Інтегральні зображення додатно визначених ядер

• Дата: 2023
• Автор: Bokhonov, Yurii
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023
• Теми: гільбертовий простір; скалярний добуток; симетричний оператор; самоспряжений оператор; додатно визначене ядро; індекс дефекту; продовження оператора; Hilbert space; scalar product; symmetric operator; self-adjoint operator; positive definite kernel; defect index; operator extension
• Онлайн доступ: http://journal.iasa.kpi.ua/article/view/279777
• Назва журналу: System research and information technologies

The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting self-adjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane.