# Naming convention: Adjective for linear operators that are endomorphisms

If a matrix has the same number of rows and columns, we call it a square matrix. The analogous concept for linear operators would be operators with the same domain and range, i.e., endomorphisms.

Is there an established adjective that can be added to the word "operator" that denotes this concept?

In other words, what could I fill in for xxx in the following example sentence: "A square matrix is invertible iff it has full rank, but not every full-rank xxx operator is invertible."

Note: In many texts, "operator" already implies that domain and range are the same, but in some texts this is not assumed. Often, this distinction is left implicit. For example, "A course in functional analysis" (Conway 1990) considers bounded operators with different domain and range, while in what is basically the follow up book "A course in operator theory" (Conway 2000), bounded operators are assumed to have same domain and range.

Short summary of suggestions

• Endomorphic? :-) – M.G. Jul 28 '19 at 10:45
• @M.G. I find very few mentions of "endomorphic operator" in Google. But if you can post your comment as an answer, people can comment more easily and we can get a feeling for the community opinion. – Dominique Unruh Jul 28 '19 at 11:35
• It was more of a joke since the immediate adjective from endomorphism would be endomorphic, but I don't think I've ever heard it being used like this (in mathematics anyway). To be honest, I doubt there is a widely used adjective for this, people just say "endomorphism of this or that". But on the other hand, if I ever see it used, then I'll most likely conclude that the author means an endomorphism. – M.G. Jul 28 '19 at 12:13
• I thought “operator” and “self-map” were basically synonyms. In other words, a linear map is a map between two vector spaces, while a linear operator is a map from a vector space to itself. – Sam Hopkins Jul 28 '19 at 13:43
• Wikipedia is a little unclear on this point: en.m.wikipedia.org/wiki/Operator_(mathematics) – Sam Hopkins Jul 28 '19 at 13:47

In Operator Theory, an operator on the space $$X$$, is quite commonly used to mean $$T:X\to X$$, denoting $$L(X)$$ or $$B(X)$$ the space of these operators in the same spirit.