Commuting operators

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

(1)

(the symbol means that is an extension of , cf. Extension of an operator). The commutation relation is denoted by and satisfies the following rules:

1) if , , then , ;

2) if , , then , ;

3) if exists, then implies that ;

4) if , then ;

5) if , then , provided that is bounded and is closed.

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

(2)

and is not required to be bounded. The generalization of (2) is justified by the fact that even a bounded operator need not commute with its inverse 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)