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Linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234502.png" />, of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234503.png" /> is of general type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234504.png" /> is bounded, and which are such that
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Linear operators $B$ and $T$, of which $T$ is of general type and $B$ is bounded, and which are such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234505.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$BT\subseteq TB\tag{1}$$
  
(the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234506.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234507.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234508.png" />, cf. [[Extension of an operator|Extension of an operator]]). The commutation relation is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c0234509.png" /> and satisfies the following rules:
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(the symbol $T\subseteq T_1$ means that $T_1$ is an extension of $T$, cf. [[Extension of an operator|Extension of an operator]]). The commutation relation is denoted by $B\cup T$ and satisfies the following rules:
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345013.png" />;
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1) if $B\cup T_1$, $B\cup T_2$, then $B\cup(T_1+T_2)$, $B\cup T_1T_2$;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345017.png" />;
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2) if $B_1\cup T$, $B_2\cup T$, then $(B_1+B_2)\cup T$, $B_1B_2\cup T$;
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345018.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345019.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345020.png" />;
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3) if $T^{-1}$ exists, then $B\cup T$ implies that $B\cup T^{-1}$;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345022.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345023.png" />;
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4) if $B\cup T_n$, $n=1,2,\dots,$ then $B\cup\lim T_n$;
  
5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345025.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345026.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345027.png" /> is bounded and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345028.png" /> is closed.
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5) if $B_n\cup T$, $n=1,2,\dots,$ then $\lim B_n\cup T$, provided that $\lim B_n$ is bounded and $T$ is closed.
  
 
If the two operators are defined on the entire space, condition 1) reduces to the usual one:
 
If the two operators are defined on the entire space, condition 1) reduces to the usual one:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$BT=TB,\tag{2}$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345030.png" /> is not required to be bounded. The generalization of (2) is justified by the fact that even a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345031.png" /> need not commute with its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023450/c02345032.png" /> if the latter is not defined on the entire space.
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and $B$ is not required to be bounded. The generalization of \ref{2} is justified by the fact that even a bounded operator $B$ need not commute with its inverse $B^{-1}$ if the latter is not defined on the entire space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>

Revision as of 09:46, 27 October 2014

Linear operators $B$ and $T$, of which $T$ is of general type and $B$ is bounded, and which are such that

$$BT\subseteq TB\tag{1}$$

(the symbol $T\subseteq T_1$ means that $T_1$ is an extension of $T$, cf. Extension of an operator). The commutation relation is denoted by $B\cup T$ and satisfies the following rules:

1) if $B\cup T_1$, $B\cup T_2$, then $B\cup(T_1+T_2)$, $B\cup T_1T_2$;

2) if $B_1\cup T$, $B_2\cup T$, then $(B_1+B_2)\cup T$, $B_1B_2\cup T$;

3) if $T^{-1}$ exists, then $B\cup T$ implies that $B\cup T^{-1}$;

4) if $B\cup T_n$, $n=1,2,\dots,$ then $B\cup\lim T_n$;

5) if $B_n\cup T$, $n=1,2,\dots,$ then $\lim B_n\cup T$, provided that $\lim B_n$ is bounded and $T$ is closed.

If the two operators are defined on the entire space, condition 1) reduces to the usual one:

$$BT=TB,\tag{2}$$

and $B$ is not required to be bounded. The generalization of \ref{2} is justified by the fact that even a bounded operator $B$ need not commute with its inverse $B^{-1}$ if the latter is not defined on the entire space.

References

[1] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian)
[2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
How to Cite This Entry:
Commuting operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commuting_operators&oldid=34095
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article