Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating operator, reformulated as convex optimization problems linear...

The norm of $C-I$ on $\ell^p$, where $C$ is the Ces\`aro operator, shown to be $1/(p-1)$ when $1<p\le2$. This verifies a recent conjecture G. J. O. Jameson. $\ell^p$ also determined $2< p<\infty$. two parts together answer question raised by Bennett in 1996. Operator norms continuous case, Hardy's averaging operator minus identity, are already known. Norms discrete and cases coincide.

We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by their matrix metrics on the matrix state spaces. A matrix metric comes from a lower semicontinuous matrix Lip-norm if and only if it is convex, midpoint balanced, and midpoint concave. The operator...

We prove that a sectorial operator admits an H∞functional calculus if and only if it has a functional model of NagyFoiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that th...