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We study the spectrum of the product of two Toeplitz operators. Assume thatthe symbols of these operators are continuous and real-valued and that one ofthem is non-negative. We prove that the spectrum of the product of finitesection Toeplitz matrices converges to the spectrum of the product of thesemi-infinite Toeplitz operators. We give an example showing that the supremumof this set is not always the supremum of the product of the two symbols.Finally, we provide an application in probability which is the first motivationof this study. More precisely, we obtain a large deviation principle forGaussian quadratic forms.

Spectrum of the product of Toeplitz matrices with application in probability