Operators similar to their restrictions to invariant subspaces
Let \(X\) be a separable infinite-dimensional Banach space and \(T\in B(X)\). The operator \(T\) is called crystal-like if, for each nonzero invariant subspace \(M\) of \(T\), there exists a positive number \(r\) such the restriction of \(T\) to \(M\) is similar to \(rT\). In this paper, the authors study several elementary properties of this new interesting operator class, including the spectrum, cyclicity, strong irreducibility, essential similarity, reflectivity and the commutativity of the commutant. A plethora of examples are provided and some open problems are also given.
scientific article; zbMATH DE number 6390681
scientific article