engleski

On bounds for discrete semigroups

This note studies the exponential decay of the powers $T^k$ of a Hilbert space operator $T$. The main result is extension on the infinite dimension of the following known result for finite matrices: while the spectral radius $spr(T)$ gives only asymptotic decay estimates the solution $X$ of the {; ; discrete Lyapunov equation}; ; $T^*XT-X=-BB^*$ yields rigorous bounds. We also present a new upper bound for the norm of the solution $X$ in the matrix case which depends on the structure of the right hand side. The new bound shows that the structure of $B$ can greatly influence $\|X\|$.

exponential decay; discrete Lyapunov equation; upper bound

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