Linear Operators and Matrices: The Peter Lancaster Anniversary Volume

278 Pages·2002·7.66 MB·English

by  Peter Lancaster (auth.)

Operator Theory: Advances and
Applications
Vol. 130
Editor:
I. Gohberg
Editorial Office:
School of Mathematical H.G. Kaper (Argonne)
Sciences ST. Kuroda (Tokyo)
Tel Aviv University P. Lancaster (Calgary)
Ramat Aviv, Israel LE. Lerer (Haifa)
B. Mityagin (Columbus)
Editorial Board: V. V. Peller (Manhattan, Kansas)
J. Arazy (Haifa) J. D. Pincus (Stony Brook)
A. Atzmon (Tel Aviv) M. Rosenblum (Charlottesville)
J. A. Ball (Blacksburg) J. Rovnyak (Charlottesville)
A. Ben-Artzi (Tel Aviv) D. E. Sarason (Berkeley)
H. Bercovici (Bloomington) H. Upmeier (Marburg)
A. Böttcher (Chemnitz) S. M. Verduyn Lunel (Amsterdam)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L. A. Coburn (Buffalo) H. Widom (Santa Cruz)
K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville)
R. G. Douglas (Stony Brook) D. Yafaev (Rennes)
H. Dym (Rehovot)
A. Dynin (Columbus) Honorary and Advisory
P. A. Fillmore (Halifax) Editorial Board:
P. A. Fuhrmann (Beer Sheva) C. Foias (Bloomington)
S. Goldberg (College Park) P. R. Haimos (Santa Clara)
B. Gramsch (Mainz) T. Kailath (Stanford)
G. Heinig (Chemnitz) P. D. Lax (New York)
J. A. Helton (La Jolla) M. S. Livsic (Beer Sheva)
M.A. Kaashoek (Amsterdam)
Linear Operators
and Matrices
The Peter Lancaster Anniversary Volume
Israel Gohberg
Heinz Langer
Editors
Springer Basel AG
Editors:
Prof. I. Gohberg Prof. H. Langer
School of Mathematical Sciences Mathematik
Raymond and Beverly Sackler Technische Universität Wien
Faculty of Exact Sciences Wiedner Hauptstrasse 8-10/1411
Tel Aviv University 1040 Wien
Ramat Aviv 69978 Austria
Israel
2000 Mathematics Subject Classification 47-06
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Linear operators and matrices : the Peter Lancaster anniversery volume /
Israel Gohberg ; Heinz Langer ed.. - Basel; Boston ; Berlin : Birkhäuser, 2001
(Operator theory ; Vol. 130)
ISBN 978-3-0348-9467-8 ISBN 978-3-0348-8181-4 (eBook)
DOI 10.1007/978-3-0348-8181-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms
or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2002 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel, Switzerland in 2002
Printed on acid-free paper produced from chlorine-free pulp. TCF °o
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9467-8
98765432 1 www.birkhauser-science.com
Contents
Preface ................................................................... Vll
Portrait of Peter Lancaster ............................................... Vlll
P. Lancaster
My Life and Mathematics 1
List of Publications of Peter Lancaster. . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 9
R.K. Guy
Forty-four Years with Peter Lancaster .................... ... ... ..... 21
I. Gohberg
Peter Lancaster, my Friend and Co-author........................... 23
M. Adam and J. Maroulas
The Joint Numerical Range of Bordered and
Tridiagonal Matrices ................................................. 29
A.L. Andrew and R.C.E. Tan
Iterative Computation of Higher Derivatives of
Repeated Eigenvalues and the Corresponding Eigenvectors ........... 43
D. Alpay, T. Ya. Azizov, A. Dijksma, and J. Rovnyak
Colligations in Pontryagin Spaces with a Symmetric
Characteristic Function .............................................. 55
H. Bart, T. Ehrhardt, and B. Silbermann
Logarithmic Residues of Fredholm Operator Valued Functions
and Sums of Finite Rank Projections ................................ 83
R. Bhatia and L. Elsner
Positive Linear Maps and the Lyapunov Equation 107
P.A. Binding and R. Hryniv
Full- and Partial-Range Completeness ............................... 121
P.A. Binding, P.J. Browne, and B.A. Watson
Spectral Isomorphisms between Generalized
Sturm-Liouville Problems ............................................ 135
E. Bohl and l. Marek
Existence and Uniqueness Results for Nonlinear
Cooperative Systems ................................................ 153
J. Erlijman, D.R. Farenick, and R. Zeng
Young's Inequality in Compact Operators. . . . . . . . . . . . . . . . . . . . . . . . . . .. 171
VI Contents
I. Feldman, N. K'T"llpnik, and A. Markus
Partial Indices of Small Perturbations of a Degenerate
Continuons Matrix Function ......................................... 185
1. Gohberg, M.A. Kaashoek, and F. van Schagen
Finite Section Method for Difference Equations ...................... 197
C.-H. Guo
Iterative Solution of a Matrix Riccati Equation Arising in
Stochastic Control ................................................... 209
W. Kliem and C. Pommer
Lyapunov Functions and Solutions of the Lyapunov
Matrix Equation for Marginally Stable Systems ...................... 223
H. Langer, A.G.M. Ran, and B.A. van de Rotten
Invariant Subspaces of Infinite Dimensional Hamiltonians and
Solutions of the Corresponding Riccati Equations .................... 235
L. Lerer and L. Rodman
Inertia Bounds for Operator Polynomials ............................ 255
P. J. Psarrakos
A Note on the Level Sets of a Matrix Polynomial
and its Numerical Range ............................................ 277
Operator Theory:
Advances and Applications, Vol. 130, 1-7
© 2001 Birkhiiuser Verlag Basel/Switzerland
My Life and Mathematics
Peter Lancaster
I was born in Appleby, a small county town in the north of England, on November
14th, 1929. I had two older brothers and was to have one younger sister. My family
moved around the north of England as my father's work in an insurance company
required. My secondary education was at Sir John Deane's Grammar School in
Northwich, Cheshire, to age 16. Then it was completed at the Collegiate School,
Liverpool, from 1946 to 1948. At school my first loves were art and mathematics.
After an unsuccessful year in the School of Architecture of Liverpool University, I
joined the honours mathematics program in the same university. Although I won
scholarships, there was some financial insecurity and, at the time of this change of
subject, my continued study was made possible by the personal intervention and
support of the curate of my parish of West Derby, Liverpool, Rev. E. W. Pugh.
I graduated from Liverpool University with an honours degree in Mathematics in
1952. While still an undergraduate I was married to Edna Hutchinson in 1951. We
were to have three daughters.
At that time military conscription was normal after graduation, but employ
ment in the aircraft industry was an acceptable alternative. So I worked as an
aerodynamicist with the English Electric Corporation (later British Aerospace)
at Warton Aerodrome, Lancashire, from 1952 to 1957. In this time I worked on
aircraft vibration and stability problems. As well as mathematical and numerical
analysis, this work included analogue computing and also (the infancy of) high
speed digital computing. This experience provided me with a fund of problems for
more careful analysis which strongly influenced my research career - to the present
day. During this time I registered for an external M.Sc. program at the University
of Liverpool under the supervision of Louis Rosenhead, FRS, Professor of Applied
Mathematics. The degree was awarded in 1956 for a dissertation on the vibration
phenomenon known as "flutter", including formulation of the model problem and
methods of solution.
At the end of the required service in industry I applied for academic positions
and was appointed Assistant Lecturer in Mathematics at the University of Malaya
(then based in Singapore; formerly Raffles College and later to become the Na
tional University of Singapore). There were strong research traditions established
in part by Professors A. Oppenheim in Pure Mathematcs (later to become Vice
Chancellor and knighted for his services), and J. C. Cooke in Applied Mathemat
ics. Both had recently left the department, but their tradition was maintained by
the current Acting Head of Department, Richard K. Guy, who was to become a
2 Peter Lancaster
friend and colleague to the present day. Though the department was small, it was
active and had the advantage of a good university library which had been built
up from the founding of Raffles College in 1929. In this period I published several
papers on vibrations, iterative methods for solution of matrix polynomial eigen
value problems, and some more general questions of spectral structure and inverse
problems for matrix polynomials (see papers 1-8). Achievements in these years
were recognized with promotions in 1959 and 1961 (to Senior Lecturer). During
this time I was strongly influenced by friendly and helpful correspondence with A.
M. Ostrowski. Indeed, Ostrowski's style, publications, and assistance were a source
of inspiration. In 1961 there seemed to be enough results for a Ph.D. dissertation,
so I registered for a Ph.D. at the University of Singapore. The dissertation "On
the Theory of Lambda-matrices with Applications to the Analysis of Vibrating
Systems" was submitted and examined in 1962. One of the external examiners,
Ian Sneddon FRS, of the University of Glasgow, recommended publication of an
extended version of the dissertation as a monograph, and this appeared in 1966
(reference Bl). After unexplained procedural delays, the Ph.D. was awarded in
1964. On the other hand, the monograph remains a frequently quoted work, par
ticularly in mechanical engineering.
An important event in this "Singapore period" was the invitation to attend
and present a paper at one of the early meetings in Gatlinburg, Tennessee, and
known at that time as a "Matrix Symposium". This was in April, 1961, and was an
excellent opportunity to meet Ostrowski personally as well as other leaders in the
field. Some of these influenced my subsequent career and tastes in research. They
included Lothar Collatz, Alston Householder, Gilbert Strang, Olga Taussky, Jack
Todd, Dick Varga, and Jim Wilkinson. This was my first visit to North America,
and therefore full of interest. The experience was coloured, however, by a tense
situation at home as our two-year old daughter, Jane, went through a dangerous
illness.
On deciding to move back to the west from Singapore in 1962, Edna and I
were presented with an interesting choice: I had offers of appointment from my
alma mater, Liverpool University, and from the fledgling Calgary campus of the
University of Alberta (to become the University of Calgary in 1965). Our love
of the outdoors, and impatience with the structured English society, swung the
balance in favour of adventure and a new institution. So, with our two daughters,
aged five and two, the move to Calgary was made in November of 1962.
In the following years my research activities continued in matrix theory, it
erative methods, perturbation and spectral theory for matrix functions (papers
10-22). These were assisted by a one-year appointment (1965-1966) as a Visiting
Associate Professor at the California Institute of Technology. This was under the
aegis of Jack Todd and Olga Taussky.
I was aware of work on matrix and operator functions in Eastern Europe, and
particularly the Soviet Union from the late fifties, and my first direct contact was
probably with P. H. Muller of the TU Dresden. From this contact I learned of the
work of his former student, Heinz Langer, and of Heinz' continuing work with M. G.
My Life and Mathematics 3
Krein. Fortunately, Israel Halperin made arrangements for Heinz to visit Canada
for the 1966-1967 year. This included a visit to Calgary and a series of lectures
on his work with Krein. At about this time the monograph B1, which includes
some early results on spectral theory and factorization of matrix functions, came
to Krein's attention and, through the good offices W. K. Hayman, an exchange of
monographs was made in 1968. I was also impressed by the Krein-Gohberg book
on non-selfadjoint operators when it appeared in 1969 (in English translation).
In 1968-1969 I was fortunate to spend a sabbatical leave with Ostrowski in
Basel, Switzerland. This did not lead to active collaboration (he was, in general, a
solitary worker), but he was a gracious host. At that time, one of my main interests
was error analysis for Newton's method. I shared this with Jon Rokne, a graduate
student who joined us in Basel (see 13, 16, 21, and 27). In November of 1968 I
was able to attend an Oberwolfach meeting organized by Collatz and made the
acquaintance of several members of his school. I was very impressed by Collatz'
analytical approach to problems of engineering and numerical analysis. Among the
connections made at that time were lifelong friends and occasional collaborators,
Erich Bohl and Ludwig Elsner. Also, in the Spring of 1969 I visited Heinz Langer
in Dresden and learned a lot about life in eastern Europe in the communist era.
Lectures in Calgary during the sixties led to my writing a senior undergraduate
text in matrix theory. This was published in 1969 (reference B2) and, in a second
edition of 1985 with Miron Tismenetsky (B6), is still in print. The first edition
was translated into Russian and 60,000 copies were printed. As a result, my name
may be better known in the former Soviet Union than it is in the west.
During this period I took an active interest in approximation theory and,
particularly, in the emerging development of spline functions and finite element
methods (papers 24, 26, 31, 35, 41, 42, 49 and, later, B7). This was undertaken
largely in collaboration with my student, David Watkins, and colleague, Kes Salka
uskas. The work was advanced by a visit with Erich Bohl (in 1974) at the University
of Munster which included a lecture series on topics of this kind.
A pivotal event in my career took place in 1975. Israel Gohberg had emigrated
from the former Soviet Union in 1974 and, due to the initiative of Chandler Davis,
visited Canada in early 1975. It was arranged that he should visit Calgary (in
a snowy February) and this began a rich friendship and collaboration. I took a
sabbatical leave in Europe in 1975-1976 and, while at the University of Dundee,
we arranged to meet again. At this time I had just completed work which, to some
degree, combined earlier ideas and showed how to organize the spectral data for
a matrix polynomial in a concise and useful way (papers 36 and 37). Later in the
year I visited Israel Gohberg and his Ph.D. student, Leiba Rodman, in Tel Aviv.
From these beginnings a series of papers (38-40,43,44,47, 50,51,56,57,61) on
matrix functions evolved, and after many visits for collaboration in Canada and
Israel, three monographs were written (B4, B5, and B8). The first was a thorough
treatment of the theory of matrix polynomials based on our earlier papers, and
the second and third (on indefinite scalar product spaces and invariant subspaces,
respectively) developed topics of wider interest which grew naturally from the
4 Peter Lancaster
first to the second volume, and from the second to the third. These collaborations
included Leiba Rodman's appointment as a Post Doctoral Fellow in Calgary from
1978 to 1980.
During the late seventies, partly influenced by discussions with Harald Wim
mer of the University of Wiirzburg, my interest in systems theory and control
developed. In particular, it was clear that our methods might apply to analysis
of Riccati equations, and our first success in this direction, with Leiba Rodman,
was the solution of a basic existence problem for algebraic Riccati equations in
paper 45 (1980). This line of investigation continued with papers 64, 66, 69, 73. A
systematic approach to systems theory using the techniques of linear algebra was
one of the main themes of the monograph B8 (1986). I remember working on this
during a long stay in Tel Aviv as part of a sabbatical leave in 1982-1983. This leave
also included some months visiting the institute of Erich Bohl at the University of
Konstanz where I lectured on systems theory, as well as some time with the lively
group led by Rien Kaashoek at the Free University of Amsterdam. The book B10
(1995) on "Algebraic Riccati Equations", written with Leiba, was a high-point for
me in this research direction. It contains a comprehensive account of necessary
mathematical tools as well as eight chapters on "applications and connections"
with problem areas in systems theory, optimal control, and statistics. This led
to an interesting analysis of Newton's method when applied to matrix equations
which was undertaken with graduate student Guo Chun Hua (115 and 119). Most
recently, an invitation to present a lecture series in 1998 at Portugal's Centro In
ternacional de Matematica, Coimbra, led to new connections and experiences (see
BI2).
A productive line of investigation began around 1984 into structured matrices
and their applications in tomography, seismic inversion, and signal processing, as
well as the design of fast algorithms for these problem areas. This was stimulated
by further collaboration with Israel Gohberg and with Post-Doctoral Fellow, Israel
Koltracht (1984-1987). The papers 63, 70, 72, 74, 75, 79, 81, 82, 83, 86, 90 come
under this heading and span the period 1986-1992.
With graduate student, Qiang Ye, we returned to some classical questions
of variational characterization of spectra of matrix pencils and numerical linear
algebra with papers 80, 85, 91, and 94. It turned out that paper 91 (1993), which
appeared after some publication delays, contained an influential idea for further
developments of factorization theory for matrix functions to be discussed below.
Qiang represented my first significant connection with PR China and Guo Chun
Hua, mentioned above, my most recent connection. The first edition of my matrix
theory text is well-known in China (through a pirated edition) so, as in the USSR,
my name was quite familiar in China. In the mid-eighties I met Sun Ji Guang in
Bielefeld. He returned to Beijing and recommended to his M.Sc. student, Qiang,
that he study with me in Calgary for the Ph.D. Later, Dai Hua of Nanjing Uni
versity spent a year with me, and we worked intensively on matrix equations and
numerical methods for multiple eigenvalues and inverse eigenvalue problems (104,
110, and 111).