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Dimension stabilization effect for the block Jacobi-type matrix of a bounded normal operator with the spectrum on an algebraic curve
MFAT 16 (2010), no. 1, 28-41
Under some natural assumptions, any bounded normal operator in an appropriate basis has a three-diagonal block Jacobi-type matrix. Just as in the case of classical Jacobi matrices (e.g. of self-adjoint operators) such a structure can be effectively used. There are two sources of difficulties: rapid growth of blocks in the Jacobi-type matrix of such operators (they act in $\mathbb C^1\oplus\mathbb C^2\oplus\mathbb C^3\oplus\cdots$) and potentially complicated spectra structure of the normal operators. The aim of this article is to show that these two aspects are closely connected: simple structure of the spectra can effectively bound the complexity of the matrix structure. The main result of the article claims that if the spectra is concentrated on an algebraic curve the dimensions of Jacobi-type matrix blocks do not grow starting with some value.