On basis properties of selfadjoint operator functions (Q1840575)

The authors consider a continuous selfadjoint operator-function \(L: [a,b]\to {\mathcal L}({\mathcal H})\), where \({\mathcal H}\) is a Hilbert space, \({\mathcal L}({\mathcal H})\) is the set of bounded closed linear operators on \({\mathcal H}\). Denote \[ {\mathcal G}(L):= \text{c.l.s.} \{\ker L(x)|x\in[a,b]\}, \] where c.l.s. stands for closed linear span. The authors give conditions [weaker then those in \textit{I. Krupnik}, Integral Equations Oper. Theory 14, No. 4, 545-551 (1991; Zbl 0767.47005)] under which every union of orthonormal bases of \(\ker L(x)\), \(x\in [a,b]\), is a Riesz basis of \({\mathcal G}(L)\). Also the authors prove Markus-Matsaev factorization theorem [\textit{A. Markus} and \textit{V. Matsaev}, Integral Equations Oper. Theory 16, No. 4, 539-564 (1993; Zbl 0778.47015)] under less restrictive conditions. Some other related results are obtained.