Lectures on Invariant Subspaces grew out of a series of lectures given gave at the University of Uppsala in the spring of 1962, and again in Berkeley the following semester. Since the subject is rather loosely defined the lecture style seemed appropriate also for this written version. The book is written for a graduate student who knows a little, but not necessarily very much, about analytic functions and about Hilbert space. The book contains 11 lectures and begins with a discussion of analytic functions. This is followed by lectures covering invariant subspaces, individual theorems, invariant subspaces in Lp, invariant subspaces in the line, and analytic vector functions. Subsequent lectures cover vectorial function theory, inner functions, range functions, and factoring of operator functions.

In recent years there has been a large amount of work on invariant subspaces, motivated by interest in the structure of non-self-adjoint of the results have been obtained in operators on Hilbert space. Some the context of certain general studies: the theory of the characteristic operator function, initiated by Livsic; the study of triangular models by Brodskii and co-workers; and the unitary dilation theory of Sz. Nagy and Foia!? Other theorems have proofs and interest independent of any particular structure theory. Since the leading workers in each of the structure theories have written excellent expositions of their work, (cf. Sz.-Nagy-Foia!? [1], Brodskii [1], and Gohberg-Krein [1], [2]), in this book we have concentrated on results independent of these theories. We hope that we have given a reasonably complete survey of such results and suggest that readers consult the above references for additional information. The table of contents indicates the material covered. We have restricted ourselves to operators on separable Hilbert space, in spite of the fact that most of the theorems are valid in all Hilbert spaces and many hold in Banach spaces as well. We felt that this restriction was sensible since it eases the exposition and since the separable-Hilbert space case of each of the theorems is generally the most interesting and potentially the most useful case.

The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.

We generalize Beurling's theorem on the shift invariant subspaces of Hardy class H[superscript]2 of the unit disk to the Hardy classes of admissible Riemann surfaces. Essentially, an open Riemann surface is admissible if it admits enough bounded multiple valued analytic functions. The class of admissible surfaces contains many infinitely connected surfaces, and all finite surfaces, but does not contain all plane regions admitting sufficiently many bounded analytic functions to sseparatepoints. We generalize the ttheorem of A.H. Read and the Cauchy integral formula to the boundary values, on the Hayashi boundary, of functions in the Hardy classes of admissible surfaces.

This book contains lecture notes in pure and applied mathematics from the proceedings of an International Conference on Nonlinear Analysis and Applications, held at Memorial University of Newfoundland in June 1981. It includes information on fractional calculus and the Stieltjes transform.

This unique book addresses advanced linear algebra from a perspective in which invariant subspaces are the central notion and main tool. It contains comprehensive coverage of geometrical, algebraic, topological, and analytic properties of invariant subspaces. The text lays clear mathematical foundations for linear systems theory and contains a thorough treatment of analytic perturbation theory for matrix functions. Audience: appropriate for students, instructors, and researchers in applied linear algebra, linear systems theory, and signal processing. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex function theory.

Deals with various aspects of the theory of bounded linear operators on Hilbert space. This book offers information on weighted shift operators with scalar weights.

Much of the importance of mathematics lies in its ability to provide theories which are useful in widely different fields of endeavour. A good example is the large and amorphous body of knowledge known as the theory of linear operators or operator theory, which came to life about a century ago as a theory to encompass properties common to matrix, differential, and integral operators. Thus, it is a primary purpose of operator theory to provide a coherent body of knowledge which can explain phenomena common to the enormous variety of problems in which such linear operators play a part. The theory is a vital part of functional analysis, whose methods and techniques are one of the major advances of twentieth century mathematics and now play a pervasive role in the modeling of phenomena in probability, imaging, signal processing, systems theory, etc, as well as in the more traditional areas of theoretical physics and mechanics. This book is based on lectures presented at a meeting on operator theory and its applications held at the Fields Institute in 1994.

This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways. Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool. This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.