Sums and products of cyclic operators

Wu, Pei Yuan

doi:10.1090/S0002-9939-1994-1203995-9

Sums and products of cyclic operators
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by Pei Yuan Wu

Proc. Amer. Math. Soc. 122 (1994), 1053-1063

DOI: https://doi.org/10.1090/S0002-9939-1994-1203995-9

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Abstract:

It is proved that every bounded linear operator on a complex separable Hilbert space is the sum of two cyclic operators. For the product, we show that an operator T is the product of finitely many cyclic operators if and only if the kernel of ${T^ \ast }$ is finite-dimensional. More precisely, if dim ker ${T^ \ast } \leq k(2 \leq k < \infty )$, then T is the product of at most $k + 2$ cyclic operators. We conjecture that in this case at most k cyclic operators would suffice and verify this for some special classes of operators.

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