View source for Intertwining operator

A continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520601.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520604.png" /> are mappings of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520605.png" /> into two topological vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520608.png" />. This concept is especially fruitful in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520609.png" /> is a group or an algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206010.png" /> are representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206011.png" />. The set of intertwining operators forms the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206012.png" />, which is a subspace of the space of all continuous linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206018.png" /> are called disjoint representations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206019.png" /> contains an operator that defines an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206023.png" /> are equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206024.png" /> are locally convex spaces, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206026.png" /> are their adjoints, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206028.png" /> are the representations contragredient to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206030.png" />, respectively (cf. [[Contragredient representation|Contragredient representation]]), then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206031.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206032.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206035.png" /> are finite-dimensional or unitary representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206036.png" /> is irreducible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206037.png" /> admits a subrepresentation equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206038.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206039.png" />. See also [[Intertwining number|Intertwining number]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Shtern,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR></table>