Yu.M. Berezansky - Böcker

This two-volume work is devoted to the methods of the spectral theory of operators and the important role they play in infinite-dimensional analysis and its applications. Central to this study is the theory of the expansion of general eigenfunctions for families of commuting self-adjoint or normal operators. This allows a consideration of commutative models which can be applied to the representation of various commutation relations. Also included is an explanation of the theory of hypercomplex systems with locally compact bases. Applications to harmonic analysis lead to a study of the infinite-dimensional moment problem which is connected to problems of axiomatic field theory, integral representations of positive definite functions and kernels with an infinite number of variables. Infinite-dimensional elliptic differential operators are also studied. Particular consideration is given to second quantization operators and their potential perturbations, as well as Dirichlet operators. Applications to quantum field theory and quantum statistical physics are described in detail.Different variants of the theory of infinite-dimensional distributions are examined and this includes a discussion of an abstract version of white noise analysis. This work should be of interest to research mathematicians and mathematical physicists with an interest in spectral theory and its applications.

This monograph is devoted to the theory of hypercomplex systems with locally compact basis. Such systems were introduced by Yu. Berezansky and S. Krein in the 1950s and are a generalization of the notion of a hypergroup (a family of generalized shift operators) which was introduced in the 1970s. The book gives an account of hypercomplex systems theory. After the introductory chapter, it treats the Lie theory of hypercomplex systems and examples. Topics covered include Fourier transforms, the Plancherel theorem, the Peter-Weyl theorem, representation theory, duality, Gelfand pairs, Sturm-Liouville operators, and Lie theory. New proofs of results concerning Tannaka-Krein duality and Gelfand pairs are given. On the basis of this theory, new approaches to the construction of harmonic analysis on well-known objects become possible.

This is the ?rst of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This?rstvolumeisdevotedtotheoperatortheoryandrelatedtopics. Itopens withthebiographypapersaboutM. G. Kreinandanumberofsurveypapersabout his work. The mainpartof the book consistsof originalresearchpaperspresenting the state of the art in operator theory and its application. The second volume of these proceedings, entitled Di?erential Operators and Mechanics, concerns other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences. The editors are sincerely grateful to the persons who contributed to the preparation of these proceedings: Sergei Marchenko, Myroslav Sushko, Kostyantyn Yusenko and Vladimir Zavalnyuk.Mark Grigorievich Krein, 1907-1989 Operator Theory: Advances and Applications, Vol. 190, xi-xx c 2009 Birkh. auser Verlag Basel/Switzerland Mark Grigorievich Krein (on his 100th birthday anniversary) V. M. Adamyan, D. Z. Arov, Yu. M. Berezansky, V. I. Gorbachuk, M. L. Gorbachuk, V. A. Mikhailets and A. M. Samoilenko April 3, 2007, is the l00th anniversary of the birth of Mark Grigorievich Krein, one of the most celebrated mathematicians of the 20th century, whose whole life was closely connected with Ukraine.

This is the second of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M.G. Krein. This second volume is devoted to the theory of di?erential operators and mechanics. It opens with the description of the conference and a number of survey papers about the work of M.G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of di?erential operators. The ?rst volume of these proceedings, entitled Operator Theory and Related Topics, concerns other aspects of the conference. The two volumes will be of - terest to a wide-rangeof readership in pure and applied mathematics, physics and engineering sciences. OperatorTheory: AdvancesandApplications,Vol.191, xi-xv c 2009Birkh. auserVerlagBasel/Switzerland The World Dimension of the Heritage of a Ukrainian Mathematician International Conference "Modern Analysis and Applications" (MAA - 2007) (April 9-14, 2007, Odessa) Yu.BerezanskyandV.Gorbachuk This forum has been dedicated to the centennial birthday anniversary of one of the most prominent mathematicians of the twentieth century Mark Gr- orievich Krein, a corresponding member of the Academy of Sciences of the Ukr. SSR (1907-1989).

This monograph is devoted to the theory of hypercomplex systems with locally compact basis. Such systems were introduced by Yu. Berezansky and S. Krein in the 1950s and are a generalisation of the notion of a hypergroup (a family of generalised shift operators) which was introduced in the 1970s. The book gives a state-of-the-art account of hypercomplex systems theory. After the introductory chapter, it treats the Lie theory of hypercomplex systems and examples. Topics covered include Fourier transforms, the Plancherel theorem, the Peter-Weyl theorem, representation theory, duality, Gelfand pairs, Sturm-Liouville operators, and Lie theory. New proofs of results concerning Tannaka-Krein duality and Gelfand pairs are given. On the basis of this theory, new approaches to the construction of harmonic analysis on well-known objects become possible. Audience: This volume will be of interest to researchers and graduate students involved in harmonic analysis and representation theory.