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Extension theory of operators in Krein, Pontryagin spaces, applications

Date:
August 18, 2016
Source:
University of Vaasa
Summary:
A new study investigates and generalizes several well-known results from Extension theory of operators. In a new doctoral dissertation, a researcher has been able to improve a couple of classical theorems known in the area, namely Shmul'yan theorem on completion of nonnegative block operators and Krein famous theorem on description of selfadjoint contractive extensions of a Hermitian contraction.  
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A new study in Mathematics from the University of Vaasa investigates and generalizes several well-known results from Extension theory of operators. In his doctoral dissertation, Dmytro Baidiuk has been able to improve a couple of classical theorems known in the area, namely Shmul'yan theorem on completion of nonnegative block operators and Krein famous theorem on description of selfadjoint contractive extensions of a Hermitian contraction.

Dmytro Baidiuk's PhD thesis in Mathematics investigates and generalizes several well-known results from Extension theory of operators.

The author has been able to improve a couple of classical theorems known in the area, namely Shmul'yan theorem on completion of nonnegative block operators and Krein famous theorem on description of selfadjoint contractive extensions of a Hermitian contraction. Using the key new results the author has proven various analogs for some other previously known results for not only wider classes of operators but also in the more general setting of Krein and Pontryagin spaces instead of a standard case of Hilbert spaces.

One of the approaches used in the dissertation is based on the notation of boundary triplets. In many cases, the modern boundary triplets' methods have appeared to offer a more convenient tool than standard methods of extension theory, for instance, when treating boundary value problems or various spectral and scattering properties of differential operators.

"Extension theory of operators offers a general framework for investigating and solving various formally other types of problems appearing in the area of mathematical analysis and mathematical physics. In particular, the results in my PhD thesis have applications in the perturbation and spectral theory of operators, in scattering theory and they provide for instance a method to solve Hamburger type moment problems and Nevanlinna-Pick type interpolation problems," says Dmytro Baidiuk who will defend his doctoral thesis at the University of Vaasa.

See the related report at: http://www.uva.fi/materiaali/pdf/isbn_978-952-476-687-6.pdf


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University of Vaasa. "Extension theory of operators in Krein, Pontryagin spaces, applications." ScienceDaily. ScienceDaily, 18 August 2016. <www.sciencedaily.com/releases/2016/08/160818090030.htm>.
University of Vaasa. (2016, August 18). Extension theory of operators in Krein, Pontryagin spaces, applications. ScienceDaily. Retrieved May 8, 2017 from www.sciencedaily.com/releases/2016/08/160818090030.htm
University of Vaasa. "Extension theory of operators in Krein, Pontryagin spaces, applications." ScienceDaily. www.sciencedaily.com/releases/2016/08/160818090030.htm (accessed May 8, 2017).