On the localization of the spectrum for quasi-selfadjoint extensions of a Carleman operator. (English) Zbl 1265.45001

Bahri, S. M.

Math. Bohem. 137, No. 3, 249-258 (2012).

The author presents results concerning the spectrum \(\Sigma =\Sigma _{A_{\omega }}\) of the quasi-selfadjoint extensions \(A_{\omega }\) of a densely defined, symmetric integral operator of Carleman type, \[ Af(x)=\int _X K(x,y)f(y)\text{d}y \text{ with } \int _X | K(x,y)| ^2 \text{d}y<\infty \text{ a.e.} \] acting on \(L^2(X)\) with \(\sigma \)-finite measure d\(y\). The extensions have defect indices \((1,1)\), namely \( A\subset A_{\omega }\subset A^{*}\) with \(\dim (\text{dom}(A_{\omega })/\text{dom}(A))=1 \), and are parametrized over the set of all bounded analytic functions \(\omega \) on the upper half-plane. While by the general theory all the selfadjoint extensions of \(A\) have the same continuous spectrum \(\Sigma _c \), the difference \(\Sigma \setminus \Sigma _c \) may however ask for a specific study. An example in this sense is the main result of the paper, Theorem 1, that gives sufficient conditions in terms of the scalar spectral function \(\rho _A\) of \(A\) for \(\Sigma _{A_\omega }\) to have only a finite number of points in certain intervals \([a_p , a_{i(p)}]\), where \(a_p\) for \(p\geq 0\) are the coefficients of \(K(x,y)=\sum _{p=0}^\infty a_p \psi _p (x)\psi _p (y)\) with respect to an orthonormal basis \(\{\psi _p \}_{p=0}^{\infty }\) of \(L^2 (X)\), and \(i(p)=q\) is that integer \(q\geq 0\) such that \(a_p, a_q\) are consecutive. A second theorem states that if \(E\) is an arbitrary closed set such that \(E\subset [a_p,a_{i(p)}]\), there is a spectral function \(\rho \) from a certain class such that the spectrum \(\Sigma =\Sigma _{A_\omega }\) of the extension \(A_\omega \) having \(\rho \) as a spectral function satisfies \(\Sigma \cap [a_p ,a_{i(p)}]=E\).

Reviewer: C. G. Ambrozie (Praha)

MSC:

45C05	Eigenvalue problems for integral equations
45P05	Integral operators
47B25	Linear symmetric and selfadjoint operators (unbounded)
58C40	Spectral theory; eigenvalue problems on manifolds

Keywords:

defect index; quasi-selfadjoint extension; spectrum; symmetric integral operator of Carleman type