The Fredholm index of a pair of commuting operators

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities in commutative algebra. Then the theory is applied to the symmetric Fock space. In particular, our results imply a satisfactory answer to Arveson’s program on developing a Fredholm theory for pure d-contractions when d = 2, including both the Fredholmness problem and the calculation of indices. We also show that Arveson’s curvature invariant is in fact always equal to the Samuel multiplicity for an arbitrary pure d-contraction with finite defect rank. It follows that the curvature is a similarity invariant. Received: October 2004 Revision: May 2005 Accepted: May 2005 Partially supported by National Science Foundation Grant DMS 0400509.     

Grothendieck’s theorem for operator spaces

We prove several versions of Grothendieck’s Theorem for completely bounded linear maps T:E→F ^*, when E and F are operator spaces. We prove that if E, F are C ^*-algebras, of which at least one is exact, then every completely bounded T:E→F ^* can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as T=T _r +T _c where T _r (resp. T _c ) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on C ^*-algebras. Moreover, our result holds more generally for any pair E, F of “exact” operator spaces. This yields a characterization of the completely bounded maps from a C ^*-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual E ^* are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class. Oblatum 31-I-2002 & 3-IV-2002?Published online: 17 June 2002     

Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1, 1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges–Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges–Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of L² (\mathbbR, dn){L^2 (\mathbb{R}, d\nu)} onto a de Branges space of entire functions which acts as multiplication by a measurable function.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

A Characterization of Group Generators on Hilbert Spaces and the H ∞ -calculus

Abstract. We give necessary and sufficient conditions for an operator A on a Hilbert space to have a bounded H ^∈ fty -calculus on a vertical strip symmetric to the imaginary axis. From this, a characterization of group generators on Hilbert spaces is obtained yielding recent results of Liu and Zwart as corollaries.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras

We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.     

Frames and their associated $\emph{H}_{{\kern-2pt}\emph{F}}^{\emph{p}}$-subspaces

Given a frame F = {f _j } for a separable Hilbert space H, we introduce the linear subspace H^p_FH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ ^p -space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as H^p_FH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in H_F^pH_{F}^{p} converges in both the Hilbert space norm and the ||·|| _{F, p} -norm which is induced by the ℓ ^p -norm.     

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Pointwise fourier inversion: A wave equation approach

Functions of the Laplace operator F(− Δ) can be synthesized from the solution operator to the wave equation. When F is the characteristic function of [0, R ² ], this gives a representation for radial Fourier inversion. A number of topics related to pointwise convergence or divergence of such inversion, as R → ∞, are studied in this article. In some cases, including analysis on Euclidean space, sphers, hyperbolic space, and certain other symmetric spaces, exact formulas for fundamental solutions to wave equations are available. In other cases, parametrices and other tools of microlocal analysis are effective.     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

Frames and bases in tensor products of Hilbert spaces and Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results. For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.     

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.     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

On implemented semigroups

We shall be concerned with implemented semigroups of the form \U(t)X := T(t)XS(t) for X ∈ \L(F,E) and t ≥ 0 , where (T(t)) _{t ≥ 0} and (S(t)) _{t ≥ 0} are C ₀ -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator' with the help of inter- and extrapolation techniques. Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results. May 15, 2000     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.