# Schur lemma

If $T, S$ are two algebraically-irreducible representations of some group or algebra in two vector spaces $X$ and $Y$, respectively, then any intertwining operator for the representations $T$ and $S$ is either zero or provides a one-to-one mapping from $X$ onto $Y$( in this case $T$ and $S$ are equivalent). The lemma was established by I. Schur

for finite-dimensional irreducible representations. The description of the family of intertwining operators for two given representations is an analogue of the Schur lemma. In particular, the following statement is often called Schur's lemma: If $T$ and $S$ are unitary irreducible representations of some group or are symmetric irreducible representations of some algebra in two Hilbert spaces $X$ and $Y$, respectively, then any closed linear operator from $X$ into $Y$ intertwining $T$ and $S$ is either zero or unitary (in this case $T$ and $S$ are unitarily equivalent). The description of the family of intertwining operators for representations that allow for an expansion in a direct integral is called the continuous analogue of Schur's lemma.

A.I. Shtern

The two following statements are generalizations of Schur's lemma to families of operators acting on infinite-dimensional spaces.

Let $T _ {x} , S _ {x}$ be two representations in Hilbert spaces ${\mathcal H} _ {T}$ and ${\mathcal H} _ {S}$ of a symmetric ring $R$. Let $A: {\mathcal H} _ {T} \rightarrow {\mathcal H} _ {S}$ be a closed linear operator with zero kernel and dense domain and range. If the relations $S _ {x} A \subset AT _ {x}$ hold for all $x \in R$, then the representations $T _ {x}$ and $S _ {x}$ are unitarily equivalent.

Let $R$ be an algebra of continuous linear operators in a locally convex space $E$ containing a non-zero compact operator and having no non-trivial closed invariant subspaces. Then any operator permutable with all operators from $R$ is a multiple of the identity operator.

How to Cite This Entry:
Schur lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_lemma&oldid=48624
This article was adapted from an original article by A.I. Shtern, V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article