@fedja The Schur test does not work: you would have to check $$\sup_{i\ge 1}\sum_{j\ge 1}\frac{1}{i+j}$$ which are all infinite. I hope that the explanations below could clarify the situation and qualify the question for MO.

@user113103 The indicatrix function of $\mathbb R_+$ is the standard Heaviside function, which is 1 on the positive half-line and 0 on the negative half-line. $J_k(a)$ appears essentially as the $k$-th derivative of a product: just apply Leibniz formula.

Yes, but just a remark here: this holomorphic method provides an explicit formula that can be numerically calculated and approximated. The algebraic proofs above require the knowledge of unknown quantities, such as the minimal polynomial, or the Jordan form. Although perfect theoretically, an algebraic method for this problem will require a very large time to provide a simple approximation of a commuting square-root: the numerical cost of the determination of the minimal polynomial or of the Jordan form is huge, compared to the simple algorithm to approximate a simple integral.

No: true only if $1\le p,q<+\infty.$