Mathematics. Articles in English (et en français dans le futur).

Elementary Properties of Cesàro Operator in L^2

We study the average of sum, in the sense of integral.
Elementary Properties of Cesàro Operator in L^2

Left Shift Semigroup and Its Infinitesimal Generator

Left shift operator

Throughout we consider the Hilbert space $L^2=L^2(\mathbb{R})$, the space of all complex-valued functions with real variable such that $f \in L^2$ if and only if

where $m$ denotes the ordinary Lebesgue measure (in fact it’s legitimate to consider Riemann integral in this context).

For each $t \geq 0$, we assign an bounded linear operator $Q(t)$ such that

This is indeed bounded since we have $\lVert Q(t)f \rVert_2 = \lVert f \rVert_2$ as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step $t$.

Left Shift Semigroup and Its Infinitesimal Generator

Several ways to prove Hardy's inequality

Several ways to prove Hardy's inequality

A Continuous Function Sending L^p Functions to L^1

A Continuous Function Sending L^p Functions to L^1