Power Regular Operators

Abstract

Abstract.We show that for a wide class of operators T on a Banach space, including the class of decomposable operators, the sequence {||T''x||1/'1}^°¡1 converges for every x in the space to the spectral radius of the restriction of T to the subspace \Z™=0{T"x} .

Local spectral theory and orbits of operators

“…But x is a cyclic vector for T , and, since T is power regular [4], this contradicts our assumption that T has spectral radius 1.…”

“…Therefore, since T | XT (K) also has property (β), we conclude that σ(T ) = σ su (T | XT (K) ) ⊂ K. Because operators with Bishop's property (β) are power regular [4], that is, lim n→∞ T n x 1/n exists for every x ∈ X, it follows that for a cyclic vector x, the sequence ( T n x 1/n ) n≥1 converges to the spectral radius of T .…”

Power Regularity of Isometric N-Jordan Operators

“…Recall that an operator S in B(H) is power regular if lim n→∞ S n h 1 n exists for every h ∈ H. It is known that compact operators, normal operators and decomposable operators are power regular (see [1] and references therein). It is important in operator theory that which operators are power regular.…”

On the local spectral properties of weighted shift operators

“…On the other hand, it follows from [5] that S is power-regular , that is, the sequence ( S n x 1/n ) n≥0 converges for all x ∈ H. In particular, ( S n e 0 1/n ) n≥0 converges; so,…”