\rightarrow AX-XB$, Trans. Amer. Math. Soc. 243 (1978), 147–168. MR 502900, DOI 10.1090/S0002-9947-1978-0502900-3
Lawrence A. Fialkow, A note on norm ideals and the operator $X\longrightarrow AX-XB$, Israel J. Math. 32 (1979), no. 4, 331–348. MR 571087, DOI 10.1007/BF02760462
Lawrence A. Fialkow, A note on the range of the operator $X\rightarrow AX-XB$, Illinois J. Math. 25 (1981), no. 1, 112–124. MR 602902
L. A. Fialkow, Elements of spectral theory for generalized derivations, J. Operator Theory 3 (1980), no. 1, 89–113. MR 565753
Lawrence A. Fialkow, Elements of spectral theory for generalized derivations. II. The semi-Fredholm domain, Canadian J. Math. 33 (1981), no. 5, 1205–1231. MR 638376, DOI 10.4153/CJM-1981-091-4

Spectral theory of elementary multiplications

I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
Donald W. Hadwin, An asymptotic double commutant theorem for $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 244 (1978), 273–297. MR 506620, DOI 10.1090/S0002-9947-1978-0506620-0
P. R. Halmos, Quasitriangular operators, Acta Sci. Math. (Szeged) 29 (1968), 283–293. MR 234310
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Gunter Lumer and Marvin Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41. MR 104167, DOI 10.1090/S0002-9939-1959-0104167-0
Richard S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. MR 0198494
Carl M. Pearcy, Some recent developments in operator theory, Regional Conference Series in Mathematics, No. 36, American Mathematical Society, Providence, R.I., 1978. MR 0487495
Marvin Rosenblum, On the operator equation $BX-XA=Q$, Duke Math. J. 23 (1956), 263–269. MR 79235
Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338