We consider the system of operator equations ABA = A² and BAB = B². 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-1,
author = {Sungeun Jung and Eungil Ko},
title = {Operator equations and subscalarity},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {97-113},
zbl = {1316.47019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-1}
}
Sungeun Jung; Eungil Ko. Operator equations and subscalarity. Studia Mathematica, Tome 223 (2014) pp. 97-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-1/