Algebraic properties of operator roots of polynomials

Decomposing algebraic m-isometric tuples

“…In [12], it was showed that if S is an isometry on any Hilbert space and N is a nilpotent operator of order n commuting with S then the sum S + N is a strict (2n − 1)-isometry. This result has been generalized to m-isometries by several authors [26,11,22].…”

“…We have provided a direct proof of Proposition 2.3. Using the hereditary functional calculus and the approach in [26], one may generalize the result to the case S being an (A, m)-isometry and N an (A, n)-nilpotent commuting with S. Under such an assumption, if S * 1 AN 1 + · · · + S * d AN d = 0, then S + N is an (A, m + 2n − 3)-isometry. We leave the details for the interested reader.…”

“…Very recently, researchers have been interested in algebraic properties, cyclicity and supercyclicity of m-isometries, among other things. See [28,24,14,16,15,18,13,12,26,11,22] and the references therein. It was showed by Agler, Helton and Stankus [4,Section 1.4] that any m-isometry T on a finite dimensional Hilbert space admits a decomposition T = S + N, where S is a unitary and N is a nilpotent operator satisfying SN = NS.…”

Power Regularity of Isometric N-Jordan Operators

“…Moreover, it is shown in [2] that if A is an m-isometry then A + Q is a (2N − m − 2)-isometry. Recently, such operators have been considered by several authors; for example, see [4,6] .…”

Some Examples of m-Isometries

“…Indeed in [15,Theorem 3], it is obtained a generalization of Theorem 1.1 for m-isometries: if T is an m-isometry, Q is an n-nilpotent operator and they commute, then T +Q is a (2n+m−2)-isometry. See also [25,28]. Moreover, the study of isometric n-Jordan operators concerning with m-isometries on Banach spaces context has been studied in [15].…”