Registration Number
0212U004172,
0107U000363
, R & D reports
Title
Investigation of analytic functions and ultra-smooth vectors on Banach spaces and their applications to the spectral operator theory.
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Head
Lopushansky O.V.,
Registration Date
31-01-2012
Organization
Pidstryhach Institute of Applied Problems of Mechanics and Mathematics NAS of Ukraine
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Object of study - algebras and Hardy spaces of analytic functions on Banach spaces, singularities of pseudo-holomorphic mappings of almost complex varieties, singular Sturm-Liouville and Dirac operators, mappings between Banach spaces, and topological semigroups Aim of the work - description of maximal ideals, differentiations, and homomorphisms of algebras of analytic functions over Banach spaces; construction of the functional calculus on the spectra of these algebras and development of the Hardy space theory of functions analytics on the Hilbert space unit balls; development of the functional calculus for unbounded operators in algebras of ultradistributions; investigation of properties of pseudo-holomorphic mappings of almost complex varieties; development of the continuation theory of holomorphic and meromorphic mappings with values on infinite-dimensional complex manifolds that is analogous to the finite-dimensional one; investigation of inverse problems for some class of operators of mathematical physics (Schr?dinger, Dirac etc); complete description of the spectral data and reconstruction algorithms for non-self-adjoint Sturm-Liouville operators, radial Schr?dinger and Dirac operators in bounded domains, and Bessel operators. The results obtained: analytic structures on the spectrum of the algebra of entire analytic functions of bounded type on a Banach space and on the spectrum of its symmetric subalgebras are described; Hardy spaces of analytic functions on the Banach space unit balls are described; polarization theorem for inhomogeneous polynomials on Banach spaces is proved; approximation of analytic functions on Banach spaces by polynomials is studied; the duality theory for ultrasmooth distributions is developed and functional calculus for unbounded operators (semigroup generators) in the convolution algebra of such distribution is constructed; theorems on singularity resolution for psuedoholomorphic curves on almost complex varieties are proved; analytic methods of the study of almost complex structures of low regularity are developed; spectral data for non-self-adjoint Sturm-Liouville operators, radial Schr?dinger and Dirac operators in bounded domains, and Bessel operators are completely described, and reconstruction algorithms for these operators are found and justified. All the results presented in the report are new and can be further continued to study operator algebras, analytic functions on Banach spaces, inverse spectral problems, in almost complex analysis and geometry of Banach spaces.
Product Description
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Івашкович С.М.
Гринів Р.О.
Загороднюк А.В.
Лозинська В.Я.
Лопушанський О.В.
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2020-04-02
