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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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##### References:
 [1] N. B. Engibaryan and A. S. Arutyunyan, ?Integal equations on a half-line with difference kernels, and nonlinear functional equations,? Mat. Sb.,97, No. 1, 35-58 (1975). · Zbl 0324.45005 [2] N. B. Engibaryan and L. G. Arabadzhyan, ?On nonlinear equations of factorization of Wiener-Hopf operators,? Preprint No. 01, Erevan State Univ., Erevan (1979). · Zbl 0582.45017 [3] L. G. Arabadzhyan and N. B. Engibaryan, ?Convolution equations and nonlinear functional equations,? Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vsesoyuznyi Institut Nauchnoi i Tekhnicheskoi Informatsii, Akad. Nauk SSSR, Moscow,22, 175-244 (1984). · Zbl 0568.45004 [4] F. Spitzer, ?The Wiener-Hopf equation whose kernel is a probability density,? Duke Math. J.,24, No. 3, 327-343 (1957);27, No. 3, 363-372 (1960). · Zbl 0082.32003 · doi:10.1215/S0012-7094-57-02439-0 [5] D. V. Lindley, ?The theory of queue with a single server,? Proc. Cambridge Philos. Soc.,48, 277-289 (1952). · Zbl 0046.35501 · doi:10.1017/S0305004100027638 [6] M. G. Krein and Yu. L. Shmul’yan, ?The Wiener-Hopf equations whose kernels admit integral representation in terms of expoentials,? Izv. Akad. Nauk Arm. SSR, Mat.,17, No. 4, 307-327; No. 5, 335-375 (1982). [7] M. G. Krein and Yu. L. Shmulyan, ?A supplement to the paper ?The Wiener-Hopf equations whose kernels admit integral representation in terms of exponentials,?? Izv. Akad. Nauk Arm. SSR, Mat.,21, No. 3, 301-305 (1986). [8] L. G. Arabadzhyan, ?On a conservative Wiener-Hopf equation,? Izv. Akad. Nauk Arm. SSR, Mat.,16, No. 1, 65-80 (1981). · Zbl 0461.45002
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