Research Projects

J. Agler introduced the notion of m-isometry and thoroughly studied their structure theory. Wold-type decomposition plays a central role in the study of structure theory of m-isometries. Wold-type decomposition of isometries was settled by Beurling, Lax and Halmos whereas that of 2-isometries was resolved by Richter. The problem of classification of m-isometries admitting Wold-type decomposition is still open for several decades and this proposal aims to find a solution of this problem. Assume that T is a bounded left-invertible operator on a complex separable Hilbert space H. Following Shimorin, we say that T admits Wold-type decomposition if $T^{\infty}(\mathcal H):=\bigcap_{n \geqslant 1}T^n(\mathcal H)$ reduces T to a unitary and $H = T^{\infty}(\mathcal H) \oplus [\ker T^*]_T.$ Given a positive integer m, we say that T is an m-isometry if $\sum_{k=0}^m (-1)^k \binom{m}{k}{T^*}^kT^k = 0.$ Shimorin asked whether a norm-increasing m-isometry admit Wold-type decomposition. It is known that the hyper-range of a norm-increasing m-isometry reduces it to a unitary. Thus, this question has an affirmative answer if and only if all analytic norm-increasing m-isometries have the wandering subspace property. We gave a family of analytic cyclic 3-isometric weighted shifts on directed graph which do not have the wandering subspace property. This partially answered the aforementioned question. Recently, we gave a class of analytic norm-increasing 3-isometric weighted shifts on rootless directed tree, which do not have the wandering subspace property. As a consequence, we answered the question of Shimorin in the negative. Thus, it is natural to wonder about the characterization of m-isometries admitting Wold-type decomposition. It is worth noting that the $\ker T^*$ of the weighted shift on rootless directed tree is infinite dimensional. This observation motivates us to ask the following question: Let T be a norm-increasing m-isometry on H. Suppose that $\ker T^*$ is finite dimensional. Is it true that T admit Wold-type decomposition? If not, then find the necessary and sufficient conditions which ensure Wold-type decomposition of T. Note that the rootless directed tree considered above has finite branching index. It turns out that a rootless directed tree of finite branching index does not support analytic weighted shifts. However, it is still not known whether a weighted shift on a rootless directed tree of finite branching index admits Wold-type decomposition. This encourages us to ask the next question: Does every norm-increasing m-isometric weighted shift on a rootless directed tree of finite branching index admit Wold-type decomposition? If not, then find the necessary and sufficient conditions which ensure Wold-type decomposition of the shift. An answer of the preceding question will yield the complete classification of norm-increasing m-isometric weighted composition operators on discrete measure space admitting Wold-type decomposition.