Fredholm and invertible $n$-tuples of operators. The deformation problem
HTML articles powered by AMS MathViewer
- by Raul E. Curto PDF
- Trans. Amer. Math. Soc. 266 (1981), 129-159 Request permission
Abstract:
Using J. L. Taylor’s definition of joint spectrum, we study Fredholm and invertible $n$-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson’s theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.References
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C^{\ast }$-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 58–128. MR 0380478
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^{\ast }$-algebras, operators with compact self-commutators, and $K$-homology, Bull. Amer. Math. Soc. 79 (1973), 973–978. MR 346540, DOI 10.1090/S0002-9904-1973-13284-7
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^*$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324. MR 458196, DOI 10.2307/1970999
- John Bunce, The joint spectrum of commuting nonnormal operators, Proc. Amer. Math. Soc. 29 (1971), 499–505. MR 283602, DOI 10.1090/S0002-9939-1971-0283602-7
- L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439. MR 322595, DOI 10.1512/iumj.1973.23.23036 R. E. Curto, Fredholm and invertible tuples of bounded linear operators, Ph.D. Dissertation, S.U.N.Y. at Stony Brook, New York, 1978.
- Raul E. Curto, On the connectedness of invertible $n$-tuples, Indiana Univ. Math. J. 29 (1980), no. 3, 393–406. MR 570688, DOI 10.1512/iumj.1980.29.29028
- Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893 —, The relation of Ext to $K$-theory, Symposia Mathematica 20 (1976), 513-529.
- R. G. Douglas and Roger Howe, On the $C^*$-algebra of Toeplitz operators on the quarterplane, Trans. Amer. Math. Soc. 158 (1971), 203–217. MR 288591, DOI 10.1090/S0002-9947-1971-0288591-1
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- J. William Helton and Roger E. Howe, Integral operators: commutators, traces, index and homology, Proceedings of a Conference Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 141–209. MR 0390829
- Jan Janas, Lifting of commutant of subnormal operators and spectral inclusion theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 6, 513–520 (English, with Russian summary). MR 511954
- A. S. Markus and I. A. Fel′dman, The index of an operator matrix, Funkcional. Anal. i Priložen. 11 (1977), no. 2, 83–84 (Russian). MR 0463965
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- Ernest Michael, Continuous selections. II, Ann. of Math. (2) 64 (1956), 562–580. MR 80909, DOI 10.2307/1969603 A. S. Mischenko, Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis, Russian Math. Surveys 31 (1976), 71-138.
- Joseph L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172–191. MR 0268706, DOI 10.1016/0022-1236(70)90055-8
- Joseph L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38. MR 271741, DOI 10.1007/BF02392329
- F.-H. Vasilescu, A characterization of the joint spectrum in Hilbert spaces, Rev. Roumaine Math. Pures Appl. 22 (1977), no. 7, 1003–1009. MR 500211 U. Venugopalkrisna, Fredholm operators associated with strongly pseudoconvex domains in ${{\mathbf {C}}^n}$, J. Funct. Anal. 9 (1973), 349-373.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 266 (1981), 129-159
- MSC: Primary 47A53
- DOI: https://doi.org/10.1090/S0002-9947-1981-0613789-6
- MathSciNet review: 613789