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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:
  Everitt WN, In: Spectral Theory and Computational Methods of Sturm-Liouville Problems (Knoxville, TN, 1996) 191 pp pp. 211–249– (1997)  DOI: 10.1112/plms/s3-64.3.524 · Zbl 0723.34022 · doi:10.1112/plms/s3-64.3.524  Everitt WN, Memoirs of the American Mathematical Society 151 pp 67, 715– (2001)  DOI: 10.1080/0003681021000035506 · Zbl 1049.47043 · doi:10.1080/0003681021000035506  DOI: 10.1017/S0308210500000366 · Zbl 0961.34046 · doi:10.1017/S0308210500000366  Carlson R, Electronic Journal of Differential Equations 6 pp 1– (1998)  Carlson R, Electronic Journal of Differential Equations 2000 pp 1– (2000)  DOI: 10.1088/0305-4470/35/1/309 · Zbl 1012.81053 · doi:10.1088/0305-4470/35/1/309  Gesztesy F, Journal fur die Reine und Angewandte Mathematik 362 pp 28– (1985)  Sobhy El-Sayed I, The Rocky Mountain Journal of Mathematics 29 pp 9– (1999) · Zbl 0947.47004  Sobhy El-Sayed I, International Journal of Mathematics and Mathematical Sciences 9 pp 557– (2003) · Zbl 1023.34076  Sokolov MS, Electronic Journal of Differential Equations 75 pp 1– (2003)  Sokolov MS, Rocky Mt. J. Math.  Ashurov RR, In: Proceedings of the International Conference ”Modern Problems of Mathematical Physics and Information Technologies” 1 pp pp. 130–134– (2003)  Sokolov MS, Methods and Applications of Analysis 10 pp 513– (2003)  Naimark MA, Linear Differential Operators (1968)  Reed M, In: Functional Analysis 1 (1972)  Dunford N, In: Spectral Theory 2 (1964)
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