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.

References

[1]	I. Schur, "Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen" Sitzungsber. Akad. Wiss. Berlin (1906) pp. 164–184
[2]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[4]	A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)
[5]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
[6]	V.I. Lomonosov, "Invariant subspaces for the family of operators which commute with a completely continuous operator" Funct. Anal. Appl. , 7 : 3 (1973) pp. 213–214 Funktsional. Anal. i Prilozhen. , 7 : 3 (1973) pp. 55–56

V.I. Lomonosov

Comments

The Schur lemma has a number of immediate consequences. An important one is that if $ T $ is an algebraically-irreducible representation in a linear space over a field $ K $, then the set $ {\mathcal C} ( T) $ of intertwining operators of $ T $ is a skew-field over $ K $. If $ K = \mathbf C $, this means that $ {\mathcal C} ( T) = \mathbf C $, i.e. every intertwining operator is a multiple of the identity. If $ K = \mathbf R $, this means that $ {\mathcal C} ( T) = \mathbf R $, $ \mathbf C $ or $ \mathbf H $, the $ \mathbf R $- algebra of quaternions.

References

[a1]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44
[a2]	J.-P. Serre, "Linear representations of finite groups" , Springer (1982) (Translated from French)
[a3]	C.E. Rickart, "General theory of Banach algebras" , v. Nostrand (1960)
[a4]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. 64
[a5]	N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1958) pp. Chapt. 8. Modules et anneaux semi-simples