Commuting operators

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)