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On the Friedrichs extension of semi-bounded difference operators. (English) Zbl 0813.47038

H. Kalf [J. Lond. Math. Soc., II. Ser. 17, 511-521 (1978; Zbl 0406.34029)] obtained a characterization of the Friedrichs extension \(T_ F\) of a general semi-bounded Sturm-Liouville operator \(T\) under the assumptions that the coefficients being those necessary for \(T\) to be bounded. In the present paper the authors have obtained an analogue of Kalf’s result for the minimal operator \(T_ 0\) defined by the second- order difference expression \[ (Mx)_ n= (1/w_ n) [-\Delta(p_{n- 1} \Delta x_{n-1})+ q_ n x_ n],\quad (n\in N_ 0,\;p_{-1}= 0) \] in the Hilbert space \(\ell^ 2_ w\) of complex sequences \(x= \{x_ n\}^ \infty_ 0\) such that \(\sum^ \infty_{n= 0} | x_ n|^ 2 w_ n< \infty\), where \(\Delta x_ n= x_{n+ 1}- x_ n\) and \(\{p_ n\}^ \infty_ 0\), \(\{q_ n\}^ \infty_ 0\) and \(\{w_ n\}^ \infty_ 0\) are real sequences with \(p_ n> 0\), \(w_ n> 0\) \((n\in N_ 0)\). The main results deal with the characterization of the Friedrichs extension of Sturm-Liouville operators in terms of a principal solution and a non-principal solution. In the process of proofs the authors have obtained some interesting discrete analogues of results given in Kalf’s paper.

MSC:

47B39 Linear difference operators
47A20 Dilations, extensions, compressions of linear operators

Citations:

Zbl 0406.34029
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References:

[1] DOI: 10.1112/jlms/s2-31.3.501 · Zbl 0615.34019 · doi:10.1112/jlms/s2-31.3.501
[2] Kauffman, The deficiency index problem for powers of ordinary differential expressions 621 (1977) · Zbl 0367.34014 · doi:10.1007/BFb0064277
[3] Atkinson, Discrete and Continuous Boundary Problems (1964) · Zbl 0117.05806
[4] DOI: 10.1016/0022-247X(78)90088-4 · Zbl 0392.39001 · doi:10.1016/0022-247X(78)90088-4
[5] Hartman, Ordinary Differential Equations (1964) · Zbl 0125.32102
[6] DOI: 10.1112/jlms/s2-17.3.511 · Zbl 0406.34029 · doi:10.1112/jlms/s2-17.3.511
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