Operators of Class $C_0$ with Spectra in Multiply Connected Regions

Operators of Class $C_0$ with Spectra in Multiply Connected Regions PDF

Author: Adele Zucchi

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 66

ISBN-13: 0821806262

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In the present paper the author studies the analogue of the class [italic capital]C0 within a class of operators having a functional calculus based on the algebra of bounded holomorphic functions in a finitely connected domain with an analytic boundary. The latter class consists of the operators having the closure of the domain as a spectral set and having no normal direct summands with spectra contained in the boundary of the domain. (If the domain is the disk the preceding class reduces to the class of completely nonunitary contractions.) The basic properties known for the case of the disk, including the model theory, are established. The extension, even the mere construction of the functional calculus, is not routine, in part because it is unknown whether the analogue of Sz.-Nagy's dilation theorem is true in the author's multiply connected setting.

Controllability, Stabilization, and the Regulator Problem for Random Differential Systems

Controllability, Stabilization, and the Regulator Problem for Random Differential Systems PDF

Author: Russell Johnson

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 63

ISBN-13: 0821808656

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This volume develops a systematic study of time-dependent control processes. The basic problem of null controllability of linear systems is first considered. Using methods of ergodic theory and topological dynamics, general local null controllability criteria are given. Then the subtle question of global null controllability is studied. Next, the random linear feedback and stabilization problem is posed and solved. Using concepts of exponential dichotomy and rotation number for linear Hamiltonian systems, a solution of the Riccati equation is obtained which has extremely good robustness properties and which also preserves all the smoothness and recurrence properties of the coefficients. Finally, a general version of the local nonlinear feedback stabilization problem is solved.

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows PDF

Author: Wenxian Shen

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 111

ISBN-13: 0821808672

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This volume is devoted to the study of almost automorphic dynamics in differential equations. By making use of techniques from abstract topological dynamics, it is shown that almost automorphy, a notion which was introduced by S. Bochner in 1955, is essential and fundamental in the qualitative study of almost periodic differential equations.

Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space

Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space PDF

Author: Peter W. Bates

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 145

ISBN-13: 0821808680

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Extends the theory for normally hyperbolic invariant manifolds to infinite dimensional dynamical systems in a Banach space, thereby providing tools for the study of PDE's and other infinite dimensional equations of evolution. In the process, the authors establish the existence of center-unstable and center-stable manifolds in a neighborhood of the unperturbed compact manifold. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Lie Groups and Subsemigroups with Surjective Exponential Function

Lie Groups and Subsemigroups with Surjective Exponential Function PDF

Author: Karl Heinrich Hofmann

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 189

ISBN-13: 0821806416

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In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under nature reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are SL(2, R) and its universal covering group, almost abelian solvable Lie groups (ie. vector groups extended by homotheties), and compact Lie groups. This text will also be of interest to those working in algebra and algebraic geometry.

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders PDF

Author: Richard Warren

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 183

ISBN-13: 082180622X

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The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called k-connected set transitivity (k-CS-transitivity), are analysed in some detail. Classification in many of the interesting cases is given. This work generlizes Droste's classification of the countable k-transitive trees (k>1). In a CFPO, the structure can be branch downwards as well as upwards, and can do so repeatedely (though it neverr returns to the starting point by a cycle). Mostly it is assumed that k>2 and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behaviour.

Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball

Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball PDF

Author: Michael A. Dritschel

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 77

ISBN-13: 0821806513

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This memoir initiates a model theory-based study of the numerical radius norm. Guided by the abstract model theory of Jim Agler, the authors propose a decomposition for operators that is particularly useful in understanding their properties with respect to the numerical radius norm. Of the topics amenable to investigation with these tools, the following are presented: a complete description of the linear extreme points of the non-matrix (numerical radius) unit ball; several equivalent characterizations of matricial extremals in the unit ball, that is, those members which do not allow a nontrivial extension remaining in the unit ball; and applications to numerical ranges of matrices, including a complete parameterization of all matrices whose numerical ranges are closed disks.

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