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On the reduction of general Wiener-Hopf operators. (English) Zbl 1476.47017

Bart, Harm (ed.) et al., Operator theory, analysis and the state space approach. In honor of Rien Kaashoek. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 271, 399-419 (2018).
Operators \(S\) and \(T\) are equivalent in extended sense if they are equivalent: \(T=ESF\) for isomorphisms \(E\) and \(F\), or they have equivalent extensions in larger spaces. Given Banach spaces \(X\) and \(Y\) and a bounded linear operator \(A\in\mathcal{L}(X,Y)\), a Wiener-Hopf operator (WHO) is defined as \(W=\left.P_2 A\right|_{P_1X}\) with \(P_1\) and \(P_2\) projectors in \(X\) and \(Y\), respectively. An equivalent reduction of \(W\) is a simpler equivalent operator (possibly after extension) where simpler means that one or more of the following properties hold: (i) \(P_1=P_2\), (ii) \(X=Y\), (iii) \(A\) is invertible or a cross factor (i.e., \(A^{-1}P_2AP_1\) and \(AQ_1A^{-1}Q_2\) are projectors where \(Q_i=I-P_i\)). This paper formulates criteria for equivalent reductions of a WHO. First, some known results are recalled from the literature which are then generalized. These include, among others, extended symmetrization, symmetrization in separable Hilbert spaces, and reduction to the truncation of a cross factor. Subsequently reduction after extension is discussed and the particular case \(X=Y\) and \(A=I\) is illustrated. Applications of the formulas in this paper include, e.g., potential theory for solutions of elliptic BVP, and matricial as well as Schur coupling. The coupling applications relate to equivalence after (one-sided) extension. The paper concludes with some open problems.
For the entire collection see [Zbl 1411.47002].

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
00A27 Lists of open problems
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