Operator Theory in Harmonic and Non-commutative Analysis: 23rd International Workshop in Operator Theory and its Applications, Sydney, July 2012

260 Pages·2014·2.49 MB·English

by  Joseph A. Ball, Michael A. Dritschel, A.F.M. ter Elst, Pierre Portal, Denis Potapov (ed.)

Operator Theory
Advances and Applications
Joseph A. Ball
Michael A. Dritschel
A.F.M. ter Elst
Pierre Portal
Denis Potapov
Editors
Operator Theory
in Harmonic and
Non-commutative
Analysis
23rd International Workshop in
Operator Theory and its Applications,
Sydney, July 2012
Operator Theory: Advances and Applications
Volume 240

Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marinus A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Wien, Austria)
Christiane Tretter (Bern, Switzerland)

Associate Editors: Honorary and Advisory Editorial Board:
Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA)
Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA)
Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA)
B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA)
Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada)
Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA)
Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA)
Leonid E. Lerer (Haifa, Israel) Bernd Silbermann (Chemnitz, Germany)
Vern Paulsen (Houston, TX, USA) Harold Widom (Santa Cruz, CA, USA)
Mihai Putinar (Santa Barbara, CA, USA)
Leiba Rodman (Williamsburg, VA, USA)
Ilya M. Spitkovsky (Williamsburg, VA, USA)

Subseries
Linear Operators and Linear Systems
Subseries editors:
Daniel Alpay (Beer Sheva, Israel)
Birgit Jacob (Wuppertal, Germany)
André C.M. Ran (Amsterdam, The Netherlands)

Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
Ingo Witt (Göttingen, Germany)
Joseph A. Ball • Michael A. Dritschel • A.F.M. ter Elst
Pierre Portal • Denis Potapov
Editors
Operator Theory in Harmonic
and Non-commutative Analysis
23rd International Workshop in Operator Theory
and its Applications, Sydney, July 2012
Editors
Joseph A. Ball Michael A. Dritschel
Department of Mathematics Department of Mathematics
Virginia Polytechnic Institute University of Newcastle upon Tyne
Blacksburg, VA, USA Newcastle upon Tyne, UK
A.F.M. ter Elst Pierre Portal
Department of Mathematics Mathematical Sciences Institute
University of Auckland Université Lille 1
Auckland, New Zealand Villeneuve d’Ascq, France
and
Denis Potapov The Australian National University
School of Mathematics and Statistics Canberra, ACT, Australia
University of New South Wales
Sydney, NSW, Australia
ISSN 0255-0156 ISSN 2296-4878 (electronic)
ISBN 978-3-319-06265-5 ISBN 978-3-319-06266-2 (eBook)
DOI 10.1007/978-3-319-06266-2
Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014942552

Mathematics Subject Classification (2010): 30E05, 30H20, 34B24, 34C25 , 34K13 , 34L05, 34L40, 35Q58,
42A45, 42B37, 46L53, 47A10, 47A13, 47A20, 47A40, 47A48, 47A55, 47A60, 47A75, 47B07, 47B10, 47B20,
47B33, 47B35, 47B38, 47B40, 47D06, 47L20, 60H15

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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
A. Amenta
Tent Spaces over Metric Measure Spaces under Doubling
and Related Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
P. Auscher and S. Stahlhut
Remarks on Functional Calculus for Perturbed First-order
Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
W. Bauer,C. Herrera Yan˜ez and N. Vasilevski
(m, λ)-Berezin Transform and Approximation of Operators on
Weighted Bergman Spaces over the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . 45
C.C. Cowen, S. Jung and E. Ko
Normal and Cohyponormal Weighted Composition
2
Operators on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
R.E. Curto, I.S. Hwang and W.Y. Lee
A Subnormal Toeplitz Completion Problem . . . . . . . . . . . . . . . . . . . . . . . . . 87
S. Dey and K.J. Haria
Generalized Repeated Interaction Model and Transfer Functions . . . . 111
F. Gesztesy and R. Weikard
Some Remarks on the Spectral Problem Underlying the
Camassa–Holm Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
G. Godefroy
Remarks on Spaces of Compact Operators between Reflexive
Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
B. Jefferies
Harmonic Analysis and Stochastic Partial Differential Equations:
The Stochastic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
vi Contents
S. Patnaik and G. Weiss
Subideals of Operators – A Survey and Introduction
to Subideal-Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
W.J. Ricker
p
Multipliers and L -operator Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
A. Skripka
Taylor Approximations of Operator Functions . . . . . . . . . . . . . . . . . . . . . . . 243
Introduction
The XXIII International Workshop on Operator Theory and its Applications
(IWOTA 2012) was held at the University of New South Wales (Sydney, Aus-
tralia) from 16 July to 20 July 2012. With 140 participants from all parts of the
world, and 22 plenary speakers representing many different branches of operator
theory, the meeting was a great success. Building on the strengths of Australian
mathematical analysis, the meeting focused on the role of operator theory in har-
monic and non-commutative analysis. Other themes were also well represented,
from pure operator theory in Banach spaces through to engineering applications.
The meeting certainly demonstrated the unity within the diversity of the field with
discussions highlighting many connections between different branches of operator
theory. It was financially supported by the Australian Mathematical Sciences Insti-
tute (AMSI), the University of New South Wales, the Centre for Mathematics and
its Applications of the Australian National University, and the National Science
Foundation of the USA. This support was vital and is gratefully acknowledged.
This volume contains the proceedings of the conference. It reflects the quality
and the diversity of the research presented at IWOTA 2012. Each paper has been
carefully refereed and has only been accepted if it meets the standards of the
journal Integral Equations and Operator Theory. We are very thankful to the
authors and the referees for their contributions.
The editors:
Joe Ball, Michael Dritschel, Tom ter Elst,
Pierre Portal, and Denis Potapov.
Operator Theory:
Advances and Applications, Vol. 240, 1–29
⃝c 2014 Springer International Publishing Switzerland
Tent Spaces over Metric Measure Spaces
under Doubling and Related Assumptions
Alex Amenta
Abstract. In this article, we define the Coifman–Meyer–Stein tent spaces
p,q,α
T (X) associated with an arbitrary metric measure space (X, d, μ) un-
der minimal geometric assumptions. While gradually strengthening our geo-
metric assumptions, we prove duality, interpolation, and change of aperture
theorems for the tent spaces. Because of the inherent technicalities in dealing
with abstract metric measure spaces, most proofs are presented in full detail.
Mathematics Subject Classification (2010). 42B35.
Keywords. Duality, change of aperture, complex interpolation, volume dou-
bling, Hardy–Littlewood maximal operator.
1. Introduction
The purpose of this article is to indicate how the theory of tent spaces, as devel-
oped by Coifman, Meyer, and Stein for Euclidean space in [7], can be extended
to more general metric measure spaces. Let X denote the metric measure space
under consideration. If X is doubling, then the methods of [7] seem at first to
carry over without much modification. However, there are some technicalities to
be considered, even in this context. This is already apparent in the proof of the
atomic decomposition given in [17].
Further still, there is an issue with the proof of the main interpolation result
of [7] (see Remark 3.20 below). Alternate proofs of the interpolation result have
since appeared in the literature – see for example [12], [4], [6], and [14] – but
these proofs are given in the Euclidean context, and no indication is given of their
general applicability. In fact, the methods of [12] and [4] can be used to obtain a
partial interpolation result under weaker assumptions than doubling. This result
relies on some tent space duality; we show in Section 3.2 that this holds once we
assume that the uncentred Hardy–Littlewood maximal operator is of strong type
1
(r, r) for all r > 1.
Supported by the Australian Research Council Discovery grant DP120103692.
1This fact is already implicit in [7].
2 A. Amenta
Finally, we consider the problem of proving the change of aperture result when
X is doubling. The proof in [7] implicitly uses a geometric property of X which
we term (NI), or ‘nice intersections’. This property is independent of doubling,
but holds for many doubling spaces which appear in applications – in particular,
all complete Riemannian manifolds have ‘nice intersections’. We provide a proof
which does not require this assumption.
2. Spatial assumptions
Throughout this article, we implicitly assume that (X, d, μ) is a metric measure
space; that is, (X, d) is a metric space and μ is a Borel measure on X. The ball
centred at x ∈ X of radius r > 0 is the set
B(x, r) := {y ∈ X : d(x, y) < r},
and we write V (x, r) := μ(B(x, r)) for the volume of this set. We assume that the
2
volume function V (x, r) is finite and positive; one can show that V is automati-
cally measurable on X × R+.
There are four geometric assumptions which we isolate for future reference:
(Proper): a subset S ⊂ X is compact if and only if it is both closed and bounded,
and the volume function V (x, r) is lower semicontinuous as a function of
3
(x, r);
(HL): the uncentred Hardy–Littlewood maximal operator M, defined for measur-
able functions f on X by
∫
1
M(f)(x) := sup |f(y)| dμ(y) (1)
B∋x μ(B) B
where the supremum is taken over all balls B containing x, is of strong type
(r, r) for all r > 1;
(Doubling): there exists a constant C > 0 such that for all x ∈ X and r > 0,
V (x, 2r) ≤ CV (x, r);
(NI): for all α, β > 0 there exists a positive constant cα,β > 0 such that for all
r > 0 and for all x, y ∈ X with d(x, y) < αr,
μ(B(x, αr) ∩ B(y, βr))
≥ cα,β.
V (x, αr)
We do not assume that X satisfies any of these assumptions unless men-
tioned otherwise. However, readers are advised to take (X, d, μ) to be a complete
Riemannian manifold with its geodesic distance and Riemannian volume if they
are not interested in such technicalities.
2
Since X is a metric space, this implies that μ is σ-finite.
3
Note that this is a strengthening of the usual definition of a proper metric space, as the usual
definition does not involve a measure. We have abused notation by using the word ‘proper’ in
this way, as it is convenient in this context.