Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

The Dirac Operator of a Commuting d-Tuple

Given a commuting d-tuple T=(T₁, …, T_d) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D_T. Significant attributes of the d-tuple are best expressed in terms of D_T, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does.     

Essentially normal Hilbert modules and <Emphasis Type="Italic">K</Emphasis>-homology

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.     

Representations of Hardy Algebras: Absolute Continuity,Intertwiners, and Superharmonic Operators

Suppose T₊(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H ^∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T₊(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H ^∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.     

Uniform version of Weyl–von Neumann theorem

We prove a “quantified” version of the Weyl–von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in Voiculescu’s theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C ₀(X), for a large class of spaces X.     

The Fredholm index of a pair of commuting operators, II

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.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Canonical Models for Representations of Hardy Algebras

We develop a model theory for completely non coisometric representations of the Hardy algebra of a W^*-correspondence defined over a von Neumann algebra. It follows very closely the model theory developed by Sz.-Nagy and Foiaş for studying single contraction operators on Hilbert space and it extends work of Popescu for row contractions.     

On free products in varieties of associative algebras

Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However, if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist infinitely many varieties for each of which a similar formula is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999.     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Characterization and structure of Finsler spaces with constant flag curvature

The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curvature along the Hilbert form on the projective sphere bundle attains identically its maximum (resp. Ricci scalar). The horizontal distributionH of this bundle is integrable if and only ifM has zero flag curvature. When a Finsler space has CFC, Hilbert form’s orthogonal complement in the horizontal distribution is also integrable. Moreover, the minimality of its foliations is equivalent to given Finsler space being Riemannian, and its first normal space is vertical Project supported by Wang KC Fundation of Hong Kong and the National Natural Science Foundation of China (Grant No. 19571005).     

Boundary Measures for Geometric Inference

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.     

Universal Jensen＇s Equations in Banach Modules over a C^＊-Algebra and Its Unitary Group

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen‘s equations in Banach modules over a unital C^*-algebra. It is applied to show the stability of universal Jensen‘s equations in a Hilbert module over a unital C^*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital C^*-algebra.     

Invariant Subspaces for Commuting Operators on a Real Banach Space

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

$${\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,$$

where ║ · ║ _e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

     

An example of a commutative basic algebra which is not an MV-algebra

Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.     

From Generalized Clifford Algebras to nambu’s formulation of dynamics

We consider a commutative part of the Generalized Clifford Algebras, denominated asalgebra of multicomplex numbers. By using the multicomplex algebra as underlying algebraic structure we construct oscillator model for the Nambu’s formulation of dynamics. We propose a new dynamicals principle which gives rise to two kinds of Hamilton-Nambu equations inD≥2-dimensional phase space. The first one is formulated with (D−1)-evolution parameter and a single Hamiltonian. The Haniltonian of the oscillator model in such dynamics is given byD-degree homogeneous form. In the second formulation, vice versa, the evolution of the system along a single evolution parameter is generated by (D−1) Hamiltonian.     

A Commutative Algebra for Oriented Matroids

Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K -algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: \circ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik—Terao, which is the main tool of our proofs. Received November 7, 2000, and in revised form May 18, 2001. Online publication November 2, 2001.     

Factorizations along commutative subspace lattices

A theorem of D. R. Larson on the factorization of positive-definite operators along complete, countable nests in a Hilbert space is generalized to the case of commutative subspace lattices: We characterize those selfadjoint, positive-definite operators which can be factored A^* A where A and A^–1 leave invariant the subspaces in a given countable, complete, commutative subspace lattice. Applications to inner-outer factorization theory and factorization of a positive-definite operator with respect an uncountable commutative subspace lattice are given. These results have applications to function theory on the polydisk, nonanticipative representations of Gaussian random fields, and multiparameter systems theory.