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Factorization of Wiener-Hopf conservative integral operators. (English. Russian original) Zbl 0724.45002
Math. Notes 46, No. 1, 501-506 (1989); translation from Mat. Zametki 46, No. 1, 3-10 (1989).
In the factorization theory of the Wiener-Hopf integral operator \({\mathcal I}-{\mathcal K}\), where \(({\mathcal K}f)(x)=\int^{\infty}_{0}K(x- t)f(x)dt,\quad K\in L_ 1(-\infty,+\infty),\) the following nonlinear equations \(V_{\pm}(x)=K(\pm x)+\int^{\infty}_{0}V_{\mp}(t)V_{\pm}(x+t)dt,\quad x>0,\) play an important role. In the so-called conservative case, i.e. when \(K\geq 0,\quad \mu =\int^{+\infty}_{-\infty}K(x)dx=1,\) we have \(V_{\pm}\geq 0,\quad \gamma_{\pm}=\int^{\infty}_{0}V_{\pm}(x)dx\leq 1,\quad (1- \gamma_ -)(1-\gamma_+)=0.\) The author studies which of the numbers \(\gamma_{\pm}\) is equal to 1 and which is smaller than 1 in the case when \(\nu_{\pm}=\int^{\infty}_{0}xK(\pm x)dx=+\infty\).

MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
45P05 Integral operators
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