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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.

##### MSC:
 34L05 General spectral theory of ordinary differential operators 47B25 Linear symmetric and selfadjoint operators (unbounded)
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##### References:
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