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Samuel multiplicity and the structure of semi-Fredholm operators. (English) Zbl 1070.47007

In this paper, the shift Samual multiplicity, denoted \(s_{mul}(T)\), and the backward shift Samual multiplicity, denoted \(bs_{mul}(T)\), are introduced for any semi-Fredholm operator \(T\) on a separable complex Hilbert space \(H\): \[ s_{mul}(T)=\text{ lim}_{t\to \infty} ({\text{dim}(H/T^k(H))/k)}, \]
\[ bs_{mul}(T)=\text{ lim}_{t\to \infty} (\text{dim} (\text{ker}(T^k)))/k). \] These numerical invariants refining the Fredholm index can be regarded as stabilized dimensions of the cokernel \(H/T(H)\) and kernel \(\ker (T)\), respectively. A geometric interpretation of \(s_{mul}(T)\) and \(bs_{mul}(T)\) leads to a local version of Apostol’s triangular representation theorem and to Gohberg’s punctured neighborhood theorem. Banach space operators are also considered.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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[1] Ambrozie, C.-G.; Vasilescu, F.-H., Banach space complexes, Mathematics and its Applications, Vol. 334 (1995), Kluwer: Kluwer Dordrecht
[2] Apostol, C., The correction by compact perturbation of the singular behavior of operators, Rev. Roumaine Math. Pures Appl., 21, 2, 155-175 (1976) · Zbl 0336.47012
[3] C. Apostol, L. Fialkow, D. Herrero, D. Voiculescu, Approximation of Hilbert Space Operators, Vol. II, Research Notes in Mathematics, Vol. 102, Pitman (Advanced Publishing Program), Boston, MA, 1984.; C. Apostol, L. Fialkow, D. Herrero, D. Voiculescu, Approximation of Hilbert Space Operators, Vol. II, Research Notes in Mathematics, Vol. 102, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0572.47001
[4] Brodmann, M. P.; Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, Vol. 60 (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0903.13006
[5] Bouldin, R., The triangular representation of C. Apostol, Proc. Amer. Math. Soc., 57, 2, 256-260 (1976) · Zbl 0336.47013
[6] Clark, D., On a similarity theory for rational Toeplitz operators, J. Reine Angew. Math., 320, 6-31 (1980) · Zbl 0441.47038
[7] Clark, D., On Toeplitz operators with loops, J. Operator Theory, 4, 1, 37-54 (1980) · Zbl 0468.47014
[8] J.B. Conway, A Course in Functional Analysis, 2nd Edition, Graduate Texts in Mathematics, Vol. 96, Springer, New York, 1990.; J.B. Conway, A Course in Functional Analysis, 2nd Edition, Graduate Texts in Mathematics, Vol. 96, Springer, New York, 1990. · Zbl 0706.46003
[9] Cowen, M.; Douglas, R., Complex geometry and operator theory, Acta Math., 141, 3-4, 187-261 (1978) · Zbl 0427.47016
[10] Cowen, M., Fredholm operators with the spanning property, Indiana Univ. Math. J., 35, 4, 855-895 (1986) · Zbl 0655.47018
[11] Curto, R. E., Fredholm and invertible \(n\)-tuples of operators. The deformation problem, Trans. Amer. Math. Soc., 266, 1, 129-159 (1981) · Zbl 0457.47017
[12] R.E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Some Recent Results in Operator Theory, Vol. II, Pitman Research Notes on Mathematics Series, Vol. 192, Longman, Harlow, 1988, pp. 25-90.; R.E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Some Recent Results in Operator Theory, Vol. II, Pitman Research Notes on Mathematics Series, Vol. 192, Longman, Harlow, 1988, pp. 25-90.
[13] Douglas, R. G.; Paulsen, V., Hilbert Modules over Function Algebras, Pitman Research Notes in Mathematics, Vol. 217 (1989), Longman: Longman London · Zbl 0686.46035
[14] Douglas, R. G.; Yan, K., Hilbert-Samuel polynomials for Hilbert modules, Indiana Univ. Math. J., 42, 811-820 (1993) · Zbl 0808.46068
[15] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry (1995), Springer: Springer New York · Zbl 0819.13001
[16] Eschmeier, J.; Putinar, M., Spectral decompositions and analytic sheaves, London Mathematical Society Monographs, New Series, Vol. 10 (1996), Oxford University Press: Oxford University Press New York · Zbl 0855.47013
[17] Fang, X., Hilbert polynomials and Arveson’s curvature invariant, J. Funct. Anal., 198, 2, 445-464 (2003) · Zbl 1040.47005
[18] X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, preprint.; X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, preprint. · Zbl 1061.46022
[19] Hartshorne, R., Local Cohomology, Lecture Notes in Mathematics, Vol. 41 (1967), Springer: Springer Berlin, New York
[20] Herrero, D., Approximation of Hilbert Space Operators, Vol. I, Research Notes in Mathematics, Vol. 72 (1982), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA
[21] Kato, T., Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 (1966), Springer: Springer New York, Inc., New York
[22] Roberts, P., Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Mathematics, Vol. 133 (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0917.13007
[23] Serre, J.-P., Local Algebra, Springer Monographs in Mathematics (2000), Springer: Springer Berlin
[24] Taylor, J., A joint spectrum for several commuting operators, J. Funct. Anal., 6, 172-191 (1970) · Zbl 0233.47024
[25] Taylor, J., The analytic-functional calculus for several commuting operators, Acta Math., 125, 1-38 (1970) · Zbl 0233.47025
[26] F.-H. Vasilescu, Analytic functional calculus and spectral decompositions (translated from the Romanian, Mathematics and its Applications (East European Series), 1. D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste Romania, Bucharest, 1982).; F.-H. Vasilescu, Analytic functional calculus and spectral decompositions (translated from the Romanian, Mathematics and its Applications (East European Series), 1. D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste Romania, Bucharest, 1982). · Zbl 0495.47013
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