Richard Rochberg - Academia.edu (original) (raw)

Papers by Richard Rochberg

Proceedings of The London Mathematical Society, 2001

Page 1. BOUNDEDNESS OF HIGHER ORDER HANKEL FORMS, FACTORIZATION IN POTENTIAL SPACES AND DERIVATIO... more Page 1. BOUNDEDNESS OF HIGHER ORDER HANKEL FORMS, FACTORIZATION IN POTENTIAL SPACES AND DERIVATIONS WILLIAM COHN, SARAH H. FERGUSON and RICHARD ROCHBERG [Received 2 January 1999; revised 10 February 2000] ...

MATHEMATICA SCANDINAVICA, 2005

The symbols of nhboxthn^{\hbox{th}}nhboxth-order Hankel forms defined on the product of certain reproducing k... more The symbols of nhboxthn^{\hbox{th}}nhboxth-order Hankel forms defined on the product of certain reproducing kernel Hilbert spaces H(ki)H(k_{i})H(ki), i=1,2i=1,2i=1,2, in the Hilbert-Schmidt class are shown to coincide with the orthogonal complement in H(k1)otimesH(k2)H(k_{1})\otimes H(k_{2})H(k1)otimesH(k2) of the ideal of polynomials which vanish up to order nnn along the diagonal. For tensor products of weighted Bergman and Dirichlet type spaces (including the Hardy space) we introduce a higher order restriction map which allows us to identify the relative quotient of the nhboxthn^{\hbox{th}}nhboxth-order ideal modulo the (n+1)hboxst(n+1)^{\hbox{st}}(n+1)hboxst-order one as a direct sum of single variable Bergman and Dirichlet-type spaces. This generalizes the well understood 0hboxth0^{\hbox{th}}0hboxth-order case.

Proceedings of the London Mathematical Society, 2001

Journal of Functional Analysis, Sep 1, 2009

Transactions of the American Mathematical Society, Aug 1, 1984

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vecto... more The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville's theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

Pacific Journal of Mathematics, May 1, 1996

Suppose {Aβ}o<o<ι is a scale of Banach spaces generated from the couple (Ao,Aι) by complex interp... more Suppose {Aβ}o<o<ι is a scale of Banach spaces generated from the couple (Ao,Aι) by complex interploation. If T is a linear operator which is bounded on the couple then T is bounded on the entire scale. Also, associated to the scale is an operator Ωi which is generally nonlinear and unbounded on Aι/2 such that the commutator [T, Ωi] is bounded on A 1 / 2. Here we extend the construction and produce a sequence Ω2, Ω 3 ,... which are increasingly nonlinear and unbounded but such that certain combinations with T are bounded. The first example is [T,Ω 2 ]-Ωi[T,Ωi].

arXiv (Cornell University), Nov 1, 2014

We recall that the fundamental theorem of complex interpolation is the Boundedness Theorem: If, f... more We recall that the fundamental theorem of complex interpolation is the Boundedness Theorem: If, for j = 0, 1, a linear operator T is a bounded map from the Banach space X j to the Banach space Y j then, for each θ, 0 < θ < 1, T is a bounded map between the complex interpolation spaces [X 0 , X 1 ] θ and [Y 0 , Y 1 ] θ. Alberto Calderón, in his foundational presentation of this material fifty-one years ago [4], also proved the following companion result: Compactness Theorem/Question: Furthermore in some cases, if T is also a compact map from X 0 to Y 0 , then, for each θ, T is a compact map from [X 0 , X 1 ] θ to [Y 0 , Y 1 ] θ. The fundamental question of exactly which cases could be covered by such a result was not resolved then, and is still not resolved. In a previous paper [10] we surveyed several of the partial answers which have been obtained to this question, with particular emphasis on the work of Nigel Kalton in a joint paper [8] with one of us. This is a preliminary version of a set of lecture notes which will be a sequel to [10]. In them, for the most part, we will amplify upon various technical details of the contents of [8]. For example we plan to give a more explicit explanation of why the positive answer in [8] to the above question when (X 0 , X 1) is a couple of lattices holds without any restriction on those lattices, and we also plan to provide more detailed versions of some of the other proofs in that paper. The main purpose of this preliminary version is to present two apparently new small results, pointing out a previously unnoticed particular case where the answer to the above question is affirmative. As our title suggests, this and future versions of these notes are intended to be more accessible to graduate students than a usual research article.

arXiv (Cornell University), Oct 16, 2014

The fundamental theorem of complex interpolation is Boundedness Theorem: If, for j = 0, 1, a line... more The fundamental theorem of complex interpolation is Boundedness Theorem: If, for j = 0, 1, a linear operator T is a bounded map from the Banach space X j to the Banach space Y j then, for each θ, 0 < θ < 1, T is a bounded map between the complex interpolation spaces [X 0 , X 1 ] θ and [Y 0 , Y 1 ] θ. Alberto Calderón, in his foundational presentation of this material fifty-one years ago [5], also proved the following companion result: Compactness Theorem/Question: Furthermore in some cases, if T is also a compact map from X 0 to Y 0 , then, for each θ, T is a compact map from [X 0 , X 1 ] θ to [Y 0 , Y 1 ] θ. The fundamental question of exactly which cases could be covered by such a result was not resolved then, and is still not resolved. The paper [20], which is the focus of this commentary, is a contribution to that question. We will not summarize in any detail here the contents of [20] or of related works. (Some of that may be done later in [27].) Rather we will take this opportunity to sketch the mathematical world, historical and current, in which that paper lives. We will see that there have been many very talented contributors and many fine contributions; however the core problem remains open. We will at least be able to announce some small new partial results.

arXiv (Cornell University), Oct 1, 2010

Suppose H is a space of functions on X. If H is a Hilbert space with reproducing kernel then that... more Suppose H is a space of functions on X. If H is a Hilbert space with reproducing kernel then that structure of H can be used to build distance functions on X. We describe some of those and their interpretations and interrelations. We also present some computational properties and examples. 2010 Mathematics Subject Classification. 46E22.

We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete... more We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete description of the analogous sequences for a discrete model of the Dirichlet space.

Proceedings of the American Mathematical Society, 2001

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and character... more We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L 2 ( R ) L^{2}(\mathbf {R}) which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro, which act as unbounded multiplication operators on Bargmann-Segal space.

Mathematical Surveys and Monographs, 2019

Pacific Journal of Mathematics, 1996

Let φ : D-» R be a subharmonic function and let AL 2 φ (Ώ)) denote the closed subspace of L 2 (D,... more Let φ : D-» R be a subharmonic function and let AL 2 φ (Ώ)) denote the closed subspace of L 2 (D, e~2 φ dA) consisting of analytic functions in the unit disk D. For a certain class of subharmonic ψ) the necessary and sufficient conditions are obtained for the Toeplitz operator T μ on AL 2 φ (D) and the Hankel operator Hb on AL^iW) in order that they belong to the Schatten ideal S p .

Proceedings of the International Conference in honour of Jaak Peetre on his 65th birthday. Lund, Sweden August 17-22, 2000, 2002

We dedicate this paper to Jaak Peetre on occasion of his 65th birthday and to the memory of Tom W... more We dedicate this paper to Jaak Peetre on occasion of his 65th birthday and to the memory of Tom Wolff. Both helped shape the mathematics of our time and profoundly influenced our mathematical thoughts. Each, through his singular humanity, helped our hearts grow.

Function Spaces in Modern Analysis, 2011

We discuss the algebraic structure of the spaces of higher-order Hankel forms and of the spaces o... more We discuss the algebraic structure of the spaces of higher-order Hankel forms and of the spaces of higher-order commutators. In both cases we find a close relationship between the space of order n + 1 and the derivations of the underlying algebra of functions into the space of order n.