The limit spectrum of a positive operator in \(L_{2}\) that is integral on some subspace
Let \((X,\Omega,\mu)\) be a finite positive measure space which is non-atomic and let \(L^2(X)\) be a separable Hilbert space with norm \(\|\cdot\|_2\) and inner product \((\cdot|\cdot)\). If \(\Delta\) is a subspace of \(L^2(X)\), then an operator \({\mathcal K}:\Delta\to L^2(X)\) is defined to be an integral operator if there is a kernel \(K\) on \(X\times X\) for which the operator \({\mathcal K}\) may be represented in the form \[ {\mathcal K}(f)(x)= \int_X K(x,y) f(y)\,d\mu(y). \] A linear operator \(M: \text{Dom}(M)\to L^2(X)\) with dense domain \(\text{Dom}(M)\) has a limit spectrum which contains \(0\) (that is, \(0\in\sigma_e(M)\)) if there is an orthonormal sequence \(\{h_j\), \(j= 0,1,2,\dots\}\) such that \(\lim_{j\to\infty}\| M(h_j)\|_2= 0\). The main results of this paper include statements that, if \(S: L^2(X)\to L^2(X)\) is representable as an integral operator on a dense subspace \(Do(S)\) of \(L^2(X)\) and \(Do(S)\) is such that \(\chi_E Do(S)\subseteq Do(S)\) for each characteristic function \(\chi_E\) of a measurable set \(E\), then \(0\in\sigma_e(S)\). Examples provided in the introduction of the paper indicate that representations as integral operators alone would not imply \(0\in\sigma_e(S)\).
scientific article; zbMATH DE number 5816259
scientific article