"Twisted" direct sums of Banach spaces

Question

It is hard to provide motivation, so I just want to state this definition "as is". Suppose I have a Banach space $E$ and two commuting, injective operators $R_0, R_1$ on $E$ which satisfy the following axiom:

• If $x,y\in E$ with $R_0(y) =R_1(x)$ then there is a sequence $(x_n)$ in $E$ with $R_0(x_n)\rightarrow x$ and $R_1(x_n)\rightarrow y$.

We can make a picture out of this:

$$ E \stackrel{R_0\oplus R_1}{\longrightarrow} E\oplus E \stackrel{R_1-R_0}{\longrightarrow} E $$ where $R_0\oplus R_1: x\mapsto (R_0(x), R_1(x))$ and $R_1-R_0:(y,z)\mapsto R_1(y)-R_0(z)$. We now see that:

• That $R_0,R_1$ commute is equivalent to $(R_1-R_0)\circ (R_0\oplus R_1)=0$
• The axiom is equivalent to the image of $R_0\oplus R_1$ being dense in the kernel of $R_1 - R_0$.

Hence we have something which looks rather like a short exact sequence (recall that $R_0, R_1$ are injective), except for this "closure" business.

A trivial example is when $R_0=R_1$ is the indentity, so $R_0\oplus R_1$ is the diagonal embedding, and the kernel of $R_1-R_0$ is exactly the image of $R_0\oplus R_1$. In this case we do have a short exact sequence.

Does anyone know of any references where this, or something quite similar, is studied?

Answer 1

The definition of the Taylor spectrum of a pair of commuting operators acting on a complex Banach space $E$ uses a "similar" notion. You can find an exposition in Chapter IV of the book [V. Müller. Spectral theory of linear operators, 2nd ed. Birkhäuser 2007]. It is usually defined for $n$-tuples of commuting operators.

A pair $(R_0,R_1)$ of commuting operators acting on $E$ is Taylor regular (split regular) if the sequence $$ 0\rightarrow E \stackrel{R_0\oplus R_1}{\rightarrow} E\oplus E \stackrel{R_1 -R_0}{\rightarrow} E \rightarrow 0 $$ is exact (is exact and splits).

Clearly splits regular implies Taylor regular, but the converse implication fails [V. Müller. The splitting spectrum differs from the Taylor spectrum. Studia Math. 123 (1997), 291-294].