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Difference between revisions of "Commuting operators"

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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
 
Linear operators $B$ and $T$, of which $T$ is of general type and $B$ is bounded, and which are such that
  
$$BT\subseteq TB\tag{1}$$
+
$$BT\subseteq TB\label{1}\tag{1}$$
  
 
(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:
 
(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:
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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:
  
$$BT=TB,\tag{2}$$
+
$$BT=TB,\label{2}\tag{2}$$
  
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.
+
and $B$ is not required to be bounded. The generalization of \eqref{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>

Latest revision as of 15:29, 14 February 2020

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

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

(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:

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

and $B$ is not required to be bounded. The generalization of \eqref{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