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On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space. (English) Zbl 1086.34067
Let \(\Omega\) be a finite or countable set of indices. On \(\Omega\), a multi-interval differential Everitt-Markus-Zettl (EMZ) system \(\{I_i, \tau_i\}_{i\in\Omega}\) can be considered, where \(I_i\) are arbitrary intervals of the real line and \(\tau_i\) are formally selfadjoint differential expressions of finite order. This EMZ system generates a family of Hilbert spaces \(\{L^2(I_i)=L_i^2\}_{i\in\Omega}\) and families of minimal \(\{T_{\text{ min},i}\}_{i\in\Omega}\) and maximal \(\{T_{\text{ max},i}\}_{i\in\Omega}\) differential operators. Consider a family \(\{T_i\}_{i\in\Omega}\) of selfadjoint extensions. Introduce the Hilbert space \(\mathbf{L}^2=\oplus_{i\in\Omega}L_i^2\), consisting of the vectors \(\mathbf{f}=\oplus_{i\in\Omega}f_i\) such that \(f_i\in L_i^2\) and \(\| \mathbf{f}\| ^2=\sum_{i\in\Omega}\| f_i\| _i^2=\sum_{i\in\Omega} \int_{I_i}| f_i| ^2\,dx<\infty\). In the space \(\mathbf{L}^2\) consider the operator \(T:\;D(T)\subseteq\mathbf{L}^2\to\text\textbf{L}^2\) defined on the domain \(D(T)=\left\{\mathbf{f}\subseteq\text\textbf{L}^2:\;\sum_{i\in\Omega}\| T_if_i\| _i^2<\infty\right\}\) by \(T\mathbf{f}=\bigoplus_ {i\in\Omega}T_if_i\). The operator \(T\) is called a selfadjoint differential vector-operator generated by the selfadjoint extensions \(T_i\), or simply a vector-operator. The operators \(T_i\) are called coordinate operators.
In this article, the authors show that spectral resolutions of differential vector-operators may be represented as a specific direct sum of integral operator with a kernel written in terms of generalized vector-operator eigenfunctions. Then, the authors prove that a generalized eigenfunction measurable with respect to the spectral parameter may be decomposed using a set of analytical defining systems of coordinate operators.

34L05 General spectral theory of ordinary differential operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI
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