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The finite fibre problem and an index formula for elementary operators


Authors: Jörg Eschmeier and Mihai Putinar
Journal: Proc. Amer. Math. Soc. 123 (1995), 743-746
MSC: Primary 47A13; Secondary 47A53, 47B47
DOI: https://doi.org/10.1090/S0002-9939-1995-1219725-1
MathSciNet review: 1219725
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Abstract: Let $ J \subset L(K,H)$ be an operator ideal between Hilbert spaces, and let $ S \in L{(H)^n},T \in L{(K)^n}$ be commuting tuples of continuous linear operators. It is shown that the elementary operator $ R:J \to J,A \to \sum\nolimits_{i = 1}^n {{S_i}A{T_i}} $ determined by S and T satisfies the finite fibre property. As a consequence it follows that an index formula proved by L. Fialkow for elementary operators under the additional assumption of the finite fibre property holds true in general.


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  • [1] R. Curto, Spectral theory of elementary operators, Elementary Operators & Applications, Proceedings of the International Workshop (M. Mathieu, ed.), World Scientific, Singapore, 1992, pp. 3-52. MR 1183936 (93i:47041)
  • [2] R. Curto and L. Fialkow, The spectral picture of $ ({L_A},{R_B})$, J. Funct. Anal. 71 (1987), 371-392. MR 880986 (88c:47006)
  • [3] -, Elementary operators with $ {H^\infty }$-symbols, Integral Equations Operator Theory 10 (1987), 707-720. MR 904485 (88j:47016)
  • [4] J. Eschmeier, Analytic index formulas for tensor product systems, Proc. Roy. Irish Acad. Sect. A 87 (1987), 121-135. MR 941707 (89f:47020)
  • [5] -, Tensor products and elementary operators, J. Reine Angew. Math. 390 (1988), 47-66. MR 953676 (89h:47048)
  • [6] A. S. Fainshtein, Joint essential spectrum of a family of linear operators, Funktsional Anal. i Prilozhen. 14 (1980), 83-84; English transl., Functional Anal. Appl. 14 (1980), 152-153. MR 575225 (81f:47005)
  • [7] L. Fialkow, The index of an elementary operator, Indiana Univ. Math. J. 35 (1986), 73-102. MR 825629 (87h:47010)
  • [8] H. Grauert and R. Remmert, Coherent analytic sheaves, Springer-Verlag, Berlin, 1984. MR 755331 (86a:32001)
  • [9] R. N. Levy, Algebraic and topological K-functors of commuting n-tuple of operators, J. Operator Theory 21 (1989), 219-253. MR 1023314 (91g:47011)
  • [10] M. Putinar, Base change and the Fredholm index, Integral Equations Operator Theory 8 (1985), 674-692. MR 813356 (87j:47020)
  • [11] F.-H. Vasilescu, Analytic functional calculus and spectral decompositions, Reidel, Dordrecht, 1982. MR 690957 (85b:47016)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1219725-1
Article copyright: © Copyright 1995 American Mathematical Society

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