Commuting operators

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

$$\begin{array}{}\text{(1)}& BT\subseteq TB\end{array}$$

(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 $B\cup (T_1+T_2)$, $B\cup T_1{T}_2$;

2) if $B_1\cup T$, $B_2\cup T$, then $(B_1+B_2)\cup T$, $B_1{B}_2\cup T$;

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:

$$\begin{array}{}\text{(2)}& BT=TB,\end{array}$$

and $B$ is not required to be bounded. The generalization of $\text{(2)}$ 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.

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