Catalan Generating Functions for Generators of Uni-parametric Families of Operators

In this paper we study solutions of the quadratic equation \(AY^2 - Y + I = 0\) where \(A\) is the generator of a one-parameter family of operators (\(C_0\)-semigroup or cosine functions) on a Banach space \(X\) with growth bound \(w_0 \le \tfrac{1}{4}\).

In the case of \(C_0\)-semigroups, we show that a solution, which we call the Catalan generating function of \(A\), \(C(A)\), is given by the Bochner integral \[ C(A)x := \int_0^\infty c(t) T(t)x \, \mathrm{d}t, \quad x \in X, \] where \(c\) is the Catalan kernel, \[ c(t) := \frac{1}{2\pi} \int_{1/4}^\infty e^{-\lambda t} \frac{\sqrt{4\lambda - 1}}{\lambda} \, \mathrm{d}\lambda, \quad t > 0. \]

Similar (and more intricate) results hold for cosine functions.
We study algebraic properties of the Catalan kernel \(c\) as an element of Banach algebras \(L^1_{\omega}(\mathbb{R}^+)\), endowed with the usual convolution product \(*\) and with the cosine convolution product \(*_c\).

The Hille–Phillips functional calculus allows transferring these properties to \(C_0\)-semigroups and cosine functions.
In particular, we obtain a spectral mapping theorem for \(C(A)\). Finally, we present examples, applications, and conjectures to illustrate our results.

Recommended citation: Mahillo, A., Miana, P.J. Catalan Generating Functions for Generators of Uni-parametric Families of Operators. Mediterr. J. Math. 19, 238 (2022).
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