Insights on the Cesàro operator: shift semigroups and invariant subspaces

Abstract

AbstractA closed subspace is invariant under the Cesàro operator$${\cal C}$$Con the classical Hardy space$${H^2}(\mathbb{D})$$H2(D)if and only if its orthogonal complement is invariant under theC0-semigroup of composition operators induced by the affine maps$${\varphi _t}(z) = {e^{ - t}}z + 1 - {e^{ - t}}$$φt(z)=e−tz+1−e−tfort≥ 0 and$$z =\mathbb{D}$$z=D. The corresponding result also holds in the Hardy spacesHp($$\mathbb{D}$$D) for 1 <p< ∞. Moreover, in the Hilbert space setting, by linking the invariant subspaces of$${\cal C}$$Cto the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weightedL2-space on the line, we exhibit a large class of non-trivial closed invariant subspaces and provide a complete characterization of the finite codimensional ones, establishing, in particular, the limits of such an approach towards describing the lattice of all invariant subspaces of$${\cal C}$$C. Finally, we present a functional calculus argument which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of$${\cal C}$$Cand discuss its invariant subspaces.