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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
Language: English Download the full text  
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
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