View source for Schur lemma

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834501.png" /> are two algebraically-irreducible representations of some group or algebra in two vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834503.png" />, respectively, then any [[Intertwining operator|intertwining operator]] for the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834505.png" /> is either zero or provides a one-to-one mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834506.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834507.png" /> (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s0834509.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345011.png" /> are unitary irreducible representations of some group or are symmetric irreducible representations of some algebra in two Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345013.png" />, respectively, then any closed linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345014.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345015.png" /> intertwining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345017.png" /> is either zero or unitary (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345020.png" /> be two representations in Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345022.png" /> of a symmetric ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345024.png" /> be a closed linear operator with zero kernel and dense domain and range. If the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345025.png" /> hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345026.png" />, then the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345028.png" /> are unitarily equivalent.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345029.png" /> be an algebra of continuous linear operators in a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345030.png" /> containing a non-zero compact operator and having no non-trivial closed invariant subspaces. Then any operator permutable with all operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345031.png" /> is a multiple of the identity operator.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,   "Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen"  ''Sitzungsber. Akad. Wiss. Berlin''  (1906)  pp. 164–184</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Shtern,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>

''V.I. Lomonosov''

====Comments====
The Schur lemma has a number of immediate consequences. An important one is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345032.png" /> is an algebraically-irreducible representation in a linear space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345033.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345034.png" /> of intertwining operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345035.png" /> is a skew-field over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345037.png" />, this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345038.png" />, i.e. every intertwining operator is a multiple of the identity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345039.png" />, this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345041.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345042.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083450/s08345043.png" />-algebra of quaternions.

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