Operator equations and subscalarity
Tom 225 / 2014
Studia Mathematica 225 (2014), 97-113 MSC: Primary 47B20, 47A62. DOI: 10.4064/sm225-2-1
We consider the system of operator equations $ABA=A^2$ and $BAB=B^2$. Let $(A,B)$ be a solution to this system. We give several connections among the operators $A$, $B$, $AB$, and $BA$. We first prove that $A$ is subscalar of finite order if and only if $B$ is, which is equivalent to the subscalarity of $AB$ or $BA$ with finite order. As a corollary, if $A$ is subscalar and its spectrum has nonempty interior, then $B$ has a nontrivial invariant subspace. We also provide examples of subscalar operator matrices. Moreover, we deal with algebraicity, power boundedness, and quasitriangularity, using some power properties obtained from the operator equations.