operatortheory and functional analysis have a long tradition, initially being guided by problems from mathematical physics and applied mathematics. Much of the work in Banach spaces from the 1930s onwards resulted fr...
ISBN:
(数字)9783764386481
ISBN:
(纸本)9783764386474
operatortheory and functional analysis have a long tradition, initially being guided by problems from mathematical physics and applied mathematics. Much of the work in Banach spaces from the 1930s onwards resulted from investigating how much real (and complex) variable function theory might be extended to fu- tions taking values in (function) spaces or operators acting in them. Many of the ?rst ideas in geometry, basis theory and the isomorphic theory of Banach spaces have vector measure-theoretic origins and can be credited (amongst others) to N. Dunford, I.M. Gelfand, B.J. Pettis and R.S. Phillips. Somewhat later came the ***,whichhavepervadedandin'uenced theshapeoffunctionalanalysisandthetheoryofvectormeasures/integrationever since. Today, each of the areas of functional analysis/operatortheory, Banach spaces, and vector measures/integration is a strong discipline in its own right. However, it is not always made clear that these areas grew up together as cousins and that they had, and still have, enormous in'uences on one another. One of the aims of this monograph is to reinforce and make transparent precisely this important point.
The book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. The main goal of the...
ISBN:
(数字)9783764387754
ISBN:
(纸本)9783764387747
The book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. The main goal of the study is to relate analytical invariants (in particular, the index) of such operators to topological invariants of the manifold itself. This problem can be solved by modern methods of noncommutative geometry. To make the book self-contained, the authors have included necessary geometric material (C*-algebras and their K-theory, cyclic homology, etc.).
Thepresentbookdealswithvarioustypesoffactorizationproblemsformatrixand operator functions. The problems appear in di'erent areasof mathematics and its applications. A uni'ed approach to treat them is developed...
详细信息
ISBN:
(数字)9783764382681
ISBN:
(纸本)9783764382674
Thepresentbookdealswithvarioustypesoffactorizationproblemsformatrixand operator functions. The problems appear in di'erent areasof mathematics and its applications. A uni'ed approach to treat them is developed. The main theorems yield explicit necessaryand su'cient conditions for the factorizations to exist and explicit formulas for the corresponding factors. Stability of the factors relative to a small perturbation of the original function is also studied in this book. The unifying theory developed in the book is based on a geometric approach which has its origins in di'erent ?elds. A number of initial steps can be found in: (1) the theory of non-selfadjoint operators, where the study of invariant s- spaces of an operator is related to factorization of the characteristic matrix or operator function of the operator involved, (2) mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a fact- ization of the associated transfer function, and (3) thefactorizationtheoryofmatrixpolynomialsintermsofinvariantsubspaces of a corresponding linearization. In all three cases a state space representation of the function to be factored is used, and the factors are expressed in state space form too. We call this approach the state space method. It hasa largenumber of *** instance, besides the areasreferred to above, Wiener-Hopf factorizations of some classes of symbols can also be treated by the state space method.
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