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On exit space extensions of symmetric operators with applications to first order symmetric systems. (English) Zbl 1289.47046

The author continues his work (V. Mogilevskii [Methods Funct. Anal. Topol. 12, No. 3, 258–280 (2006; Zbl 1125.47015)]) on the method of abstract boundary conditions in the extension theory of symmetric operators and relations with non-equal deficiency numbers. In particular, he gives a description, in terms of the Weyl functions, of the densely defined operators and the exit space operator-extensions (among relations-extensions). The general results are applied to (possibly, non-Hamiltonian) symmetric differential systems \(Jy'-B(t)y=\Delta (t)f(t)\), \(t\in [a,b]\), where \(B(t)=B(t)^*\), \(\Delta (t)\geq 0\). The system is considered in the quotient space corresponding to the quasi-inner product \(\langle f,g\rangle =\int\limits_a^b \Delta (t)f(t)[g(t)]^*\,dt\).

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47E05 General theory of ordinary differential operators
34B08 Parameter dependent boundary value problems for ordinary differential equations
34L05 General spectral theory of ordinary differential operators
47A06 Linear relations (multivalued linear operators)

Citations:

Zbl 1125.47015
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