7 results
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2. The theory of pure operations
- Author
-
C. M. Edwards
- Subjects
Discrete mathematics ,Linear operators ,Physical system ,Complex system ,Statistical and Nonlinear Physics ,Combinatorics ,Linear map ,symbols.namesake ,Von Neumann algebra ,81.46 ,Norm (mathematics) ,Ordered vector space ,46A40 ,symbols ,Mathematical Physics ,Mathematics ,Quantum computer - Abstract
This paper continues the study started in [13] where classes of operations were investigated in the partially ordered vector space approach to the theory of statistical physical systems. In this approach the set of states is represented by a norm closed generating coneK in a complete base norm space (V, K, B) and the set of operations is represented by the setP of positive norm non-increasing linear operators onV. In actual physical experiments it is usually the case that only certain subsets ofK are available and it is supposed that the set Γ(K) of such subsets is the set of split faces ofK. The properties of two important classes of operation are examined. The first classP of strong operations has the property that each member leaves every element of Γ(K) invariant and therefore can be measured in every restricted situation. The second classPP of pure operations has the property above and also sends pure states into pure states. A study is made, in terms of the structure of Γ(K), of when such operations are physically relevant. The paper ends with an examination of Γ(K),P, PP in the Von Neumann algebra model.
- Published
- 1972
3. Multiplicative perturbation of semigroup generators
- Author
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Günter Lumer and Karl Gustafson
- Subjects
Discrete mathematics ,Pure mathematics ,Multiplicative perturbation ,Semigroup ,General Mathematics ,Infinitesimal ,Linear operators ,Banach space ,47A55 ,47B44 ,Bounded function ,47D05 ,Infinitesimal generator ,Contraction (operator theory) ,Mathematics - Abstract
Contraction semigroups of linear operators on a Banach space appear in a wide variety of physical and mathematical situations, and for this reason there has been much interest attached to studying the properties of such semigroups and their infinitesimal generators. In this paper we examine in some detail the question of when one can (left or right) multiply the infinitesimal generator A by another (bounded or unbounded) operator B and still preserve the generator property. Also, we state when polynomials in a generator retain the generator property. Our treatment is not exhaustive but is meant to be representative of the type of results which may be obtained.
- Published
- 1972
4. Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras
- Author
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F. F. Bonsall, J. Duncan, F. F. Bonsall, and J. Duncan
- Subjects
- Linear operators, Banach algebras, Normed linear spaces
- Abstract
The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.
- Published
- 1971
5. Numerical Ranges II
- Author
-
F. F. Bonsall, J. Duncan, F. F. Bonsall, and J. Duncan
- Subjects
- Normed linear spaces, Banach algebras, Linear operators
- Abstract
Numerical Ranges II is a sequel to Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras written by the same authors and published in this series in 1971. The present volume reflects the progress made in the subject, expanding and discussing topics under the general headings of spatial and algebra numerical ranges and further ranges.
- Published
- 1973
6. Topics in Operator Theory
- Author
-
Pearcy, Carl M. and Pearcy, Carl M.
- Subjects
- Linear operators
- Abstract
The five articles in this volume are expository in nature, and they all deal with various aspects of the theory of bounded linear operators on Hilbert space. The volume is very timely, because in the last year or two great progress has been made on hard problems in this field, and thus operator theory today is a very exciting part of mathematical research. One particular problem on which considerable progress has been made recently is the invariant subspace problem. This is the question whether every bounded linear operator on a separable, infinite-dimensional, complex Hilbert space $\mathcal H$ has a nontrivial invariant subspace. Even though this problem remains unresolved, there are some operators T on $\mathcal H$ for which the structure of a lattice of all invariant subspaces of T is even, and the first article in this volume, “invariant subspaces” by Donald Sarason, is added to a discussion of such operators. One of the interesting features of this lucid presentation is the interplay between operator theory and classical analysis. The second article is entitled “Weighted shift operators and analytic function theory” and was written by Allen Shields. He has taken essentially all of the information presently given about weighted shift operators (with scalar weights) and incorporated it into this comprehensive article. A central theme of the composition is the interaction between weighted shift operators and analytic function theory, and in an added bonus for the reader, the article contains a list of thirty-two interesting research problems. The third article in the volume is a treatise called “A version of multiplicity theory” by Arlen Brown. The problem treated is how to decide when two normal operators are unitarily equivalent. (Unitary equivalence is the analog for operators of the concept of isomorphism for groups, rings, etc.) The unitary equivalence problem for arbitrary operators is exceedingly difficult, but the theory of spectral multiplicity, which can be approached in several different ways, furnishes a reasonable complete set of unitary invariants for normal operators. The author focuses attention on the concept of a spectral measure, and his clear presentation of this circle of ideas should lead to a better understanding of multiplicity theory by beginners and experts alike. The fourth article in this volume, “Canonical models” by R. G. Douglas, is concerned with the theory of canonical models for operators on Hilbert space. The central underlying idea is that if T is any contraction operator on $\mathcal H$ (i.e., if the norm of T is at most 1), then there is a canonical construction that associates with T an operator $\mathrm{M}_\mathrm{T}$ that is unitarily equivalent to T, called its “canonical model”. One can therefore study T by studying $\mathrm{M}_\mathrm{T}$ instead, and this theory has made significant progress in the past ten years. The author, who has contributed substantially to the geometrization of this theory, exposes in his article various important components of the theory, and thereby gives the reader much insight into its successes and failures. The final article in this volume, “A survey of the Lomonosov technique in the theory of invariant subspaces” by Carl Pearcy and Allen Shields, is a survey of some new invariant-subspace theorems that resulted from the brilliant and elegant method of proof introduced by Victor Lomonosov early in 1973. Further study and refinement of this technique should lead to additional progress on the invariant subspace problem.
- Published
- 1974
7. Banach Lattices and Positive Operators
- Author
-
H.H. Schaefer and H.H. Schaefer
- Subjects
- Linear operators, Positive operators, Linear topological spaces, Banach lattices, Lattice theory
- Published
- 1974
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