Robin Harte (TCD)

will speak on

More Spectral Disjointness

Time: 4:00PM
Date: Tue 4th February 2020
Location: Seminar Room SCN 1.25 [map]

Abstract: Spectral disjointness confers a certain ``independence'' upon linear operators. If $G$ is a ring with identity $I$ then an idempotent $Q=Q^2\in G$
gives the ring $G$ a block structure
$$G\cong\pmatrix{A&M\cr N&B\cr}$$
where for example $A=QGQ$; then
$$T=\pmatrix{a&m\cr n&b\cr}\in G$$
commutes with $Q$ iff it is a ``block diagonal'':
$$TQ=QT\Longleftrightarrow T=\pmatrix{a&0\cr 0&b\cr}\ .$$
Specialising to complex Banach algebras, for block diagonals there is two way implication
$$\sigma_A(a)\cap\sigma_B(b)=\emptyset\Longleftrightarrow Q\in{\rm Holo}(T)\ :$$
$Q=f(T)$ with $f:U\to G$ holomorphic on an open neighbourhood of $\sigma_G(T)$. Weaker spectral disjointness gives a little less:
$$\sigma_A^{left}(a)\cap\sigma_B^{right}(b)=\emptyset=\sigma_A^{right}(a)\cap\sigma_B^{left}(b)\Longrightarrow Q\in{\rm comm}^2(T)\ :$$
the block structure idempotent $Q$ ``double commutes'' with $T\in G$. Specializing to $G=B(X)$, the bounded operators on a Banach space,
closed complemented subspaces $Y\subseteq X$ give us again the block structure, and operators $T\in G$ for which $Y$ is ``invariant'' become ``block triangles'':
$$T(Y)\subseteq Y\Longleftrightarrow T=\pmatrix{a&m\cr 0&b\cr}\ .$$
When $Y\subseteq X$ is not complemented then the block structure is missing and we must resort to the restriction and the quotient:
$$a=T_Y\in A=B(Y)\ ;\ b=T_{/Y}\in B(X/Y)\ .$$
Now spectral disjointness
$$\sigma_A(a)\cap\sigma_B(b)=\emptyset$$
ensures that the subspace $Y\subseteq X$ is both {\sl hyperinvariant} and {\sl reducing}, in particular complemented.

(This talk is part of the Analysis series.)

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