Samuel multiplicity and the structure of semi-Fredholm operators

Fang, Xiang

doi:10.1016/j.aim.2003.07.013

Cited by 20 publications

(35 citation statements)

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“…The space was first considered by Drury in [23] to generalize Von Neumann's inequality. From the point of view of Hilbert modules, the module has been comprehensively investigated by Arveson, we refer the reader to the references [1][2][3][4][5]7] and [26][27][28] for a far-reaching operatortheoretic and index-theoretic developments of the Hilbert module. Arveson raised a conjecture about the essentially normality or p-essentially normality ( p > d) of graded submodules of H 2 d ⊗ C r with finite multiplicity r [3][4][5][6][7].…”

Section: Each Graded Principal Submodule Is P-essentially Normalmentioning

confidence: 99%

Essentially normal Hilbert modules and K-homology

Guo

Wang

2007

Math. Ann.

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This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal-including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson's conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K -homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K -homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K -homology elements, and gives an explicit expression for K -homology invariant in dimension d = 2.

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Section: Each Graded Principal Submodule Is P-essentially Normalmentioning

confidence: 99%

Essentially normal Hilbert modules and K-homology

Guo

Wang

2007

Math. Ann.

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“…Roughly speaking, the one variable case [15] exhibits both algebraic and analytic aspects. For two variables, the first step taken in [18] deals with the algebraic aspect.…”

Section: Introductionmentioning

confidence: 99%

The Fredholm index of a pair of commuting operators, II

Fang

2009

Journal of Functional Analysis

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We first show that an inequality on Hilbert modules, obtained by Douglas and Yan in 1993, is always an equality. This allows us to establish the semi-continuity of the generalized Samuel multiplicities for a pair of commuting operators. Then we discuss the general structure of a Fredholm pair, aiming at developing a model theory. For application we prove that the Samuel additivity formula on Hilbert spaces of holomorphic functions is equivalent to a generalized Gleason problem. As a consequence it follows the additivity of Samuel multiplicity, in its full generality, on the symmetric Fock space. During the course we discover that a variant e (·) of the classic algebraic Samuel multiplicity might be more suitable for Hilbert modules and can lead to better results. Published by Elsevier Inc.

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“…Recall that a pure shift is a left-invertible operator which is also analytic [2]. The proof of Corollary 3 is essentially contained in the proof of Theorem 2.…”

Section: Corollary 3 If T ∈ B(h) Is An Operator With the Cellular Inmentioning

confidence: 99%

“…We first recall the 4 × 4 upper-triangular matrix model of semi-Fredholm operators developed in [2] which we rely on heavily.…”

Section: Corollary 3 If T ∈ B(h) Is An Operator With the Cellular Inmentioning

confidence: 99%