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DOI: 10.46698/w5172-0182-0041-c

# Partial Integral Operators of Fredholm Type on Kaplansky-Hilbert Module over $$L_0$$

Eshkabilov, Y. Kh. , Kucharov, R. R.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 3.
Abstract:
The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky--Hilbert module $$L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$$ over $$L_{0}\left(\Omega_{2}\right)$$. Some mathematical tools from the theory of Kaplansky--Hilbert module are used. In the Kaplansky--Hilbert module $$L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$$ over $$L_{0} \left (\Omega _ {2} \right)$$ we consider the partially integral operator of Fredholm type $$T_{1}$$ ($$\Omega_{1}$$ and $$\Omega_{2}$$ are closed bounded sets in $${\mathbb R}^{\nu_{1}}$$ and $${\mathbb R}^{\nu_{2}},$$ $$\nu_{1}, \nu_{2} \in {\mathbb N}$$, respectively). The existence of $$L_{0} \left (\Omega _ {2} \right)$$ nonzero eigenvalues for any self-adjoint partially integral operator $$T_{1}$$ is proved; moreover, it is shown that $$T_{1}$$ has finite and countable number of real $$L_{0}(\Omega_{2})$$-eigenvalues. In the latter case, the sequence $$L_{0}(\Omega_{2})$$-eigenvalues is order convergent to the zero function. It is also established that the operator $$T_{1}$$ admits an expansion into a series of $$\nabla_{1}$$-one-dimensional operators.
Keywords: partial integral operator, Kaplansky-Hilbert module, $$L_0$$-eigenvalue
For citation: Eshkabilov, Yu. Kh. and Kucharov, R. R. Partial Integral Operators of Fredholm Type on Kaplansky-Hilbert Module  over $$L_0$$,  Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 80-90. DOI 10.46698/w5172-0182-0041-c