Decay estimates for Cayley transforms and inverses of semigroup generators via the B-calculus

Let $$-A$$ - A be the generator of a bounded C₀ C 0 -semigroup (e-tA)t ≥ 0 ( e - t A ) t ≥ 0 on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform Vω(A) :=(A-ω I) (A+ω I)⁻¹ V ω ( A ) : = ( A - ω I ) ( A + ω I ) - 1 with ω >0 ω > 0 . We give a decay estimate for Vω(A)ⁿA⁻¹ ‖ V ω ( A ) n A - 1 ‖ when (e-tA)t ≥ 0 ( e - t A ) t ≥ 0 is polynomially stable. Considering the case where the parameter ω ω varies, we estimate (∏ ₖ₌₁ⁿ Vω ₖ(A))A⁻¹ ‖ ( ∏ k = 1 n V ω k ( A ) ) A - 1 ‖ for exponentially stable C₀ C 0 -semigroups (e-tA)t ≥ 0 ( e - t A ) t ≥ 0 . Next we show that if the generator $$-A$$ - A of the bounded C₀ C 0 -semigroup has a bounded inverse, then t ≥ 0 e^-tA⁻¹ A-α < ∞ sup t ≥ 0 ‖ e - t A - 1 A - α ‖ < ∞ for all α >0 α > 0 . We also present an estimate for the rate of decay of e^-tA⁻¹ A⁻¹ ‖ e - t A - 1 A - 1 ‖ , assuming that (e-tA)t ≥ 0 ( e - t A ) t ≥ 0 is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the B B -calculus.