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Block-Toeplitz matrices and associated properties of a Gaussian model on a half-axis. (English. Russian original) Zbl 0608.47026
Theor. Math. Phys. 63, 427-431 (1985); translation from Teor. Mat. Fiz. 63, No. 1, 154-160 (1985).
Let $$\{s_ j\}^{+\infty}_{-\infty}$$ be a sequence of $$m\times m$$ matrices (it describes in the paper the interaction in a Gaussian model on a half-axis). Suppose that $$S_ n=(s_{j-k})^ n_{j,k=0}>0$$, $$n=1,2,...$$. Split the matrix $$S_ n^{-1}$$ into $$m\times m$$ blocks: $$S_ n^{-1}=(v_{jk}^{(n)})$$. The paper contains proofs of the expressions for $$\lim_{n}v_{jk}^{(n)}=v_{jk}$$ and $$\lim_{n}v_ n=v$$, where $$v_ n=\sum^{n}_{i,j=0}v_{jk}^{(n)}.$$
Let us describe the first result (Theorem 1) of the paper. If $$s_ j=(2\pi)^{-1}\int^{\pi}_{-\pi}e^{-ij\theta}d\sigma (\theta)$$, where the measure $$\sigma$$ has a decomposition into parts $$d\sigma (\theta)=G(\theta)d\theta +d\sigma_ s(\theta)$$ where ln det $$G\in L^ 1$$ and $$\sigma_ s$$ is singular with respect to the Lebesgue measure then $$G(\theta)=F_+(e^{i\theta})F_+(e^{i\theta})^*$$ a.e. and denoting by $$g_ j$$ the Taylor coefficients of $$F_+^{-1}$$ one has:
Theorem. Under the above conditions the lim $$v_{jk}$$ exists and $$v_{jk}=\sum^{\min (j,k)}_{r=0}g^*_{j-r}g_{k-r}$$.
Reviewer: J.Janas

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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